Most Recent Proofs - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer Most Recent Proofs
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Most recent proofs These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

See the MPE Most Recent Proofs page for news and some useful links.

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Last updated on 12-Feb-2022 at 12:00 PM ET.

**Recent Additions to the Intuitionistic Logic Explorer**
Date	Label	Description
Theorem

10-Feb-2022	ltmininf 10116	Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < inf({𝐵, 𝐶}, ℝ, < ) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))

10-Feb-2022	maxltsup 10104	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶)))

9-Feb-2022	maxleastlt 10101	The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ, < ))) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))

6-Feb-2022	unsnfidcel 6386	The ¬ 𝐵 ∈ 𝐴 condition in unsnfi 6384. This is intended to show that unsnfi 6384 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ 𝐴)

6-Feb-2022	unsnfidcex 6385	The 𝐵 ∈ 𝑉 condition in unsnfi 6384. This is intended to show that unsnfi 6384 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ 𝐴 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ V)

5-Feb-2022	funrnfi 6392	The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((Rel 𝐴 ∧ Fun ^◡𝐴 ∧ 𝐴 ∈ Fin) → ran 𝐴 ∈ Fin)

5-Feb-2022	relcnvfi 6391	If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ^◡𝐴 ∈ Fin)

5-Feb-2022	fundmfi 6389	The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ∈ Fin)

5-Feb-2022	infiexmid 6362	If the intersection of any finite set and any other set is finite, excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.)
⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝑦) ∈ Fin) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

3-Feb-2022	unsnfi 6384	Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin)

3-Feb-2022	domfiexmid 6363	If any set dominated by a finite set is finite, excluded middle follows. (Contributed by Jim Kingdon, 3-Feb-2022.)
⊢ ((𝑥 ∈ Fin ∧ 𝑦 ≼ 𝑥) → 𝑦 ∈ Fin) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

3-Feb-2022	ssfilem 6360	Lemma for ssfiexmid 6361. (Contributed by Jim Kingdon, 3-Feb-2022.)
⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

1-Feb-2022	maxclpr 10108	The maximum of two real numbers is one of those numbers if and only if dichotomy (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) holds. For example, this can be combined with zletric 8395 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 1-Feb-2022.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)))

31-Jan-2022	znege1 10556	The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≠ 𝐵) → 1 ≤ (abs‘(𝐴 − 𝐵)))

31-Jan-2022	maxleastb 10100	Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Jim Kingdon, 31-Jan-2022.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (sup({𝐴, 𝐵}, ℝ, < ) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)))

30-Jan-2022	max0addsup 10105	The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
⊢ (𝐴 ∈ ℝ → (sup({𝐴, 0}, ℝ, < ) + sup({-𝐴, 0}, ℝ, < )) = (abs‘𝐴))

29-Jan-2022	expcanlem 9643	Lemma for expcan 9644. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 1 < 𝐴) ⇒ ⊢ (𝜑 → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) → 𝑀 ≤ 𝑁))

28-Jan-2022	exfzdc 9249	Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓) ⇒ ⊢ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)

25-Jan-2022	1nen2 6347	One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
⊢ ¬ 1_𝑜 ≈ 2_𝑜

24-Jan-2022	divmulasscomap 7784	An associative/commutative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = (𝐵 · ((𝐴 · 𝐶) / 𝐷)))

24-Jan-2022	divmulassap 7783	An associative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = ((𝐴 · 𝐵) · (𝐶 / 𝐷)))

21-Jan-2022	lcmmndc 10444	Decidablity lemma used in various proofs related to lcm. (Contributed by Jim Kingdon, 21-Jan-2022.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))

20-Jan-2022	infssuzcldc 10347	The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑆 = {𝑛 ∈ (ℤ_≥‘𝑀) ∣ 𝜓} & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓) ⇒ ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆)

19-Jan-2022	suprnubex 8031	An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 ¬ 𝐵 < 𝑧))

19-Jan-2022	suprlubex 8030	The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐵 < 𝑧))

18-Jan-2022	suprubex 8029	A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))

17-Jan-2022	zdvdsdc 10216	Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ∥ 𝑁)

17-Jan-2022	suplub2ti 6414	Bidirectional form of suplubti 6413. (Contributed by Jim Kingdon, 17-Jan-2022.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

16-Jan-2022	zssinfcl 10344	The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐵 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐵 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐵 ⊆ ℤ) & ⊢ (𝜑 → inf(𝐵, ℝ, < ) ∈ ℤ) ⇒ ⊢ (𝜑 → inf(𝐵, ℝ, < ) ∈ 𝐵)

16-Jan-2022	supelti 6415	Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐶 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)

15-Jan-2022	infsupneg 8684	If a set of real numbers has a greatest lower bound, the set of the negation of those numbers has a least upper bound. To go in the other direction see supinfneg 8683. (Contributed by Jim Kingdon, 15-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑦 < 𝑧)))

15-Jan-2022	supinfneg 8683	If a set of real numbers has a least upper bound, the set of the negation of those numbers has a greatest lower bound. For a theorem which is similar but only for the boundedness part, see ublbneg 8698. (Contributed by Jim Kingdon, 15-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦)))

14-Jan-2022	supminfex 8685	A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))

14-Jan-2022	infrenegsupex 8682	The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))

13-Jan-2022	infssuzledc 10346	The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑆 = {𝑛 ∈ (ℤ_≥‘𝑀) ∣ 𝜓} & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓) ⇒ ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴)

13-Jan-2022	infssuzex 10345	Existence of the infimum of a subset of an upper set of integers. (Contributed by Jim Kingdon, 13-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑆 = {𝑛 ∈ (ℤ_≥‘𝑀) ∣ 𝜓} & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑆 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝑆 𝑧 < 𝑦)))

11-Jan-2022	eucialg 10441	Euclid's Algorithm computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0. Theorem 1.15 in [ApostolNT] p. 20. Upon halting, the 1st member of the final state (𝑅‘𝑁) is equal to the gcd of the values comprising the input state ⟨𝑀, 𝑁⟩. This is Metamath 100 proof #69 (greatest common divisor algorithm). (Contributed by Jim Kingdon, 11-Jan-2022.)
⊢ 𝐸 = (𝑥 ∈ ℕ₀, 𝑦 ∈ ℕ₀ ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩)) & ⊢ 𝑅 = seq0((𝐸 ∘ 1^st ), (ℕ₀ × {𝐴}), (ℕ₀ × ℕ₀)) & ⊢ 𝑁 = (2^nd ‘𝐴) & ⊢ 𝐴 = ⟨𝑀, 𝑁⟩ ⇒ ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (1^st ‘(𝑅‘𝑁)) = (𝑀 gcd 𝑁))

11-Jan-2022	eucialgcvga 10440	Once Euclid's Algorithm halts after 𝑁 steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.)
⊢ 𝐸 = (𝑥 ∈ ℕ₀, 𝑦 ∈ ℕ₀ ↦ if(𝑦 = 0, ⟨𝑥, 𝑦⟩, ⟨𝑦, (𝑥 mod 𝑦)⟩)) & ⊢ 𝑅 = seq0((𝐸 ∘ 1^st ), (ℕ₀ × {𝐴}), (ℕ₀ × ℕ₀)) & ⊢ 𝑁 = (2^nd ‘𝐴) ⇒ ⊢ (𝐴 ∈ (ℕ₀ × ℕ₀) → (𝐾 ∈ (ℤ_≥‘𝑁) → (2^nd ‘(𝑅‘𝐾)) = 0))

11-Jan-2022	ifcldadc 3378	Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → DECID 𝜓) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

9-Jan-2022	bezoutlemsup 10398	Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the supremum of divisors of both 𝐴 and 𝐵. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) & ⊢ (𝜑 → ∀𝑧 ∈ ℤ (𝑧 ∥ 𝐷 ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵))) & ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ⇒ ⊢ (𝜑 → 𝐷 = sup({𝑧 ∈ ℤ ∣ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)}, ℝ, < ))

9-Jan-2022	bezoutlemle 10397	Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the largest number which divides both 𝐴 and 𝐵. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) & ⊢ (𝜑 → ∀𝑧 ∈ ℤ (𝑧 ∥ 𝐷 ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵))) & ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ ℤ ((𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵) → 𝑧 ≤ 𝐷))

9-Jan-2022	bezoutlemeu 10396	Lemma for Bézout's identity. There is exactly one nonnegative integer meeting the greatest common divisor condition. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) & ⊢ (𝜑 → ∀𝑧 ∈ ℤ (𝑧 ∥ 𝐷 ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵))) ⇒ ⊢ (𝜑 → ∃!𝑑 ∈ ℕ₀ ∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)))

9-Jan-2022	bezoutlemmo 10395	Lemma for Bézout's identity. There is at most one nonnegative integer meeting the greatest common divisor condition. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) & ⊢ (𝜑 → ∀𝑧 ∈ ℤ (𝑧 ∥ 𝐷 ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵))) & ⊢ (𝜑 → 𝐸 ∈ ℕ₀) & ⊢ (𝜑 → ∀𝑧 ∈ ℤ (𝑧 ∥ 𝐸 ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵))) ⇒ ⊢ (𝜑 → 𝐷 = 𝐸)

8-Jan-2022	bezoutlembi 10394	Lemma for Bézout's identity. Like bezoutlembz 10393 but the greatest common divisor condition is a biconditional, not just an implication. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∃𝑑 ∈ ℕ₀ (∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 ↔ (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))

8-Jan-2022	bezoutlembz 10393	Lemma for Bézout's identity. Like bezoutlemaz 10392 but where ' B ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∃𝑑 ∈ ℕ₀ (∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))

8-Jan-2022	bezoutlemaz 10392	Lemma for Bézout's identity. Like bezoutlemzz 10391 but where ' A ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ₀) → ∃𝑑 ∈ ℕ₀ (∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))

8-Jan-2022	bezoutlemzz 10391	Lemma for Bézout's identity. Like bezoutlemex 10390 but where ' z ' is any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ₀) → ∃𝑑 ∈ ℕ₀ (∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))

6-Jan-2022	bezoutlemnewy 10385	Lemma for Bézout's identity. The is-bezout predicate holds for (𝑦 mod 𝑊). (Contributed by Jim Kingdon, 6-Jan-2022.)
⊢ (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) & ⊢ (𝜃 → 𝐴 ∈ ℕ₀) & ⊢ (𝜃 → 𝐵 ∈ ℕ₀) & ⊢ (𝜃 → 𝑊 ∈ ℕ) & ⊢ (𝜃 → [𝑦 / 𝑟]𝜑) & ⊢ (𝜃 → 𝑦 ∈ ℕ₀) & ⊢ (𝜃 → [𝑊 / 𝑟]𝜑) ⇒ ⊢ (𝜃 → [(𝑦 mod 𝑊) / 𝑟]𝜑)

3-Jan-2022	bezoutlemex 10390	Lemma for Bézout's identity. Existence of a number which we will later show to be the greater common divisor and its decomposition into cofactors. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jan-2022.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ₀) → ∃𝑑 ∈ ℕ₀ (∀𝑧 ∈ ℕ₀ (𝑧 ∥ 𝑑 → (𝑧 ∥ 𝐴 ∧ 𝑧 ∥ 𝐵)) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑑 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))))

3-Jan-2022	bezoutlemstep 10386	Lemma for Bézout's identity. This is the induction step for the proof by induction. (Contributed by Jim Kingdon, 3-Jan-2022.)
⊢ (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) & ⊢ (𝜃 → 𝐴 ∈ ℕ₀) & ⊢ (𝜃 → 𝐵 ∈ ℕ₀) & ⊢ (𝜃 → 𝑊 ∈ ℕ) & ⊢ (𝜃 → [𝑦 / 𝑟]𝜑) & ⊢ (𝜃 → 𝑦 ∈ ℕ₀) & ⊢ (𝜃 → [𝑊 / 𝑟]𝜑) & ⊢ (𝜓 ↔ ∀𝑧 ∈ ℕ₀ (𝑧 ∥ 𝑟 → (𝑧 ∥ 𝑥 ∧ 𝑧 ∥ 𝑦))) & ⊢ ((𝜃 ∧ [(𝑦 mod 𝑊) / 𝑟]𝜑) → ∃𝑟 ∈ ℕ₀ ([(𝑦 mod 𝑊) / 𝑥][𝑊 / 𝑦]𝜓 ∧ 𝜑)) & ⊢ Ⅎ𝑥𝜃 & ⊢ Ⅎ𝑟𝜃 ⇒ ⊢ (𝜃 → ∃𝑟 ∈ ℕ₀ ([𝑊 / 𝑥]𝜓 ∧ 𝜑))

1-Jan-2022	fvifdc 5217	Move a conditional outside of a function. (Contributed by Jim Kingdon, 1-Jan-2022.)
⊢ (DECID 𝜑 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)))

1-Jan-2022	ifeq1dadc 3379	Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) & ⊢ (𝜑 → DECID 𝜓) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

1-Jan-2022	ifsbdc 3363	Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷) & ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ (DECID 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))

31-Dec-2021	dfgcd3 10399	Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (℩𝑑 ∈ ℕ₀ ∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 ↔ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁))))

30-Dec-2021	bezoutlemb 10389	Lemma for Bézout's identity. The is-bezout condition is satisfied by 𝐵. (Contributed by Jim Kingdon, 30-Dec-2021.)
⊢ (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) & ⊢ (𝜃 → 𝐴 ∈ ℕ₀) & ⊢ (𝜃 → 𝐵 ∈ ℕ₀) ⇒ ⊢ (𝜃 → [𝐵 / 𝑟]𝜑)

30-Dec-2021	bezoutlema 10388	Lemma for Bézout's identity. The is-bezout condition is satisfied by 𝐴. (Contributed by Jim Kingdon, 30-Dec-2021.)
⊢ (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) & ⊢ (𝜃 → 𝐴 ∈ ℕ₀) & ⊢ (𝜃 → 𝐵 ∈ ℕ₀) ⇒ ⊢ (𝜃 → [𝐴 / 𝑟]𝜑)

30-Dec-2021	bezoutlemmain 10387	Lemma for Bézout's identity. This is the main result which we prove by induction and which represents the application of the Extended Euclidean algorithm. (Contributed by Jim Kingdon, 30-Dec-2021.)
⊢ (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) & ⊢ (𝜓 ↔ ∀𝑧 ∈ ℕ₀ (𝑧 ∥ 𝑟 → (𝑧 ∥ 𝑥 ∧ 𝑧 ∥ 𝑦))) & ⊢ (𝜃 → 𝐴 ∈ ℕ₀) & ⊢ (𝜃 → 𝐵 ∈ ℕ₀) ⇒ ⊢ (𝜃 → ∀𝑥 ∈ ℕ₀ ([𝑥 / 𝑟]𝜑 → ∀𝑦 ∈ ℕ₀ ([𝑦 / 𝑟]𝜑 → ∃𝑟 ∈ ℕ₀ (𝜓 ∧ 𝜑))))

22-Dec-2021	maxleast 10099	The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)) → sup({𝐴, 𝐵}, ℝ, < ) ≤ 𝐶)

22-Dec-2021	maxcl 10096	The maximum of two real numbers is a real number. (Contributed by Jim Kingdon, 22-Dec-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) ∈ ℝ)

22-Dec-2021	maxabslemval 10094	Lemma for maxabs 10095. Value of the supremum. (Contributed by Jim Kingdon, 22-Dec-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) ∈ ℝ ∧ ∀𝑥 ∈ {𝐴, 𝐵} ¬ (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) < 𝑥 ∧ ∀𝑥 ∈ ℝ (𝑥 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) → ∃𝑧 ∈ {𝐴, 𝐵}𝑥 < 𝑧)))

21-Dec-2021	dfabsmax 10103	Absolute value of a real number in terms of maximum. Definition 3.13 of [Geuvers], p. 11. (Contributed by BJ and Jim Kingdon, 21-Dec-2021.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = sup({𝐴, -𝐴}, ℝ, < ))

21-Dec-2021	maxleb 10102	Equivalence of ≤ and being equal to the maximum of two reals. Lemma 3.12 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ sup({𝐴, 𝐵}, ℝ, < ) = 𝐵))

21-Dec-2021	maxle2 10098	The maximum of two reals is no smaller than the second real. Lemma 3.10 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ sup({𝐴, 𝐵}, ℝ, < ))

21-Dec-2021	maxle1 10097	The maximum of two reals is no smaller than the first real. Lemma 3.10 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ, < ))

21-Dec-2021	maxabslemab 10092	Lemma for maxabs 10095. A variation of maxleim 10091- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 21-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵)

21-Dec-2021	maxleim 10091	Value of maximum when we know which number is larger. (Contributed by Jim Kingdon, 21-Dec-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → sup({𝐴, 𝐵}, ℝ, < ) = 𝐵))

21-Dec-2021	maxcom 10089	The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
⊢ sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < )

20-Dec-2021	maxabs 10095	Maximum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 20-Dec-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2))

20-Dec-2021	maxabslemlub 10093	Lemma for maxabs 10095. A least upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 20-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2)) ⇒ ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))

20-Dec-2021	maxabsle 10090	An upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 20-Dec-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2))

20-Dec-2021	suprzclex 8445	The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) & ⊢ (𝜑 → 𝐴 ⊆ ℤ) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)

20-Dec-2021	r19.12sn 3458	Special case of r19.12 2466 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑))

19-Dec-2021	infisoti 6445	Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) & ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) ⇒ ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))

19-Dec-2021	infsnti 6443	The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵)

19-Dec-2021	infeuti 6442	An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))

18-Dec-2021	infmoti 6441	Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))

18-Dec-2021	infminti 6440	The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

18-Dec-2021	infnlbti 6439	A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))

18-Dec-2021	infglbti 6438	An infimum is the greatest lower bound. See also infclti 6436 and inflbti 6437. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))

18-Dec-2021	inflbti 6437	An infimum is a lower bound. See also infclti 6436 and infglbti 6438. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))

17-Dec-2021	infclti 6436	An infimum belongs to its base class (closure law). See also inflbti 6437 and infglbti 6438. (Contributed by Jim Kingdon, 17-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

17-Dec-2021	infvalti 6435	Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))

17-Dec-2021	cnvti 6432	If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) ⇒ ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢^◡𝑅𝑣 ∧ ¬ 𝑣^◡𝑅𝑢)))

17-Dec-2021	cnvinfex 6431	Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥^◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧)))

16-Dec-2021	eqinftid 6434	Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

16-Dec-2021	eqinfti 6433	Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))

13-Dec-2021	orandc 880	Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)
⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))

12-Dec-2021	gcdsupex 10349	Existence of the supremum used in defining gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ ¬ (𝑋 = 0 ∧ 𝑌 = 0)) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑋 ∧ 𝑛 ∥ 𝑌)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑋 ∧ 𝑛 ∥ 𝑌)}𝑦 < 𝑧)))

12-Dec-2021	gcdmndc 10340	Decidablity lemma used in various proofs related to gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∧ 𝑁 = 0))

12-Dec-2021	ddifstab 3104	A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)
⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥 ∈ 𝐴)

11-Dec-2021	gcdsupcl 10350	Closure of the supremum used in defining gcd. A lemma for gcdval 10351 and gcdn0cl 10354. (Contributed by Jim Kingdon, 11-Dec-2021.)
⊢ (((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) ∧ ¬ (𝑋 = 0 ∧ 𝑌 = 0)) → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑋 ∧ 𝑛 ∥ 𝑌)}, ℝ, < ) ∈ ℕ)

11-Dec-2021	dvdsbnd 10348	There is an upper bound to the divisors of a nonzero integer. (Contributed by Jim Kingdon, 11-Dec-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) → ∃𝑛 ∈ ℕ ∀𝑚 ∈ (ℤ_≥‘𝑛) ¬ 𝑚 ∥ 𝐴)

7-Dec-2021	zsupcl 10343	Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at 𝑀 (which corresponds to the non-empty condition of classical supremum theorems), (b) decidable at each value after 𝑀, and (c) be false after 𝑗 (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → DECID 𝜓) & ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑀)∀𝑛 ∈ (ℤ_≥‘𝑗) ¬ 𝜓) ⇒ ⊢ (𝜑 → sup({𝑛 ∈ ℤ ∣ 𝜓}, ℝ, < ) ∈ (ℤ_≥‘𝑀))

7-Dec-2021	zsupcllemex 10342	Lemma for zsupcl 10343. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → DECID 𝜓) & ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑀)∀𝑛 ∈ (ℤ_≥‘𝑗) ¬ 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))

7-Dec-2021	zsupcllemstep 10341	Lemma for zsupcl 10343. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → DECID 𝜓) ⇒ ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))

5-Dec-2021	divalglemqt 10319	Lemma for divalg 10324. The 𝑄 = 𝑇 case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.)
⊢ (𝜑 → 𝐷 ∈ ℤ) & ⊢ (𝜑 → 𝑅 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℤ) & ⊢ (𝜑 → 𝑄 ∈ ℤ) & ⊢ (𝜑 → 𝑇 ∈ ℤ) & ⊢ (𝜑 → 𝑄 = 𝑇) & ⊢ (𝜑 → ((𝑄 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑆)) ⇒ ⊢ (𝜑 → 𝑅 = 𝑆)

4-Dec-2021	divalglemeuneg 10323	Lemma for divalg 10324. Uniqueness for a negative denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))

4-Dec-2021	divalglemeunn 10321	Lemma for divalg 10324. Uniqueness for a positive denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))

4-Dec-2021	divalglemnqt 10320	Lemma for divalg 10324. The 𝑄 < 𝑇 case involved in showing uniqueness. (Contributed by Jim Kingdon, 4-Dec-2021.)
⊢ (𝜑 → 𝐷 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℤ) & ⊢ (𝜑 → 𝑄 ∈ ℤ) & ⊢ (𝜑 → 𝑇 ∈ ℤ) & ⊢ (𝜑 → 0 ≤ 𝑆) & ⊢ (𝜑 → 𝑅 < 𝐷) & ⊢ (𝜑 → ((𝑄 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑆)) ⇒ ⊢ (𝜑 → ¬ 𝑄 < 𝑇)

2-Dec-2021	rspc2gv 2712	Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓))

30-Nov-2021	divalglemex 10322	Lemma for divalg 10324. The quotient and remainder exist. (Contributed by Jim Kingdon, 30-Nov-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → ∃𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))

30-Nov-2021	divalglemnn 10318	Lemma for divalg 10324. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))

27-Nov-2021	ioom 9269	An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.)
⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐴 < 𝐵))

26-Nov-2021	supisoti 6423	Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))) & ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) ⇒ ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))

26-Nov-2021	isoti 6420	An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))

26-Nov-2021	isotilem 6419	Lemma for isoti 6420. (Contributed by Jim Kingdon, 26-Nov-2021.)
⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))

26-Nov-2021	supsnti 6418	The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

24-Nov-2021	supmaxti 6417	The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

24-Nov-2021	suplubti 6413	A supremum is the least upper bound. See also supclti 6411 and supubti 6412. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

24-Nov-2021	supubti 6412	A supremum is an upper bound. See also supclti 6411 and suplubti 6413. This proof demonstrates how to expand an iota-based definition (df-iota 4887) using riotacl2 5501. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))

24-Nov-2021	supclti 6411	A supremum belongs to its base class (closure law). See also supubti 6412 and suplubti 6413. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)

24-Nov-2021	eqsuptid 6410	Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

23-Nov-2021	eqsupti 6409	Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))

23-Nov-2021	supval2ti 6408	Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))

23-Nov-2021	supeuti 6407	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

23-Nov-2021	supmoti 6406	Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 7191) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

17-Nov-2021	sqne2sq 10555	The square of a natural number can never be equal to two times the square of a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴↑2) ≠ (2 · (𝐵↑2)))

17-Nov-2021	2sqpwodd 10554	The greatest power of two dividing twice the square of an integer is an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ₀ ↦ ((2↑𝑦) · 𝑥)) ⇒ ⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (2^nd ‘(^◡𝐹‘(2 · (𝐴↑2)))))

17-Nov-2021	sqpweven 10553	The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ₀ ↦ ((2↑𝑦) · 𝑥)) ⇒ ⊢ (𝐴 ∈ ℕ → 2 ∥ (2^nd ‘(^◡𝐹‘(𝐴↑2))))

17-Nov-2021	oddpwdclemndvds 10549	Lemma for oddpwdc 10552. A natural number is not divisible by one more than the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ (𝐴 ∈ ℕ → ¬ (2↑((℩𝑧 ∈ ℕ₀ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)) + 1)) ∥ 𝐴)

17-Nov-2021	oddpwdclemdvds 10548	Lemma for oddpwdc 10552. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ (𝐴 ∈ ℕ → (2↑(℩𝑧 ∈ ℕ₀ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))) ∥ 𝐴)

17-Nov-2021	pw2dvdseulemle 10545	Lemma for pw2dvdseu 10546. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℕ₀) & ⊢ (𝜑 → 𝐵 ∈ ℕ₀) & ⊢ (𝜑 → (2↑𝐴) ∥ 𝑁) & ⊢ (𝜑 → ¬ (2↑(𝐵 + 1)) ∥ 𝑁) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵)

16-Nov-2021	oddpwdclemdc 10551	Lemma for oddpwdc 10552. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.)
⊢ ((((𝑋 ∈ ℕ ∧ ¬ 2 ∥ 𝑋) ∧ 𝑌 ∈ ℕ₀) ∧ 𝐴 = ((2↑𝑌) · 𝑋)) ↔ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (2↑(℩𝑧 ∈ ℕ₀ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ₀ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))))

16-Nov-2021	oddpwdclemodd 10550	Lemma for oddpwdc 10552. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.)
⊢ (𝐴 ∈ ℕ → ¬ 2 ∥ (𝐴 / (2↑(℩𝑧 ∈ ℕ₀ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))))

16-Nov-2021	oddpwdclemxy 10547	Lemma for oddpwdc 10552. Another way of stating that decomposing a natural number into a power of two and an odd number is unique. (Contributed by Jim Kingdon, 16-Nov-2021.)
⊢ ((((𝑋 ∈ ℕ ∧ ¬ 2 ∥ 𝑋) ∧ 𝑌 ∈ ℕ₀) ∧ 𝐴 = ((2↑𝑌) · 𝑋)) → (𝑋 = (𝐴 / (2↑(℩𝑧 ∈ ℕ₀ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ₀ ((2↑𝑧) ∥ 𝐴 ∧ ¬ (2↑(𝑧 + 1)) ∥ 𝐴))))

16-Nov-2021	pw2dvdseu 10546	A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.)
⊢ (𝑁 ∈ ℕ → ∃!𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))

14-Nov-2021	pw2dvds 10544	A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.)
⊢ (𝑁 ∈ ℕ → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))

14-Nov-2021	pw2dvdslemn 10543	Lemma for pw2dvds 10544. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ ¬ (2↑𝐴) ∥ 𝑁) → ∃𝑚 ∈ ℕ₀ ((2↑𝑚) ∥ 𝑁 ∧ ¬ (2↑(𝑚 + 1)) ∥ 𝑁))

14-Nov-2021	dvdsdc 10203	Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ∥ 𝑁)

11-Nov-2021	fz01or 10278	An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)
⊢ (𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1))

11-Nov-2021	muldivdirap 7795	Distribution of division over addition with a multiplication. (Contributed by Jim Kingdon, 11-Nov-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (((𝐶 · 𝐴) + 𝐵) / 𝐶) = (𝐴 + (𝐵 / 𝐶)))

10-Nov-2021	zeoxor 10268	An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.)
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ⊻ ¬ 2 ∥ 𝑁))

9-Nov-2021	qlttri2 8726	Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 9-Nov-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))

8-Nov-2021	dvdslelemd 10243	Lemma for dvdsle 10244. (Contributed by Jim Kingdon, 8-Nov-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑁 < 𝑀) ⇒ ⊢ (𝜑 → (𝐾 · 𝑀) ≠ 𝑁)

8-Nov-2021	qabsord 9962	The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))

8-Nov-2021	qabsor 9961	The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)
⊢ (𝐴 ∈ ℚ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))

6-Nov-2021	ibcval5 9690	Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 · (𝑁 − 1) · ... · ((𝑁 − 𝐾) + 1)) / 𝐾! explicitly. In this form, it is valid even for 𝑁 < 𝐾, although it is no longer valid for nonpositive 𝐾. (Contributed by Jim Kingdon, 6-Nov-2021.)
⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁 − 𝐾) + 1)( · , I , ℂ)‘𝑁) / (!‘𝐾)))

3-Nov-2021	qavgle 9267	The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵))

31-Oct-2021	nn0opth2d 9650	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 9649. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ₀) & ⊢ (𝜑 → 𝐵 ∈ ℕ₀) & ⊢ (𝜑 → 𝐶 ∈ ℕ₀) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) ⇒ ⊢ (𝜑 → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

31-Oct-2021	nn0opthd 9649	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers 𝐴 and 𝐵 by (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 3407 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ₀) & ⊢ (𝜑 → 𝐵 ∈ ℕ₀) & ⊢ (𝜑 → 𝐶 ∈ ℕ₀) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) ⇒ ⊢ (𝜑 → ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

31-Oct-2021	nn0opthlem2d 9648	Lemma for nn0opth2 9651. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ₀) & ⊢ (𝜑 → 𝐵 ∈ ℕ₀) & ⊢ (𝜑 → 𝐶 ∈ ℕ₀) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵)))

31-Oct-2021	nn0opthlem1d 9647	A rather pretty lemma for nn0opth2 9651. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ₀) & ⊢ (𝜑 → 𝐶 ∈ ℕ₀) ⇒ ⊢ (𝜑 → (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶)))

31-Oct-2021	nn0le2msqd 9646	The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ₀) & ⊢ (𝜑 → 𝐵 ∈ ℕ₀) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))

28-Oct-2021	mulle0r 8022	Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 0 ∧ 0 ≤ 𝐵)) → (𝐴 · 𝐵) ≤ 0)

26-Oct-2021	modqeqmodmin 9396	A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → (𝐴 mod 𝑀) = ((𝐴 − 𝑀) mod 𝑀))

26-Oct-2021	modqsubdir 9395	Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → ((𝐵 mod 𝐶) ≤ (𝐴 mod 𝐶) ↔ ((𝐴 − 𝐵) mod 𝐶) = ((𝐴 mod 𝐶) − (𝐵 mod 𝐶))))

26-Oct-2021	modqdi 9394	Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐵 mod 𝐶)) = ((𝐴 · 𝐵) mod (𝐴 · 𝐶)))

26-Oct-2021	modqaddmulmod 9393	The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐶 ∈ ℤ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 + ((𝐵 mod 𝑀) · 𝐶)) mod 𝑀) = ((𝐴 + (𝐵 · 𝐶)) mod 𝑀))

26-Oct-2021	modqmulmodr 9392	The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 · (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))

25-Oct-2021	modqmulmod 9391	The product of a rational number modulo a modulus and an integer equals the product of the rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) · 𝐵) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))

25-Oct-2021	q2submod 9387	If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴 − 𝐵))

25-Oct-2021	q2txmodxeq0 9386	Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) mod 𝑋) = 0)

25-Oct-2021	modqsubmodmod 9385	The difference of a number modulo a modulus and another number modulo the same modulus equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) − (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 − 𝐵) mod 𝑀))

25-Oct-2021	modqsubmod 9384	The difference of a number modulo a modulus and another number equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) − 𝐵) mod 𝑀) = ((𝐴 − 𝐵) mod 𝑀))

25-Oct-2021	modqsub12d 9383	Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 𝐸 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐸) & ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) & ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐶) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸))

25-Oct-2021	modqadd12d 9382	Additive property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 𝐸 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐸) & ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) & ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸))

25-Oct-2021	syldc 45	Syllogism deduction. Commuted form of syld 44. (Contributed by BJ, 25-Oct-2021.)
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜓 → (𝜑 → 𝜃))

25-Oct-2021	syl11 31	A syllogism inference. Commuted form of an instance of syl 14. (Contributed by BJ, 25-Oct-2021.)
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → 𝜑) ⇒ ⊢ (𝜓 → (𝜃 → 𝜒))

24-Oct-2021	ax-ddkcomp 10784	Axiom of Dedekind completeness for Dedekind real numbers: every inhabited upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then the axiom ax-ddkcomp 10784 should be used in place of construction specific results. In particular, axcaucvg 7066 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝐵) → 𝑥 ≤ 𝐵)))

24-Oct-2021	modqnegd 9381	Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐶) & ⊢ (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶)) ⇒ ⊢ (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶))

24-Oct-2021	modqmul12d 9380	Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℤ) & ⊢ (𝜑 → 𝐸 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐸) & ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) & ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸))

24-Oct-2021	modqmul1 9379	Multiplication property of the modulo operation. Note that the multiplier 𝐶 must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐷) & ⊢ (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐷) = ((𝐵 · 𝐶) mod 𝐷))

24-Oct-2021	modqltm1p1mod 9378	If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ (𝐴 mod 𝑀) < (𝑀 − 1)) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 + 1) mod 𝑀) = ((𝐴 mod 𝑀) + 1))

24-Oct-2021	modqm1p1mod0 9377	If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = (𝑀 − 1) → ((𝐴 + 1) mod 𝑀) = 0))

24-Oct-2021	modqadd2mod 9376	The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀) = ((𝐵 + 𝐴) mod 𝑀))

24-Oct-2021	qnegmod 9371	The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (-𝐴 mod 𝑁) = ((𝑁 − 𝐴) mod 𝑁))

23-Oct-2021	modqmuladdnn0 9370	Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ₀ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))

23-Oct-2021	modqmuladdim 9369	Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))

23-Oct-2021	modqmuladd 9368	Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ (0[,)𝑀)) & ⊢ (𝜑 → 𝑀 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝑀) ⇒ ⊢ (𝜑 → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))

23-Oct-2021	mulqaddmodid 9366	The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℚ) ∧ (𝐴 ∈ ℚ ∧ 𝐴 ∈ (0[,)𝑀))) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴)

23-Oct-2021	modqaddmod 9365	The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the same modulus. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) + 𝐵) mod 𝑀) = ((𝐴 + 𝐵) mod 𝑀))

22-Oct-2021	modqaddabs 9364	Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (((𝐴 mod 𝐶) + (𝐵 mod 𝐶)) mod 𝐶) = ((𝐴 + 𝐵) mod 𝐶))

22-Oct-2021	modqadd1 9363	Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐷) & ⊢ (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷))

21-Oct-2021	modqcyc2 9362	The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) ∧ (𝐵 ∈ ℚ ∧ 0 < 𝐵)) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = (𝐴 mod 𝐵))

21-Oct-2021	modqcyc 9361	The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) ∧ (𝐵 ∈ ℚ ∧ 0 < 𝐵)) → ((𝐴 + (𝑁 · 𝐵)) mod 𝐵) = (𝐴 mod 𝐵))

21-Oct-2021	modqabs2 9360	Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) mod 𝐵) = (𝐴 mod 𝐵))

21-Oct-2021	modqabs 9359	Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐵) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 mod 𝐵) mod 𝐶) = (𝐴 mod 𝐵))

21-Oct-2021	q1mod 9358	Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝑁 ∈ ℚ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1)

21-Oct-2021	q0mod 9357	Special case: 0 modulo a positive real number is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (0 mod 𝑁) = 0)

21-Oct-2021	modqid2 9353	Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 𝐴 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)))

21-Oct-2021	modqid0 9352	A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 mod 𝑁) = 0)

21-Oct-2021	modqid 9351	Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)

20-Oct-2021	modqvalp1 9345	The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Jim Kingdon, 20-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 + 𝐵) − (((⌊‘(𝐴 / 𝐵)) + 1) · 𝐵)) = (𝐴 mod 𝐵))

20-Oct-2021	onunsnss 6383	Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴)

19-Oct-2021	snon0 6387	An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)

18-Oct-2021	qdencn 10785	The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 10088 (and also would hold for ℝ × ℝ with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ 𝑄 = {𝑧 ∈ ℂ ∣ ((ℜ‘𝑧) ∈ ℚ ∧ (ℑ‘𝑧) ∈ ℚ)} ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ⁺) → ∃𝑥 ∈ 𝑄 (abs‘(𝑥 − 𝐴)) < 𝐵)

18-Oct-2021	modqmulnn 9344	Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) ≤ ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)))

18-Oct-2021	intqfrac 9341	Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1)))

18-Oct-2021	flqmod 9340	The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))

18-Oct-2021	modqfrac 9339	The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (𝐴 mod 1) = (𝐴 − (⌊‘𝐴)))

18-Oct-2021	modqdifz 9338	The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ)

18-Oct-2021	modqdiffl 9337	The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵)))

18-Oct-2021	modqelico 9336	Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ (0[,)𝐵))

18-Oct-2021	modqlt 9335	The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵)

18-Oct-2021	modqge0 9334	The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))

18-Oct-2021	negqmod0 9333	𝐴 is divisible by 𝐵 iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (-𝐴 mod 𝐵) = 0))

18-Oct-2021	mulqmod0 9332	The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 · 𝑀) mod 𝑀) = 0)

18-Oct-2021	flqdiv 9323	Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ) → (⌊‘((⌊‘𝐴) / 𝑁)) = (⌊‘(𝐴 / 𝑁)))

18-Oct-2021	intqfrac2 9321	Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ 𝑍 = (⌊‘𝐴) & ⊢ 𝐹 = (𝐴 − 𝑍) ⇒ ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))

17-Oct-2021	modq0 9331	𝐴 mod 𝐵 is zero iff 𝐴 is evenly divisible by 𝐵. (Contributed by Jim Kingdon, 17-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (𝐴 / 𝐵) ∈ ℤ))

16-Oct-2021	modqcld 9330	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℚ)

16-Oct-2021	flqpmodeq 9329	Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴)

16-Oct-2021	modqcl 9328	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)

16-Oct-2021	modqvalr 9327	The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵)))

16-Oct-2021	modqval 9326	The value of the modulo operation. The modulo congruence notation of number theory, 𝐽≡𝐾 (modulo 𝑁), can be expressed in our notation as (𝐽 mod 𝑁) = (𝐾 mod 𝑁). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive numbers to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) As with flqcl 9277 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))

15-Oct-2021	qdenre 10088	The rational numbers are dense in ℝ: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9265. (Contributed by BJ, 15-Oct-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ∃𝑥 ∈ ℚ (abs‘(𝑥 − 𝐴)) < 𝐵)

14-Oct-2021	qbtwnrelemcalc 9264	Lemma for qbtwnre 9265. Calculations involved in showing the constructed rational number is less than 𝐵. (Contributed by Jim Kingdon, 14-Oct-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑀 < (𝐴 · (2 · 𝑁))) & ⊢ (𝜑 → (1 / 𝑁) < (𝐵 − 𝐴)) ⇒ ⊢ (𝜑 → ((𝑀 + 2) / (2 · 𝑁)) < 𝐵)

13-Oct-2021	rebtwn2z 9263	A real number can be bounded by integers above and below which are two apart. The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)
⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))

13-Oct-2021	rebtwn2zlemshrink 9262	Lemma for rebtwn2z 9263. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐽 ∈ (ℤ_≥‘2) ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))

13-Oct-2021	rebtwn2zlemstep 9261	Lemma for rebtwn2z 9263. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + 𝐾)))

11-Oct-2021	flqeqceilz 9320	A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))

11-Oct-2021	flqleceil 9319	The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ (⌈‘𝐴))

11-Oct-2021	ceilqidz 9318	A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))

11-Oct-2021	ceilqle 9316	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (⌈‘𝐴) ≤ 𝐵)

11-Oct-2021	ceiqle 9315	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵)

11-Oct-2021	ceilqm1lt 9314	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ((⌈‘𝐴) − 1) < 𝐴)

11-Oct-2021	ceiqm1l 9313	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (-(⌊‘-𝐴) − 1) < 𝐴)

11-Oct-2021	ceilqge 9312	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))

11-Oct-2021	ceiqge 9311	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 ≤ -(⌊‘-𝐴))

11-Oct-2021	ceilqcl 9310	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)

11-Oct-2021	ceiqcl 9309	The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)

11-Oct-2021	qdceq 9256	Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵)

11-Oct-2021	qltlen 8725	Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7730 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)))

10-Oct-2021	ceilqval 9308	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))

10-Oct-2021	flqmulnn0 9301	Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))

10-Oct-2021	flqzadd 9300	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℚ) → (⌊‘(𝑁 + 𝐴)) = (𝑁 + (⌊‘𝐴)))

10-Oct-2021	flqaddz 9299	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))

10-Oct-2021	flqge1nn 9296	The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)

10-Oct-2021	flqge0nn0 9295	The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)

10-Oct-2021	4ap0 8138	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ 4 # 0

10-Oct-2021	3ap0 8135	The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ 3 # 0

9-Oct-2021	flqbi2 9293	A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1)))

9-Oct-2021	flqbi 9292	A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1))))

9-Oct-2021	flqword2 9291	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ_≥‘(⌊‘𝐴)))

9-Oct-2021	flqwordi 9290	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))

9-Oct-2021	flqltnz 9289	If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴)

9-Oct-2021	flqidz 9288	A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ))

8-Oct-2021	flqidm 9287	The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))

8-Oct-2021	flqlt 9285	The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵))

8-Oct-2021	flqge 9284	The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴)))

8-Oct-2021	qfracge0 9283	The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴)))

8-Oct-2021	qfraclt1 9282	The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1)

8-Oct-2021	flqltp1 9281	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1))

8-Oct-2021	flqle 9280	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)

8-Oct-2021	flqcld 9279	The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → (⌊‘𝐴) ∈ ℤ)

8-Oct-2021	flqlelt 9278	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)))

8-Oct-2021	flqcl 9277	The floor (greatest integer) function yields an integer when applied to a rational (closure law). It would presumably be possible to prove a similar result for some real numbers (for example, those apart from any integer), but not real numbers in general. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)

8-Oct-2021	qbtwnz 9260	There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

8-Oct-2021	qbtwnzlemex 9259	Lemma for qbtwnz 9260. Existence of the integer. The proof starts by finding two integers which are less than and greater than the given rational number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on rational number trichotomy, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

8-Oct-2021	qbtwnzlemshrink 9258	Lemma for qbtwnz 9260. Shrinking the range around the given rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐽 ∈ ℕ ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))

8-Oct-2021	qbtwnzlemstep 9257	Lemma for qbtwnz 9260. Induction step. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ ((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐾)))

8-Oct-2021	qltnle 9255	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

7-Oct-2021	qlelttric 9254	Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴))

6-Oct-2021	qletric 9253	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))

6-Oct-2021	qtri3or 9252	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
⊢ ((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))

3-Oct-2021	reg3exmid 4322	If any inhabited set satisfying df-wetr 4089 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
⊢ (( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

3-Oct-2021	reg3exmidlemwe 4321	Lemma for reg3exmid 4322. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} ⇒ ⊢ E We 𝐴

2-Oct-2021	sqrt2irrap 10558	The square root of 2 is irrational. That is, for any rational number, (√‘2) is apart from it. In the absence of excluded middle, we can distinguish between this and "the square root of 2 is not rational" which is sqrt2irr 10541. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ (𝑄 ∈ ℚ → (√‘2) # 𝑄)

2-Oct-2021	sqrt2irraplemnn 10557	Lemma for sqrt2irrap 10558. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (√‘2) # (𝐴 / 𝐵))

2-Oct-2021	reg2exmid 4279	If any inhabited set has a minimal element (when expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ ∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

2-Oct-2021	reg2exmidlema 4277	Lemma for reg2exmid 4279. If 𝐴 has a minimal element (expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} ⇒ ⊢ (∃𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 → (𝜑 ∨ ¬ 𝜑))

30-Sep-2021	fin0or 6370	A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴))

30-Sep-2021	wessep 4320	A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → E We 𝐵)

28-Sep-2021	frind 4107	Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑)) & ⊢ (𝜒 → 𝑅 Fr 𝐴) & ⊢ (𝜒 → 𝐴 ∈ 𝑉) ⇒ ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → 𝜑)

26-Sep-2021	wetriext 4319	A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶)

25-Sep-2021	nnwetri 6382	A natural number is well-ordered by E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
⊢ (𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))

23-Sep-2021	wepo 4114	A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 Po 𝐴)

23-Sep-2021	df-wetr 4089	Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals don't have that as seen at ordtriexmid 4265). Given excluded middle, well-ordering is usually defined to require trichotomy (and the defintion of Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.)
⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

22-Sep-2021	frforeq3 4102	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))

22-Sep-2021	frforeq2 4100	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))

22-Sep-2021	frforeq1 4098	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))

22-Sep-2021	df-frfor 4086	Define the well-founded relation predicate where 𝐴 might be a proper class. By passing in 𝑆 we allow it potentially to be a proper class rather than a set. (Contributed by Jim Kingdon and Mario Carneiro, 22-Sep-2021.)
⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆))

21-Sep-2021	frirrg 4105	A well-founded relation is irreflexive. This is the case where 𝐴 exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

21-Sep-2021	df-frind 4087	Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because 𝑠 is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.)
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)

17-Sep-2021	ontr2exmid 4268	An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
⊢ ∀𝑥 ∈ On ∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

16-Sep-2021	decrmac 8534	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by AV, 16-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐺 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝑁) = ;𝐺𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹

16-Sep-2021	decrmanc 8533	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by AV, 16-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ (𝐴 · 𝑃) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝑁) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹

16-Sep-2021	nnsseleq 6102	For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

15-Sep-2021	fientri3 6381	Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴))

15-Sep-2021	nntri2or2 6099	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

14-Sep-2021	findcard2sd 6376	Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏)) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → 𝜂)

13-Sep-2021	php5fin 6366	A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) → ¬ 𝐴 ≈ (𝐴 ∪ {𝐵}))

13-Sep-2021	fiunsnnn 6365	Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → (𝐴 ∪ {𝐵}) ≈ suc 𝑁)

12-Sep-2021	fisbth 6367	Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵)

11-Sep-2021	diffisn 6377	Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ∈ Fin)

10-Sep-2021	fin0 6369	A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
⊢ (𝐴 ∈ Fin → (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴))

9-Sep-2021	fidifsnid 6356	If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3531 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

9-Sep-2021	fidifsnen 6355	All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
⊢ ((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))

8-Sep-2021	3dvdsdec 10264	A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if 𝐴 and 𝐵 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴 and 𝐵, especially if 𝐴 is itself a decimal number, e.g. 𝐴 = ;𝐶𝐷. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵))

8-Sep-2021	1lt10 8615	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 1 < ;10

8-Sep-2021	2lt10 8614	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 2 < ;10

8-Sep-2021	3lt10 8613	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 3 < ;10

8-Sep-2021	4lt10 8612	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 4 < ;10

8-Sep-2021	5lt10 8611	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 5 < ;10

8-Sep-2021	6lt10 8610	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 6 < ;10

8-Sep-2021	7lt10 8609	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 7 < ;10

8-Sep-2021	8lt10 8608	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 8 < ;10

8-Sep-2021	9lt10 8607	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
⊢ 9 < ;10

8-Sep-2021	decltdi 8515	Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 < ;𝐴𝐵

8-Sep-2021	decleh 8511	Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷

8-Sep-2021	decle 8510	Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐵 ≤ 𝐶 ⇒ ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶

8-Sep-2021	3declth 8508	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐸 ≤ 9 ⇒ ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹

8-Sep-2021	declth 8506	Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 < ;𝐵𝐷

8-Sep-2021	le9lt10 8503	A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐴 ≤ 9 ⇒ ⊢ 𝐴 < ;10

8-Sep-2021	10re 8495	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ ;10 ∈ ℝ

8-Sep-2021	10pos 8493	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ 0 < ;10

8-Sep-2021	diffifi 6378	Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ Fin)

8-Sep-2021	diffitest 6371	If subtracting any set from a finite set gives a finite set, any proposition of the form ¬ 𝜑 is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove 𝐴 ∈ Fin → (𝐴 ∖ 𝐵) ∈ Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
⊢ ∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin ⇒ ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)

6-Sep-2021	9t11e99 8606	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
⊢ (9 · ;11) = ;99

6-Sep-2021	8t6e48 8595	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 6) = ;48

6-Sep-2021	8t5e40 8594	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 · 5) = ;40

6-Sep-2021	6t6e36 8584	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 · 6) = ;36

6-Sep-2021	6t5e30 8583	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 · 5) = ;30

6-Sep-2021	5t5e25 8579	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 5) = ;25

6-Sep-2021	5t4e20 8578	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 4) = ;20

6-Sep-2021	5t3e15 8577	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (5 · 3) = ;15

6-Sep-2021	10m1e9 8572	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
⊢ (;10 − 1) = 9

6-Sep-2021	10p10e20 8571	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + ;10) = ;20

6-Sep-2021	9p2e11 8563	9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (9 + 2) = ;11

6-Sep-2021	8p3e11 8557	8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (8 + 3) = ;11

6-Sep-2021	8p2e10 8556	8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (8 + 2) = ;10

6-Sep-2021	7p4e11 8552	7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (7 + 4) = ;11

6-Sep-2021	7p3e10 8551	7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (7 + 3) = ;10

6-Sep-2021	6p5e11 8549	6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (6 + 5) = ;11

6-Sep-2021	6p4e10 8548	6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (6 + 4) = ;10

6-Sep-2021	5p5e10 8547	5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
⊢ (5 + 5) = ;10

6-Sep-2021	decmul10add 8545	A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑀 ∈ ℕ₀ & ⊢ 𝐸 = (𝑀 · 𝐴) & ⊢ 𝐹 = (𝑀 · 𝐵) ⇒ ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹)

6-Sep-2021	decmul2c 8542	The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 & ⊢ (𝑃 · 𝐵) = ;𝐸𝐷 ⇒ ⊢ (𝑃 · 𝑁) = ;𝐶𝐷

6-Sep-2021	decmul1c 8541	The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 & ⊢ (𝐵 · 𝑃) = ;𝐸𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷

6-Sep-2021	decmul1 8540	The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ₀ & ⊢ (𝐴 · 𝑃) = 𝐶 & ⊢ (𝐵 · 𝑃) = 𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷

6-Sep-2021	decsubi 8539	Difference between a numeral 𝑀 and a nonnegative integer 𝑁 (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 − 𝑁) = 𝐶 ⇒ ⊢ (𝑀 − 𝑁) = ;𝐴𝐶

6-Sep-2021	decaddci2 8538	Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 + 𝑁) = ;10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐷0

6-Sep-2021	decaddc2 8532	Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ (𝐵 + 𝐷) = ;10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸0

6-Sep-2021	decaddc 8531	Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ 𝐹 ∈ ℕ₀ & ⊢ (𝐵 + 𝐷) = ;1𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹

6-Sep-2021	decadd 8530	Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ (𝐴 + 𝐶) = 𝐸 & ⊢ (𝐵 + 𝐷) = 𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹

6-Sep-2021	decma2c 8529	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplier 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐺 ∈ ℕ₀ & ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝑃 · 𝐵) + 𝐷) = ;𝐺𝐹 ⇒ ⊢ ((𝑃 · 𝑀) + 𝑁) = ;𝐸𝐹

6-Sep-2021	decmac 8528	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐺 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = ;𝐺𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹

6-Sep-2021	decma 8527	Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ₀ & ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹

6-Sep-2021	dec10p 8519	Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (;10 + 𝐴) = ;1𝐴

6-Sep-2021	decsucc 8517	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ (𝐴 + 1) = 𝐵 & ⊢ 𝑁 = ;𝐴9 ⇒ ⊢ (𝑁 + 1) = ;𝐵0

6-Sep-2021	declti 8514	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 < ;10 ⇒ ⊢ 𝐶 < ;𝐴𝐵

6-Sep-2021	3decltc 8509	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐸 ∈ ℕ₀ & ⊢ 𝐹 ∈ ℕ₀ & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 < ;10 & ⊢ 𝐸 < ;10 ⇒ ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹

6-Sep-2021	decsuc 8507	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ (𝐵 + 1) = 𝐶 & ⊢ 𝑁 = ;𝐴𝐵 ⇒ ⊢ (𝑁 + 1) = ;𝐴𝐶

6-Sep-2021	decltc 8505	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐷 ∈ ℕ₀ & ⊢ 𝐶 < ;10 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 < ;𝐵𝐷

6-Sep-2021	declt 8504	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ ;𝐴𝐵 < ;𝐴𝐶

6-Sep-2021	decnncl2 8500	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ ⇒ ⊢ ;𝐴0 ∈ ℕ

6-Sep-2021	dec0h 8498	Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ 𝐴 = ;0𝐴

6-Sep-2021	dec0u 8497	Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ ⇒ ⊢ (;10 · 𝐴) = ;𝐴0

6-Sep-2021	decnncl 8496	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ;𝐴𝐵 ∈ ℕ

6-Sep-2021	10nn0 8494	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ₀

6-Sep-2021	10nn 8492	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ

6-Sep-2021	deccl 8491	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ ;𝐴𝐵 ∈ ℕ₀

6-Sep-2021	deceq2 8482	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵)

6-Sep-2021	deceq1 8481	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶)

6-Sep-2021	phpelm 6352	Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ≈ 𝐵)

5-Sep-2021	ex-gcd 10568	Example for df-gcd 10339. (Contributed by AV, 5-Sep-2021.)
⊢ (-6 gcd 9) = 3

5-Sep-2021	fidceq 6354	Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that {𝐵, 𝐶} is finite would require showing it is equinumerous to 1_𝑜 or to 2_𝑜 but to show that you'd need to know 𝐵 = 𝐶 or ¬ 𝐵 = 𝐶, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)

5-Sep-2021	phplem4on 6353	Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

4-Sep-2021	ex-bc 10566	Example for df-bc 9675. (Contributed by AV, 4-Sep-2021.)
⊢ (5C3) = ;10

4-Sep-2021	ex-fac 10565	Example for df-fac 9653. (Contributed by AV, 4-Sep-2021.)
⊢ (!‘5) = ;;120

4-Sep-2021	ex-ceil 10564	Example for df-ceil 9275. (Contributed by AV, 4-Sep-2021.)
⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

4-Sep-2021	5t2e10 8576	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
⊢ (5 · 2) = ;10

3-Sep-2021	0elsucexmid 4308	If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥 ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

1-Sep-2021	php5dom 6349	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ (𝐴 ∈ ω → ¬ suc 𝐴 ≼ 𝐴)

1-Sep-2021	phplem4dom 6348	Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≼ suc 𝐵 → 𝐴 ≼ 𝐵))

1-Sep-2021	snnen2oprc 6346	A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a set, see snnen2og 6345. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2_𝑜)

1-Sep-2021	snnen2og 6345	A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a proper class, see snnen2oprc 6346. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} ≈ 2_𝑜)

1-Sep-2021	phplem3g 6342	A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6340 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))

31-Aug-2021	nndifsnid 6103	If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3531 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

30-Aug-2021	carden2bex 6458	If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))

30-Aug-2021	cardval3ex 6454	The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})

30-Aug-2021	cardcl 6450	The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)

30-Aug-2021	onintrab2im 4262	An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)

30-Aug-2021	onintonm 4261	The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
⊢ ((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ On)

29-Aug-2021	onintexmid 4315	If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

29-Aug-2021	ordtri2or2exmid 4314	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

29-Aug-2021	ordtri2or2exmidlem 4269	A set which is 2_𝑜 if 𝜑 or ∅ if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On

29-Aug-2021	2ordpr 4267	Version of 2on 6032 with the definition of 2_𝑜 expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ Ord {∅, {∅}}

25-Aug-2021	nnap0d 8084	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐴 # 0)

24-Aug-2021	climcaucn 10188	A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 10184 but adds the part that (𝐹‘𝑘) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))

24-Aug-2021	climcvg1nlem 10186	Lemma for climcvg1n 10187. We construct sequences of the real and imaginary parts of each term of 𝐹, show those converge, and use that to show that 𝐹 converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℝ⁺) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛)) & ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ (ℜ‘(𝐹‘𝑥))) & ⊢ 𝐻 = (𝑥 ∈ ℕ ↦ (ℑ‘(𝐹‘𝑥))) & ⊢ 𝐽 = (𝑥 ∈ ℕ ↦ (i · (𝐻‘𝑥))) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )

23-Aug-2021	climcvg1n 10187	A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within 𝐶 / 𝑛 of the nth term, where 𝐶 is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℝ⁺) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )

23-Aug-2021	climrecvg1n 10185	A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within 𝐶 / 𝑛 of the nth term, where 𝐶 is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ⁺) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )

22-Aug-2021	climserile 10183	The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by Jim Kingdon, 22-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ≤ 𝐴)

22-Aug-2021	iserige0 10181	The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴)

22-Aug-2021	iserile 10180	Comparison of the limits of two infinite series. (Contributed by Jim Kingdon, 22-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴) & ⊢ (𝜑 → seq𝑀( + , 𝐺, ℂ) ⇝ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵)

22-Aug-2021	serile 9474	Comparison of partial sums of two infinite series of reals. (Contributed by Jim Kingdon, 22-Aug-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ≤ (seq𝑀( + , 𝐺, ℂ)‘𝑁))

22-Aug-2021	serige0 9473	A finite sum of nonnegative terms is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))

21-Aug-2021	divalgmodcl 10328	The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor. Variant of divalgmod 10327. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ 𝑅 ∈ ℕ₀) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅))))

21-Aug-2021	divalgmod 10327	The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor (compare divalg2 10326 and divalgb 10325). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 ∈ ℕ₀ ∧ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))))

21-Aug-2021	clim2iser2 10176	The limit of an infinite series with an initial segment added. (Contributed by Jim Kingdon, 21-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹, ℂ) ⇝ 𝐴) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ (𝐴 + (seq𝑀( + , 𝐹, ℂ)‘𝑁)))

21-Aug-2021	iseqdistr 9470	The distributive property for series. (Contributed by Jim Kingdon, 21-Aug-2021.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥))) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

21-Aug-2021	iseqhomo 9468	Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 21-Aug-2021.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))

20-Aug-2021	oddprmge3 10516	An odd prime is greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 20-Aug-2021.)
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘3))

20-Aug-2021	oddprmgt2 10515	An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.)
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)

20-Aug-2021	clim2iser 10175	The limit of an infinite series with an initial segment removed. (Contributed by Jim Kingdon, 20-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴) ⇒ ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹, ℂ) ⇝ (𝐴 − (seq𝑀( + , 𝐹, ℂ)‘𝑁)))

20-Aug-2021	iseqex 9433	Existence of the sequence builder operation. (Contributed by Jim Kingdon, 20-Aug-2021.)
⊢ seq𝑀( + , 𝐹, 𝑆) ∈ V

20-Aug-2021	frecex 6004	Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
⊢ frec(𝐹, 𝐴) ∈ V

20-Aug-2021	tfrfun 5978	Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
⊢ Fun recs(𝐹)

19-Aug-2021	modremain 10329	The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ₀ ∧ 𝑅 < 𝐷)) → ((𝑁 mod 𝐷) = 𝑅 ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁))

19-Aug-2021	iserclim0 10144	The zero series converges to zero. (Contributed by Jim Kingdon, 19-Aug-2021.)
⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ_≥‘𝑀) × {0}), ℂ) ⇝ 0)

19-Aug-2021	shftval4g 9725	Value of a sequence shifted by -𝐴. (Contributed by Jim Kingdon, 19-Aug-2021.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵)))

19-Aug-2021	iser0f 9472	A zero-valued infinite series is equal to the constant zero function. (Contributed by Jim Kingdon, 19-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0}), ℂ) = (𝑍 × {0}))

19-Aug-2021	iser0 9471	The value of the partial sums in a zero-valued infinite series. (Contributed by Jim Kingdon, 19-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}), ℂ)‘𝑁) = 0)

19-Aug-2021	ovexg 5559	Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ V)

18-Aug-2021	iseqid3s 9466	A sequence that consists of zeroes up to 𝑁 sums to zero at 𝑁. In this case by "zero" we mean whatever the identity 𝑍 is for the operation +). (Contributed by Jim Kingdon, 18-Aug-2021.)
⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍) & ⊢ (𝜑 → 𝑍 ∈ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)

18-Aug-2021	iseqid3 9465	A sequence that consists entirely of zeroes (or whatever the identity 𝑍 is for operation +) sums to zero. (Contributed by Jim Kingdon, 18-Aug-2021.)
⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) = 𝑍) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, {𝑍})‘𝑁) = 𝑍)

18-Aug-2021	iseqss 9446	Specifying a larger universe for seq. As long as 𝐹 and + are closed over 𝑆, then any set which contains 𝑆 can be used as the last argument to seq. This theorem does not allow 𝑇 to be a proper class, however. It also currently requires that + be closed over 𝑇 (as well as 𝑆). (Contributed by Jim Kingdon, 18-Aug-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝑇) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥 + 𝑦) ∈ 𝑇) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))

17-Aug-2021	iseqcaopr 9462	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Jim Kingdon, 17-Aug-2021.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

17-Aug-2021	declei 8512	Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 ≤ ;𝐴𝐵

16-Aug-2021	oddge22np1 10281	An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)
⊢ (𝑁 ∈ (ℤ_≥‘2) → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ ((2 · 𝑛) + 1) = 𝑁))

16-Aug-2021	iseqcaopr3 9460	Lemma for iseqcaopr2 . (Contributed by Jim Kingdon, 16-Aug-2021.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘))) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1))))) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

16-Aug-2021	iseq1p 9459	Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 + 1))) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = ((𝐹‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))

16-Aug-2021	iseqsplit 9458	Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 + 1))) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝐾)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (𝐹‘𝑥) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))

15-Aug-2021	n2dvdsm1 10313	2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
⊢ ¬ 2 ∥ -1

15-Aug-2021	shftfibg 9708	Value of a fiber of the relation 𝐹. (Contributed by Jim Kingdon, 15-Aug-2021.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵 − 𝐴)}))

15-Aug-2021	ovshftex 9707	Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) ∈ V)

15-Aug-2021	isermono 9457	The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘𝑀)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝐾)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, ℝ)‘𝐾) ≤ (seq𝑀( + , 𝐹, ℝ)‘𝑁))

15-Aug-2021	iserf 9453	An infinite series of complex terms is a function from ℕ to ℂ. (Contributed by Jim Kingdon, 15-Aug-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)

15-Aug-2021	iseqshft2 9452	Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾))) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))

15-Aug-2021	iseqfeq 9451	Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) = (𝐺‘𝑘)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))

13-Aug-2021	absdivapd 10081	Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

13-Aug-2021	absrpclapd 10074	The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ⁺)

13-Aug-2021	absdivapzi 10040	Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

13-Aug-2021	absgt0ap 9985	The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ 0 < (abs‘𝐴)))

13-Aug-2021	recvalap 9983	Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) = ((∗‘𝐴) / ((abs‘𝐴)↑2)))

13-Aug-2021	sqap0 9542	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) # 0 ↔ 𝐴 # 0))

13-Aug-2021	leltapd 7737	'<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴))

13-Aug-2021	leltap 7724	'<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴))

12-Aug-2021	evennn2n 10283	A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.)
⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (2 · 𝑛) = 𝑁))

12-Aug-2021	evennn02n 10282	A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)
⊢ (𝑁 ∈ ℕ₀ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ₀ (2 · 𝑛) = 𝑁))

12-Aug-2021	abssubap0 9976	If the absolute value of a complex number is less than a real, its difference from the real is apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ (abs‘𝐴) < 𝐵) → (𝐵 − 𝐴) # 0)

12-Aug-2021	ltabs 9973	A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < (abs‘𝐴)) → 𝐴 < 0)

12-Aug-2021	absext 9949	Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) # (abs‘𝐵) → 𝐴 # 𝐵))

12-Aug-2021	sq11ap 9639	Analogue to sq11 9548 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) # (𝐵↑2) ↔ 𝐴 # 𝐵))

12-Aug-2021	subap0d 7740	Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) # 0)

12-Aug-2021	ltapd 7736	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 # 𝐵)

12-Aug-2021	gtapd 7735	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 # 𝐴)

12-Aug-2021	ltapi 7734	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 # 𝐴)

12-Aug-2021	ltapii 7733	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 # 𝐵

12-Aug-2021	gtapii 7732	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐵 # 𝐴

12-Aug-2021	ltap 7731	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴)

11-Aug-2021	absexpzap 9966	Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))

11-Aug-2021	absdivap 9956	Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

11-Aug-2021	abs00ap 9948	The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) # 0 ↔ 𝐴 # 0))

11-Aug-2021	absrpclap 9947	The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) ∈ ℝ⁺)

11-Aug-2021	sqrt11ap 9924	Analogue to sqrt11 9925 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) # (√‘𝐵) ↔ 𝐴 # 𝐵))

11-Aug-2021	cjap0d 9835	A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (∗‘𝐴) # 0)

11-Aug-2021	ap0gt0d 7739	A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → 0 < 𝐴)

11-Aug-2021	ap0gt0 7738	A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ 0 < 𝐴))

10-Aug-2021	rersqrtthlem 9916	Lemma for resqrtth 9917. (Contributed by Jim Kingdon, 10-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴)))

10-Aug-2021	rersqreu 9914	Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))

10-Aug-2021	rsqrmo 9913	Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃*𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))

9-Aug-2021	dvdsdivcl 10250	The complement of a divisor of 𝑁 is also a divisor of 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝐴) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})

9-Aug-2021	resqrexlemoverl 9907	Lemma for resqrex 9912. Every term in the sequence is an overestimate compared with the limit 𝐿. Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) & ⊢ (𝜑 → 𝐾 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐿 ≤ (𝐹‘𝐾))

8-Aug-2021	isoddgcd1 10538	The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.)
⊢ (𝑍 ∈ ℤ → (¬ 2 ∥ 𝑍 ↔ (2 gcd 𝑍) = 1))

8-Aug-2021	isevengcd2 10537	The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.)
⊢ (𝑍 ∈ ℤ → (2 ∥ 𝑍 ↔ (2 gcd 𝑍) = 2))

8-Aug-2021	gcdeq 10412	𝐴 is equal to its gcd with 𝐵 if and only if 𝐴 divides 𝐵. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by AV, 8-Aug-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) = 𝐴 ↔ 𝐴 ∥ 𝐵))

8-Aug-2021	dfgcd2 10403	Alternate definition of the gcd operator, see definition in [ApostolNT] p. 15. (Contributed by AV, 8-Aug-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐷 = (𝑀 gcd 𝑁) ↔ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷))))

8-Aug-2021	resqrexlemga 9909	Lemma for resqrex 9912. The sequence formed by squaring each term of 𝐹 converges to 𝐴. (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) & ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)) ⇒ ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐺‘𝑘) < (𝐴 + 𝑒) ∧ 𝐴 < ((𝐺‘𝑘) + 𝑒)))

8-Aug-2021	resqrexlemglsq 9908	Lemma for resqrex 9912. The sequence formed by squaring each term of 𝐹 converges to (𝐿↑2). (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) & ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2)) ⇒ ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐺‘𝑘) < ((𝐿↑2) + 𝑒) ∧ (𝐿↑2) < ((𝐺‘𝑘) + 𝑒)))

8-Aug-2021	recvguniqlem 9880	Lemma for recvguniq 9881. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐴 < ((𝐹‘𝐾) + ((𝐴 − 𝐵) / 2))) & ⊢ (𝜑 → (𝐹‘𝐾) < (𝐵 + ((𝐴 − 𝐵) / 2))) ⇒ ⊢ (𝜑 → ⊥)

8-Aug-2021	ifcldcd 3381	Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → DECID 𝜓) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

8-Aug-2021	ifbothdc 3380	A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃)

7-Aug-2021	divconjdvds 10249	If a nonzero integer 𝑀 divides another integer 𝑁, the other integer 𝑁 divided by the nonzero integer 𝑀 (i.e. the divisor conjugate of 𝑁 to 𝑀) divides the other integer 𝑁. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)
⊢ ((𝑀 ∥ 𝑁 ∧ 𝑀 ≠ 0) → (𝑁 / 𝑀) ∥ 𝑁)

7-Aug-2021	dvdseq 10248	If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.)
⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀)) → 𝑀 = 𝑁)

7-Aug-2021	dvdsabseq 10247	If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)
⊢ ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → (abs‘𝑀) = (abs‘𝑁))

7-Aug-2021	resqrexlemsqa 9910	Lemma for resqrex 9912. The square of a limit is 𝐴. (Contributed by Jim Kingdon, 7-Aug-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) ⇒ ⊢ (𝜑 → (𝐿↑2) = 𝐴)

7-Aug-2021	resqrexlemgt0 9906	Lemma for resqrex 9912. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒))) ⇒ ⊢ (𝜑 → 0 ≤ 𝐿)

7-Aug-2021	recvguniq 9881	Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) < (𝐿 + 𝑥) ∧ 𝐿 < ((𝐹‘𝑘) + 𝑥))) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) < (𝑀 + 𝑥) ∧ 𝑀 < ((𝐹‘𝑘) + 𝑥))) ⇒ ⊢ (𝜑 → 𝐿 = 𝑀)

6-Aug-2021	resqrexlemcvg 9905	Lemma for resqrex 9912. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝑟 + 𝑥) ∧ 𝑟 < ((𝐹‘𝑖) + 𝑥)))

6-Aug-2021	cvg1nlemcxze 9868	Lemma for cvg1n 9872. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
⊢ (𝜑 → 𝐶 ∈ ℝ⁺) & ⊢ (𝜑 → 𝑋 ∈ ℝ⁺) & ⊢ (𝜑 → 𝑍 ∈ ℕ) & ⊢ (𝜑 → 𝐸 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → ((((𝐶 · 2) / 𝑋) / 𝑍) + 𝐴) < 𝐸) ⇒ ⊢ (𝜑 → (𝐶 / (𝐸 · 𝑍)) < (𝑋 / 2))

4-Aug-2021	coprmdvds1 10473	If two positive integers are coprime, i.e. their greatest common divisor is 1, the only positive integer that divides both of them is 1. (Contributed by AV, 4-Aug-2021.)
⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐼 = 1))

1-Aug-2021	3dvds2dec 10265	A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if 𝐴, 𝐵 and 𝐶 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴, 𝐵 and 𝐶. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ & ⊢ 𝐶 ∈ ℕ₀ ⇒ ⊢ (3 ∥ ;;𝐴𝐵𝐶 ↔ 3 ∥ ((𝐴 + 𝐵) + 𝐶))

1-Aug-2021	cvg1n 9872	Convergence of real sequences. This is a version of caucvgre 9867 with a constant multiplier 𝐶 on the rate of convergence. That is, all terms after the nth term must be within 𝐶 / 𝑛 of the nth term. (Contributed by Jim Kingdon, 1-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ⁺) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛)))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹‘𝑖) + 𝑥)))

1-Aug-2021	cvg1nlemres 9871	Lemma for cvg1n 9872. The original sequence 𝐹 has a limit (turns out it is the same as the limit of the modified sequence 𝐺). (Contributed by Jim Kingdon, 1-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ⁺) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛)))) & ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍))) & ⊢ (𝜑 → 𝑍 ∈ ℕ) & ⊢ (𝜑 → 𝐶 < 𝑍) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹‘𝑖) + 𝑥)))

1-Aug-2021	cvg1nlemcau 9870	Lemma for cvg1n 9872. By selecting spaced out terms for the modified sequence 𝐺, the terms are within 1 / 𝑛 (without the constant 𝐶). (Contributed by Jim Kingdon, 1-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ⁺) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛)))) & ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍))) & ⊢ (𝜑 → 𝑍 ∈ ℕ) & ⊢ (𝜑 → 𝐶 < 𝑍) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛))))

1-Aug-2021	cvg1nlemf 9869	Lemma for cvg1n 9872. The modified sequence 𝐺 is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ⁺) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛)))) & ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍))) & ⊢ (𝜑 → 𝑍 ∈ ℕ) & ⊢ (𝜑 → 𝐶 < 𝑍) ⇒ ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)

1-Aug-2021	3dec 9642	A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶)

1-Aug-2021	sq10e99m1 9641	The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
⊢ (;10↑2) = (;99 + 1)

1-Aug-2021	sq10 9640	The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
⊢ (;10↑2) = ;;100

1-Aug-2021	dfdec10 8480	Version of the definition of the "decimal constructor" using ;10 instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)

1-Aug-2021	9p1e10 8479	9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
⊢ (9 + 1) = ;10

1-Aug-2021	df-dec 8478	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 10562. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)

31-Jul-2021	resqrexlemnm 9904	Lemma for resqrex 9912. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ≤ 𝑀) ⇒ ⊢ (𝜑 → ((𝐹‘𝑁) − (𝐹‘𝑀)) < ((((𝐹‘1)↑2) · 2) / (2↑(𝑁 − 1))))

31-Jul-2021	resqrexlemdecn 9898	Lemma for resqrex 9912. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 < 𝑀) ⇒ ⊢ (𝜑 → (𝐹‘𝑀) < (𝐹‘𝑁))

30-Jul-2021	resqrexlemnmsq 9903	Lemma for resqrex 9912. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ≤ 𝑀) ⇒ ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − ((𝐹‘𝑀)↑2)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1))))

30-Jul-2021	decaddm10 8535	The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ (;𝐴0 + ;𝐵0) = ;(𝐴 + 𝐵)0

30-Jul-2021	divmuldivapd 7918	Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐷 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)))

30-Jul-2021	muladd11r 7264	A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 1) · (𝐵 + 1)) = (((𝐴 · 𝐵) + (𝐴 + 𝐵)) + 1))

29-Jul-2021	resqrexlemcalc3 9902	Lemma for resqrex 9912. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘𝑁)↑2) − 𝐴) ≤ (((𝐹‘1)↑2) / (4↑(𝑁 − 1))))

29-Jul-2021	resqrexlemcalc2 9901	Lemma for resqrex 9912. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘(𝑁 + 1))↑2) − 𝐴) ≤ ((((𝐹‘𝑁)↑2) − 𝐴) / 4))

29-Jul-2021	resqrexlemcalc1 9900	Lemma for resqrex 9912. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘(𝑁 + 1))↑2) − 𝐴) = (((((𝐹‘𝑁)↑2) − 𝐴)↑2) / (4 · ((𝐹‘𝑁)↑2))))

29-Jul-2021	resqrexlemlo 9899	Lemma for resqrex 9912. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (1 / (2↑𝑁)) < (𝐹‘𝑁))

29-Jul-2021	resqrexlemdec 9897	Lemma for resqrex 9912. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) < (𝐹‘𝑁))

28-Jul-2021	resqrexlemp1rp 9892	Lemma for resqrex 9912. Applying the recursion rule yields a positive real (expressed in a way that will help apply iseqf 9444 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ (𝐵 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵(𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2))𝐶) ∈ ℝ⁺)

28-Jul-2021	resqrexlem1arp 9891	Lemma for resqrex 9912. 1 + 𝐴 is a positive real (expressed in a way that will help apply iseqf 9444 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ⁺)

28-Jul-2021	rpap0d 8779	A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ⁺) ⇒ ⊢ (𝜑 → 𝐴 # 0)

27-Jul-2021	resqrexlemex 9911	Lemma for resqrex 9912. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))

27-Jul-2021	resqrexlemover 9896	Lemma for resqrex 9912. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 < ((𝐹‘𝑁)↑2))

27-Jul-2021	resqrexlemfp1 9895	Lemma for resqrex 9912. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) = (((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2))

27-Jul-2021	resqrexlemf1 9894	Lemma for resqrex 9912. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘1) = (1 + 𝐴))

27-Jul-2021	resqrexlemf 9893	Lemma for resqrex 9912. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ⁺) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐹:ℕ⟶ℝ⁺)

26-Jul-2021	nndivides 10202	Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (𝑛 · 𝑀) = 𝑁))

25-Jul-2021	summodnegmod 10226	The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 𝐵) mod 𝑁) = 0 ↔ (𝐴 mod 𝑁) = (-𝐵 mod 𝑁)))

24-Jul-2021	dvdsnprmd 10507	If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.)
⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → 𝐴 < 𝑁) & ⊢ (𝜑 → 𝐴 ∥ 𝑁) ⇒ ⊢ (𝜑 → ¬ 𝑁 ∈ ℙ)

23-Jul-2021	iseqf 9444	Range of the recursive sequence builder. (Contributed by Jim Kingdon, 23-Jul-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆):𝑍⟶𝑆)

22-Jul-2021	ialgrlemconst 10425	Lemma for ialgr0 10426. Closure of a constant function, in a form suitable for theorems such as iseq1 9442 or iseqfn 9441. (Contributed by Jim Kingdon, 22-Jul-2021.)
⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)

22-Jul-2021	ialgrlem1st 10424	Lemma for ialgr0 10426. Expressing algrflemg 5871 in a form suitable for theorems such as iseq1 9442 or iseqfn 9441. (Contributed by Jim Kingdon, 22-Jul-2021.)
⊢ (𝜑 → 𝐹:𝑆⟶𝑆) ⇒ ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1^st )𝑦) ∈ 𝑆)

22-Jul-2021	algrflemg 5871	Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1^st )𝐶) = (𝐹‘𝐵))

21-Jul-2021	mod2eq0even 10277	An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example 2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ↔ 2 ∥ 𝑁))

21-Jul-2021	modmulconst 10227	Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) ∧ 𝑀 ∈ ℕ) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) ↔ ((𝐶 · 𝐴) mod (𝐶 · 𝑀)) = ((𝐶 · 𝐵) mod (𝐶 · 𝑀))))

21-Jul-2021	zmod1congr 9343	Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 mod 1) = (𝐵 mod 1))

20-Jul-2021	caucvgrelemcau 9866	Lemma for caucvgre 9867. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛)))) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ ℕ (𝑛 <_ℝ 𝑘 → ((𝐹‘𝑛) <_ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <_ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))

20-Jul-2021	caucvgrelemrec 9865	Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (℩𝑟 ∈ ℝ (𝐴 · 𝑟) = 1) = (1 / 𝐴))

19-Jul-2021	sqoddm1div8z 10286	A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (((𝑁↑2) − 1) / 8) ∈ ℤ)

19-Jul-2021	mulsucdiv2z 10285	An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.)
⊢ (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ)

19-Jul-2021	caucvgre 9867	Convergence of real sequences. A Cauchy sequence (as defined here, which has a rate of convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / 𝑛 of the nth term. (Contributed by Jim Kingdon, 19-Jul-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛)))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹‘𝑖) + 𝑥)))

19-Jul-2021	sqoddm1div8 9625	A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 = ((2 · 𝑁) + 1)) → (((𝑀↑2) − 1) / 8) = ((𝑁 · (𝑁 + 1)) / 2))

19-Jul-2021	mulbinom2 9589	The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐶 · 𝐴) + 𝐵)↑2) = ((((𝐶 · 𝐴)↑2) + ((2 · 𝐶) · (𝐴 · 𝐵))) + (𝐵↑2)))

19-Jul-2021	nzadd 8403	The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.)
⊢ ((𝐴 ∈ (ℝ ∖ ℤ) ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ (ℝ ∖ ℤ))

19-Jul-2021	ax-caucvg 7096	Completeness. Axiom for real and complex numbers, justified by theorem axcaucvg 7066. A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / 𝑛 of the nth term. This axiom should not be used directly; instead use caucvgre 9867 (which is the same, but stated in terms of the ℕ and 1 / 𝑛 notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} & ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <_ℝ 𝑘 → ((𝐹‘𝑛) <_ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <_ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <_ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <_ℝ 𝑘 → ((𝐹‘𝑘) <_ℝ (𝑦 + 𝑥) ∧ 𝑦 <_ℝ ((𝐹‘𝑘) + 𝑥)))))

18-Jul-2021	mulnqpr 6767	Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <_Q (𝐴 ·_Q 𝐵)}, {𝑢 ∣ (𝐴 ·_Q 𝐵) <_Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ·_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩))

18-Jul-2021	mulnqprlemfu 6766	Lemma for mulnqpr 6767. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2^nd ‘⟨{𝑙 ∣ 𝑙 <_Q (𝐴 ·_Q 𝐵)}, {𝑢 ∣ (𝐴 ·_Q 𝐵) <_Q 𝑢}⟩) ⊆ (2^nd ‘(⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ·_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)))

18-Jul-2021	mulnqprlemfl 6765	Lemma for mulnqpr 6767. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1^st ‘⟨{𝑙 ∣ 𝑙 <_Q (𝐴 ·_Q 𝐵)}, {𝑢 ∣ (𝐴 ·_Q 𝐵) <_Q 𝑢}⟩) ⊆ (1^st ‘(⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ·_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)))

18-Jul-2021	mulnqprlemru 6764	Lemma for mulnqpr 6767. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2^nd ‘(⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ·_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)) ⊆ (2^nd ‘⟨{𝑙 ∣ 𝑙 <_Q (𝐴 ·_Q 𝐵)}, {𝑢 ∣ (𝐴 ·_Q 𝐵) <_Q 𝑢}⟩))

18-Jul-2021	mulnqprlemrl 6763	Lemma for mulnqpr 6767. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1^st ‘(⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ·_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)) ⊆ (1^st ‘⟨{𝑙 ∣ 𝑙 <_Q (𝐴 ·_Q 𝐵)}, {𝑢 ∣ (𝐴 ·_Q 𝐵) <_Q 𝑢}⟩))

18-Jul-2021	lt2mulnq 6595	Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <_Q 𝐵 ∧ 𝐶 <_Q 𝐷) → (𝐴 ·_Q 𝐶) <_Q (𝐵 ·_Q 𝐷)))

17-Jul-2021	recidpirqlemcalc 7025	Lemma for recidpirq 7026. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ P) & ⊢ (𝜑 → 𝐵 ∈ P) & ⊢ (𝜑 → (𝐴 ·_P 𝐵) = 1_P) ⇒ ⊢ (𝜑 → ((((𝐴 +_P 1_P) ·_P (𝐵 +_P 1_P)) +_P (1_P ·_P 1_P)) +_P 1_P) = ((((𝐴 +_P 1_P) ·_P 1_P) +_P (1_P ·_P (𝐵 +_P 1_P))) +_P (1_P +_P 1_P)))

17-Jul-2021	recidpipr 7024	Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)
⊢ (𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑁, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑁, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ·_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑁, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑁, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) = 1_P)

16-Jul-2021	2teven 10287	A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = (2 · 𝐴)) → 2 ∥ 𝐵)

16-Jul-2021	2tp1odd 10284	A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = ((2 · 𝐴) + 1)) → ¬ 2 ∥ 𝐵)

15-Jul-2021	mulp1mod1 9367	The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘2)) → (((𝑁 · 𝐴) + 1) mod 𝑁) = 1)

15-Jul-2021	rereceu 7055	The reciprocal from axprecex 7046 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 0 <_ℝ 𝐴) → ∃!𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)

15-Jul-2021	recidpirq 7026	A real number times its reciprocal is one, where reciprocal is expressed with *_Q. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ (𝑁 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑁, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑁, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑁, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑁, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩) = 1)

15-Jul-2021	recnnre 7019	Embedding the reciprocal of a natural number into ℝ. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑁, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑁, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ ∈ ℝ)

15-Jul-2021	pitore 7018	Embedding from N to ℝ. Similar to pitonn 7016 but separate in the sense that we have not proved nnssre 8043 yet. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑁, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑁, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ ∈ ℝ)

15-Jul-2021	pitoregt0 7017	Embedding from N to ℝ yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
⊢ (𝑁 ∈ N → 0 <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑁, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑁, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩)

14-Jul-2021	m1modge3gt1 9373	Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.)
⊢ (𝑀 ∈ (ℤ_≥‘3) → 1 < (-1 mod 𝑀))

14-Jul-2021	m1modnnsub1 9372	Minus one modulo a positive integer is equal to the integer minus one. (Contributed by AV, 14-Jul-2021.)
⊢ (𝑀 ∈ ℕ → (-1 mod 𝑀) = (𝑀 − 1))

14-Jul-2021	adddivflid 9294	The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴))

14-Jul-2021	nnindnn 7059	Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8055 designed for real number axioms which involve natural numbers (notably, axcaucvg 7066). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} & ⊢ (𝑧 = 1 → (𝜑 ↔ 𝜓)) & ⊢ (𝑧 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑧 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑧 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑘 ∈ 𝑁 → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ 𝑁 → 𝜏)

14-Jul-2021	peano5nnnn 7058	Peano's inductive postulate. This is a counterpart to peano5nni 8042 designed for real number axioms which involve natural numbers (notably, axcaucvg 7066). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⇒ ⊢ ((1 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑧 + 1) ∈ 𝐴) → 𝑁 ⊆ 𝐴)

14-Jul-2021	peano2nnnn 7021	A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8051 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7066). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⇒ ⊢ (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ 𝑁)

14-Jul-2021	peano1nnnn 7020	One is an element of ℕ. This is a counterpart to 1nn 8050 designed for real number axioms which involve natural numbers (notably, axcaucvg 7066). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⇒ ⊢ 1 ∈ 𝑁

14-Jul-2021	addvalex 7012	Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 7098. (Contributed by Jim Kingdon, 14-Jul-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V)

13-Jul-2021	cncongrprm 10536	Corollary 2 of Cancellability of Congruences: Two products with a common factor are congruent modulo a prime number not dividing the common factor iff the other factors are congruent modulo the prime number. (Contributed by AV, 13-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶)) → (((𝐴 · 𝐶) mod 𝑃) = ((𝐵 · 𝐶) mod 𝑃) ↔ (𝐴 mod 𝑃) = (𝐵 mod 𝑃)))

13-Jul-2021	prmndvdsfaclt 10535	A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑁 < 𝑃 → ¬ 𝑃 ∥ (!‘𝑁)))

13-Jul-2021	cncongrcoprm 10488	Corollary 1 of Cancellability of Congruences: Two products with a common factor are congruent modulo an integer being coprime to the common factor iff the other factors are congruent modulo the integer. (Contributed by AV, 13-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ (𝐶 gcd 𝑁) = 1)) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) ↔ (𝐴 mod 𝑁) = (𝐵 mod 𝑁)))

13-Jul-2021	cncongr 10487	Cancellability of Congruences (see ProofWiki "Cancellability of Congruences, https://proofwiki.org/wiki/Cancellability_of_Congruences, 10-Jul-2021): Two products with a common factor are congruent modulo a positive integer iff the other factors are congruent modulo the integer divided by the greates common divisor of the integer and the common factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) ↔ (𝐴 mod 𝑀) = (𝐵 mod 𝑀)))

13-Jul-2021	cncongr1 10485	One direction of the bicondition in cncongr 10487. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) → (𝐴 mod 𝑀) = (𝐵 mod 𝑀)))

13-Jul-2021	nntopi 7060	Mapping from ℕ to N. (Contributed by Jim Kingdon, 13-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⇒ ⊢ (𝐴 ∈ 𝑁 → ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑧, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑧, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ = 𝐴)

13-Jul-2021	ltrennb 7022	Ordering of natural numbers with <_N or <_ℝ. (Contributed by Jim Kingdon, 13-Jul-2021.)
⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩))

12-Jul-2021	divgcdcoprmex 10484	Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1))

12-Jul-2021	divgcdcoprm0 10483	Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)

12-Jul-2021	ltrenn 7023	Ordering of natural numbers with <_N or <_ℝ. (Contributed by Jim Kingdon, 12-Jul-2021.)
⊢ (𝐽 <_N 𝐾 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩)

11-Jul-2021	cncongr2 10486	The other direction of the bicondition in cncongr 10487. (Contributed by AV, 11-Jul-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) → ((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁)))

11-Jul-2021	congr 10482	Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer 𝐴 is congruent to an integer 𝐵 modulo 𝑀 if their difference is a multiple of 𝑀. See also the definition in [ApostolNT] p. 104: "... 𝑎 is congruent to 𝑏 modulo 𝑚, and we write 𝑎≡𝑏 (mod 𝑚) if 𝑚 divides the difference 𝑎 − 𝑏", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℕ) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = (𝐴 − 𝐵)))

11-Jul-2021	divgcdz 10363	An integer divided by the gcd of it and a nonzero integer is an integer. (Contributed by AV, 11-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ)

11-Jul-2021	zeqzmulgcd 10362	An integer is the product of an integer and the gcd of it and another integer. (Contributed by AV, 11-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∃𝑛 ∈ ℤ 𝐴 = (𝑛 · (𝐴 gcd 𝐵)))

11-Jul-2021	recriota 7056	Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
⊢ (𝑁 ∈ N → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑁, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑁, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑁, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑁, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩)

11-Jul-2021	elrealeu 6998	The real number mapping in elreal 6997 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
⊢ (𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R ⟨𝑥, 0_R⟩ = 𝐴)

10-Jul-2021	coprmdvds 10474	Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 · 𝑁) ∧ (𝐾 gcd 𝑀) = 1) → 𝐾 ∥ 𝑁))

10-Jul-2021	divgcdnnr 10367	A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ) → (𝐴 / (𝐵 gcd 𝐴)) ∈ ℕ)

10-Jul-2021	divgcdnn 10366	A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ)

10-Jul-2021	gcd2n0cl 10361	Closure of the gcd operator if the second operand is not 0. (Contributed by AV, 10-Jul-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀 gcd 𝑁) ∈ ℕ)

10-Jul-2021	axcaucvglemres 7065	Lemma for axcaucvg 7066. Mapping the limit from N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} & ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <_ℝ 𝑘 → ((𝐹‘𝑛) <_ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <_ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))) & ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑗, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑗, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩) = ⟨𝑧, 0_R⟩)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <_ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <_ℝ 𝑘 → ((𝐹‘𝑘) <_ℝ (𝑦 + 𝑥) ∧ 𝑦 <_ℝ ((𝐹‘𝑘) + 𝑥)))))

10-Jul-2021	axcaucvglemval 7063	Lemma for axcaucvg 7066. Value of sequence when mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} & ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <_ℝ 𝑘 → ((𝐹‘𝑛) <_ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <_ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))) & ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑗, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑗, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩) = ⟨𝑧, 0_R⟩)) ⇒ ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩) = ⟨(𝐺‘𝐽), 0_R⟩)

10-Jul-2021	axcaucvglemcl 7061	Lemma for axcaucvg 7066. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} & ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) ⇒ ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩) = ⟨𝑧, 0_R⟩) ∈ R)

9-Jul-2021	axcaucvglemcau 7064	Lemma for axcaucvg 7066. The result of mapping to N and R satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} & ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <_ℝ 𝑘 → ((𝐹‘𝑛) <_ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <_ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))) & ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑗, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑗, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩) = ⟨𝑧, 0_R⟩)) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐺‘𝑛) <_R ((𝐺‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐺‘𝑘) <_R ((𝐺‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ))))

9-Jul-2021	axcaucvglemf 7062	Lemma for axcaucvg 7066. Mapping to N and R yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} & ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <_ℝ 𝑘 → ((𝐹‘𝑛) <_ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <_ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))) & ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑗, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑗, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩) = ⟨𝑧, 0_R⟩)) ⇒ ⊢ (𝜑 → 𝐺:N⟶R)

8-Jul-2021	fldiv4p1lem1div2 9307	The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ_≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))

8-Jul-2021	divfl0 9298	The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0))

8-Jul-2021	div4p1lem1div2 8284	An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))

8-Jul-2021	axcaucvg 7066	Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). Because we are stating this axiom before we have introduced notations for ℕ or division, we use 𝑁 for the natural numbers and express a reciprocal in terms of ℩. This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7096. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)
⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} & ⊢ (𝜑 → 𝐹:𝑁⟶ℝ) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <_ℝ 𝑘 → ((𝐹‘𝑛) <_ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <_ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <_ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <_ℝ 𝑘 → ((𝐹‘𝑘) <_ℝ (𝑦 + 𝑥) ∧ 𝑦 <_ℝ ((𝐹‘𝑘) + 𝑥)))))

7-Jul-2021	ltadd1sr 6953	Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
⊢ (𝐴 ∈ R → 𝐴 <_R (𝐴 +_R 1_R))

7-Jul-2021	lteupri 6807	The difference from ltexpri 6803 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
⊢ (𝐴<_P 𝐵 → ∃!𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

7-Jul-2021	sbcimdv 2879	Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1386). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))

7-Jul-2021	eqsbc3r 2874	eqsbc3 2853 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴))

4-Jul-2021	flodddiv4t2lthalf 10337	The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2))

4-Jul-2021	flodddiv4lt 10336	The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4))

4-Jul-2021	fldivndvdslt 10335	The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.)
⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℤ ∧ 𝐿 ≠ 0) ∧ ¬ 𝐿 ∥ 𝐾) → (⌊‘(𝐾 / 𝐿)) < (𝐾 / 𝐿))

4-Jul-2021	4dvdseven 10317	An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.)
⊢ (4 ∥ 𝑁 → 2 ∥ 𝑁)

4-Jul-2021	z4even 10316	4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
⊢ 2 ∥ 4

4-Jul-2021	caucvgsrlemasr 6966	Lemma for caucvgsr 6978. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <_R (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → 𝐴 ∈ R)

3-Jul-2021	caucvgsrlemoffres 6976	Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +_R 1_R) +_R (𝐴 ·_R -1_R))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0_R <_R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <_N 𝑘 → ((𝐹‘𝑘) <_R (𝑦 +_R 𝑥) ∧ 𝑦 <_R ((𝐹‘𝑘) +_R 𝑥)))))

3-Jul-2021	caucvgsrlemoffgt1 6975	Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +_R 1_R) +_R (𝐴 ·_R -1_R))) ⇒ ⊢ (𝜑 → ∀𝑚 ∈ N 1_R <_R (𝐺‘𝑚))

3-Jul-2021	caucvgsrlemoffcau 6974	Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +_R 1_R) +_R (𝐴 ·_R -1_R))) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐺‘𝑛) <_R ((𝐺‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐺‘𝑘) <_R ((𝐺‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ))))

3-Jul-2021	caucvgsrlemofff 6973	Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +_R 1_R) +_R (𝐴 ·_R -1_R))) ⇒ ⊢ (𝜑 → 𝐺:N⟶R)

3-Jul-2021	caucvgsrlemoffval 6972	Lemma for caucvgsr 6978. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +_R 1_R) +_R (𝐴 ·_R -1_R))) ⇒ ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐺‘𝐽) +_R 𝐴) = ((𝐹‘𝐽) +_R 1_R))

2-Jul-2021	caucvgsrlemcl 6965	Lemma for caucvgsr 6978. Terms of the sequence from caucvgsrlemgt1 6971 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑚 ∈ N 1_R <_R (𝐹‘𝑚)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +_P 1_P), 1_P⟩] ~_R ) ∈ P)

1-Jul-2021	halfleoddlt 10294	An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑀 ∈ ℤ) → ((𝑁 / 2) ≤ 𝑀 ↔ (𝑁 / 2) < 𝑀))

1-Jul-2021	zltaddlt1le 9028	The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (0(,)1)) → ((𝑀 + 𝐴) < 𝑁 ↔ (𝑀 + 𝐴) ≤ 𝑁))

1-Jul-2021	addlelt 8839	If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.)
⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((𝑀 + 𝐴) ≤ 𝑁 → 𝑀 < 𝑁))

29-Jun-2021	ltoddhalfle 10293	An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑀 ∈ ℤ) → (𝑀 < (𝑁 / 2) ↔ 𝑀 ≤ ((𝑁 − 1) / 2)))

29-Jun-2021	ledivge1le 8803	If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺ ∧ (𝐶 ∈ ℝ⁺ ∧ 1 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐴 / 𝐶) ≤ 𝐵))

29-Jun-2021	divle1le 8802	A real number divided by a positive real number is less than or equal to 1 iff the real number is less than or equal to the positive real number. (Contributed by AV, 29-Jun-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ 𝐵))

29-Jun-2021	caucvgsrlemfv 6967	Lemma for caucvgsr 6978. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1_R <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +_P 1_P), 1_P⟩] ~_R )) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +_P 1_P), 1_P⟩] ~_R = (𝐹‘𝐴))

29-Jun-2021	prsrriota 6964	Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
⊢ ((𝐴 ∈ R ∧ 0_R <_R 𝐴) → [⟨((℩𝑥 ∈ P [⟨(𝑥 +_P 1_P), 1_P⟩] ~_R = 𝐴) +_P 1_P), 1_P⟩] ~_R = 𝐴)

29-Jun-2021	prsrlt 6963	Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<_P 𝐵 ↔ [⟨(𝐴 +_P 1_P), 1_P⟩] ~_R <_R [⟨(𝐵 +_P 1_P), 1_P⟩] ~_R ))

29-Jun-2021	prsradd 6962	Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → [⟨((𝐴 +_P 𝐵) +_P 1_P), 1_P⟩] ~_R = ([⟨(𝐴 +_P 1_P), 1_P⟩] ~_R +_R [⟨(𝐵 +_P 1_P), 1_P⟩] ~_R ))

28-Jun-2021	nnoddm1d2 10310	A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ))

28-Jun-2021	nn0oddm1d2 10309	A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
⊢ (𝑁 ∈ ℕ₀ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℕ₀))

28-Jun-2021	nnehalf 10304	The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ)

28-Jun-2021	nn0ehalf 10303	The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.)
⊢ ((𝑁 ∈ ℕ₀ ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ₀)

26-Jun-2021	m1expo 10300	Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1)

25-Jun-2021	m1expe 10299	Exponentiation of -1 by an even power. Variant of m1expeven 9523. (Contributed by AV, 25-Jun-2021.)
⊢ (2 ∥ 𝑁 → (-1↑𝑁) = 1)

25-Jun-2021	caucvgsrlembound 6970	Lemma for caucvgsr 6978. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1_R <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +_P 1_P), 1_P⟩] ~_R )) ⇒ ⊢ (𝜑 → ∀𝑚 ∈ N 1_P<_P (𝐺‘𝑚))

25-Jun-2021	prsrpos 6961	Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)
⊢ (𝐴 ∈ P → 0_R <_R [⟨(𝐴 +_P 1_P), 1_P⟩] ~_R )

25-Jun-2021	prsrcl 6960	Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
⊢ (𝐴 ∈ P → [⟨(𝐴 +_P 1_P), 1_P⟩] ~_R ∈ R)

25-Jun-2021	srpospr 6959	Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
⊢ ((𝐴 ∈ R ∧ 0_R <_R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +_P 1_P), 1_P⟩] ~_R = 𝐴)

23-Jun-2021	z2even 10314	2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
⊢ 2 ∥ 2

23-Jun-2021	z0even 10311	0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
⊢ 2 ∥ 0

23-Jun-2021	caucvgsrlemcau 6969	Lemma for caucvgsr 6978. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1_R <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +_P 1_P), 1_P⟩] ~_R )) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐺‘𝑛)<_P ((𝐺‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐺‘𝑘)<_P ((𝐺‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩))))

23-Jun-2021	caucvgsrlemf 6968	Lemma for caucvgsr 6978. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1_R <_R (𝐹‘𝑚)) & ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +_P 1_P), 1_P⟩] ~_R )) ⇒ ⊢ (𝜑 → 𝐺:N⟶P)

22-Jun-2021	oddm1d2 10292	An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021.) (Proof shortened by AV, 22-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℤ))

22-Jun-2021	oddp1d2 10290	An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo 8452 and zeo2 8453. (Contributed by AV, 22-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ))

22-Jun-2021	evend2 10289	An integer is even iff its quotient with 2 is an integer. This is a representation of even numbers without using the divides relation, see zeo 8452 and zeo2 8453. (Contributed by AV, 22-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ (𝑁 / 2) ∈ ℤ))

22-Jun-2021	zeneo 10270	No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 8448 follows immediately from the fact that a contradiction implies anything, see pm2.21i 607. (Contributed by AV, 22-Jun-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵))

22-Jun-2021	evenelz 10266	An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 10200. (Contributed by AV, 22-Jun-2021.)
⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ)

22-Jun-2021	caucvgsrlemgt1 6971	Lemma for caucvgsr 6978. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 1_R <_R (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0_R <_R 𝑥 → ∃𝑗 ∈ N ∀𝑖 ∈ N (𝑗 <_N 𝑖 → ((𝐹‘𝑖) <_R (𝑦 +_R 𝑥) ∧ 𝑦 <_R ((𝐹‘𝑖) +_R 𝑥)))))

20-Jun-2021	m1exp1 10301	Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)
⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) = 1 ↔ 2 ∥ 𝑁))

20-Jun-2021	caucvgsr 6978	A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). This is similar to caucvgprpr 6902 but is for signed reals rather than positive reals. Here is an outline of how we prove it: 1. Choose a lower bound for the sequence (see caucvgsrlembnd 6977). 2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 6973). 3. Since a signed real (element of R) which is greater than zero can be mapped to a positive real (element of P), perform that mapping on each element of the sequence and invoke caucvgprpr 6902 to get a limit (see caucvgsrlemgt1 6971). 4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6971). 5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6976). (Contributed by Jim Kingdon, 20-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0_R <_R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <_N 𝑘 → ((𝐹‘𝑘) <_R (𝑦 +_R 𝑥) ∧ 𝑦 <_R ((𝐹‘𝑘) +_R 𝑥)))))

19-Jun-2021	prm2orodd 10508	A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.)
⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))

19-Jun-2021	oddnn02np1 10280	A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.)
⊢ (𝑁 ∈ ℕ₀ → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ₀ ((2 · 𝑛) + 1) = 𝑁))

19-Jun-2021	2tnp1ge0ge0 9303	Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁))

19-Jun-2021	nn0ledivnn 8838	Division of a nonnegative integer by a positive integer is less than or equal to the integer. (Contributed by AV, 19-Jun-2021.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≤ 𝐴)

19-Jun-2021	nnledivrp 8837	Division of a positive integer by a positive number is less than or equal to the integer iff the number is greater than or equal to 1. (Contributed by AV, 19-Jun-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ⁺) → (1 ≤ 𝐵 ↔ (𝐴 / 𝐵) ≤ 𝐴))

19-Jun-2021	caucvgsrlembnd 6977	Lemma for caucvgsr 6978. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶R) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_R ((𝐹‘𝑘) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ) ∧ (𝐹‘𝑘) <_R ((𝐹‘𝑛) +_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <_R (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0_R <_R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <_N 𝑘 → ((𝐹‘𝑘) <_R (𝑦 +_R 𝑥) ∧ 𝑦 <_R ((𝐹‘𝑘) +_R 𝑥)))))

19-Jun-2021	caucvgprprlemclphr 6895	Lemma for caucvgprpr 6902. The putative limit is a positive real. Like caucvgprprlemcl 6894 but without a distinct variable constraint between 𝜑 and 𝑟. (Contributed by Jim Kingdon, 19-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → 𝐿 ∈ P)

19-Jun-2021	ltnqpr 6783	We can order fractions via <_Q or <_P. (Contributed by Jim Kingdon, 19-Jun-2021.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <_Q 𝐵 ↔ ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩))

17-Jun-2021	flodddiv4 10334	The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = ((2 · 𝑀) + 1)) → (⌊‘(𝑁 / 4)) = if(2 ∥ 𝑀, (𝑀 / 2), ((𝑀 − 1) / 2)))

17-Jun-2021	zeo4 10269	An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

17-Jun-2021	zeo3 10267	An integer is even or odd. (Contributed by AV, 17-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ ¬ 2 ∥ 𝑁))

17-Jun-2021	caucvgprprlemnbj 6883	Lemma for caucvgprpr 6902. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → 𝐵 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) ⇒ ⊢ (𝜑 → ¬ (((𝐹‘𝐵) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝐵, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝐵, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)<_P (𝐹‘𝐽))

16-Jun-2021	caucvgprprlemexbt 6896	Lemma for caucvgprpr 6902. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝑇 ∈ P) & ⊢ (𝜑 → (𝐿 +_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑄}, {𝑞 ∣ 𝑄 <_Q 𝑞}⟩)<_P 𝑇) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ N (((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩) +_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑄}, {𝑞 ∣ 𝑄 <_Q 𝑞}⟩)<_P 𝑇)

16-Jun-2021	prplnqu 6810	Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
⊢ (𝜑 → 𝑋 ∈ P) & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝐴 ∈ (2^nd ‘(𝑋 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑄}, {𝑢 ∣ 𝑄 <_Q 𝑢}⟩))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ (2^nd ‘𝑋)(𝑦 +_Q 𝑄) = 𝐴)

15-Jun-2021	11multnc 8544	The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
⊢ 𝑁 ∈ ℕ₀ ⇒ ⊢ (𝑁 · ;11) = ;𝑁𝑁

15-Jun-2021	decmulnc 8543	The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
⊢ 𝑁 ∈ ℕ₀ & ⊢ 𝐴 ∈ ℕ₀ & ⊢ 𝐵 ∈ ℕ₀ ⇒ ⊢ (𝑁 · ;𝐴𝐵) = ;(𝑁 · 𝐴)(𝑁 · 𝐵)

15-Jun-2021	caucvgprprlemexb 6897	Lemma for caucvgprpr 6902. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ & ⊢ (𝜑 → 𝑄 ∈ P) & ⊢ (𝜑 → 𝑅 ∈ N) ⇒ ⊢ (𝜑 → (((𝐿 +_P 𝑄) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑅, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑅, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑅) +_P 𝑄) → ∃𝑏 ∈ N (((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩) +_P (𝑄 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑅, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑅, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩))<_P ((𝐹‘𝑅) +_P 𝑄)))

5-Jun-2021	caucvgprprlemaddq 6898	Lemma for caucvgprpr 6902. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ & ⊢ (𝜑 → 𝑋 ∈ P) & ⊢ (𝜑 → 𝑄 ∈ P) & ⊢ (𝜑 → ∃𝑟 ∈ N (𝑋 +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ((𝐹‘𝑟) +_P 𝑄)) ⇒ ⊢ (𝜑 → 𝑋<_P (𝐿 +_P 𝑄))

5-Apr-2021	simpl2im 378	Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)
⊢ (𝜑 → (𝜓 ∧ 𝜒)) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜑 → 𝜃)

27-Mar-2021	notnotd 592	Deduction associated with notnot 591 and notnoti 606. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ¬ ¬ 𝜓)

20-Mar-2021	addmodlteq 9400	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. (Contributed by AV, 20-Mar-2021.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))

20-Mar-2021	subfzo0 9251	The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → (-𝑁 < (𝐼 − 𝐽) ∧ (𝐼 − 𝐽) < 𝑁))

19-Mar-2021	modsumfzodifsn 9398	The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ (1..^𝑁)) → ((𝐾 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽}))

19-Mar-2021	modfzo0difsn 9397	For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑖 ∈ (1..^𝑁)𝐾 = ((𝑖 + 𝐽) mod 𝑁))

19-Mar-2021	addmodidr 9375	The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝐴 + 𝑀) mod 𝑀) = 𝐴)

19-Mar-2021	ltaddnegr 7529	Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐴 + 𝐵) < 𝐵))

14-Mar-2021	addmodlteqALT 10259	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 9400 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))

14-Mar-2021	modlteq 9399	Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐼 mod 𝑁) = (𝐽 mod 𝑁) ↔ 𝐼 = 𝐽))

3-Mar-2021	caucvgprprlemval 6878	Lemma for caucvgprpr 6902. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) ⇒ ⊢ ((𝜑 ∧ 𝐴 <_N 𝐵) → ((𝐹‘𝐴)<_P ((𝐹‘𝐵) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝐴, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝐴, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩) ∧ (𝐹‘𝐵)<_P ((𝐹‘𝐴) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝐴, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝐴, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)))

28-Feb-2021	n2dvds3 10315	2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
⊢ ¬ 2 ∥ 3

27-Feb-2021	recnnpr 6738	The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
⊢ (𝐴 ∈ N → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝐴, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝐴, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩ ∈ P)

12-Feb-2021	caucvgprprlemnjltk 6881	Lemma for caucvgprpr 6902. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ ((𝜑 ∧ 𝐽 <_N 𝐾) → ¬ (⟨{𝑝 ∣ 𝑝 <_Q (𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑆}, {𝑞 ∣ 𝑆 <_Q 𝑞}⟩))

12-Feb-2021	caucvgprprlemnkeqj 6880	Lemma for caucvgprpr 6902. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <_Q (𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑆}, {𝑞 ∣ 𝑆 <_Q 𝑞}⟩))

12-Feb-2021	caucvgprprlemnkltj 6879	Lemma for caucvgprpr 6902. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ ((𝜑 ∧ 𝐾 <_N 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <_Q (𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑆}, {𝑞 ∣ 𝑆 <_Q 𝑞}⟩))

12-Feb-2021	caucvgprprlemcbv 6877	Lemma for caucvgprpr 6902. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) ⇒ ⊢ (𝜑 → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <_N 𝑏 → ((𝐹‘𝑎)<_P ((𝐹‘𝑏) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑏)<_P ((𝐹‘𝑎) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑎, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩))))

9-Feb-2021	min2inf 10114	The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ≤ 𝐵)

8-Feb-2021	min1inf 10113	The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) ≤ 𝐴)

8-Feb-2021	minmax 10112	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = -sup({-𝐴, -𝐵}, ℝ, < ))

8-Feb-2021	mincom 10111	The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < )

8-Feb-2021	caucvgprprlemloccalc 6874	Lemma for caucvgprpr 6902. Rearranging some expressions for caucvgprprlemloc 6893. (Contributed by Jim Kingdon, 8-Feb-2021.)
⊢ (𝜑 → 𝑆 <_Q 𝑇) & ⊢ (𝜑 → 𝑌 ∈ Q) & ⊢ (𝜑 → (𝑆 +_Q 𝑌) = 𝑇) & ⊢ (𝜑 → 𝑋 ∈ Q) & ⊢ (𝜑 → (𝑋 +_Q 𝑋) <_Q 𝑌) & ⊢ (𝜑 → 𝑀 ∈ N) & ⊢ (𝜑 → (_Q‘[⟨𝑀, 1_𝑜⟩] ~_Q ) <_Q 𝑋) ⇒ ⊢ (𝜑 → (⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q (_Q‘[⟨𝑀, 1_𝑜⟩] ~_Q ))}, {𝑢 ∣ (𝑆 +_Q (_Q‘[⟨𝑀, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑀, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (*_Q‘[⟨𝑀, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)<_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑇}, {𝑢 ∣ 𝑇 <_Q 𝑢}⟩)

5-Feb-2021	fvinim0ffz 9250	The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))

28-Jan-2021	caucvgprprlemelu 6876	Lemma for caucvgprpr 6902. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝑋 ∈ (2^nd ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑋}, {𝑞 ∣ 𝑋 <_Q 𝑞}⟩))

21-Jan-2021	caucvgprprlemell 6875	Lemma for caucvgprpr 6902. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝑋 ∈ (1^st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑋 +_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑋 +_Q (_Q‘[⟨𝑏, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑏)))

20-Jan-2021	caucvgprprlemnkj 6882	Lemma for caucvgprpr 6902. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → ¬ (⟨{𝑝 ∣ 𝑝 <_Q (𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑆}, {𝑞 ∣ 𝑆 <_Q 𝑞}⟩))

17-Jan-2021	addn0nid 7478	Adding a nonzero number to a complex number does not yield the complex number. (Contributed by AV, 17-Jan-2021.)
⊢ ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ 𝑌 ≠ 0) → (𝑋 + 𝑌) ≠ 𝑋)

17-Jan-2021	addid0 7477	If adding a number to a another number yields the other number, the added number must be 0. This shows that 0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)
⊢ ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑋 + 𝑌) = 𝑋 ↔ 𝑌 = 0))

8-Jan-2021	ltnqpri 6784	We can order fractions via <_Q or <_P. (Contributed by Jim Kingdon, 8-Jan-2021.)
⊢ (𝐴 <_Q 𝐵 → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)

29-Dec-2020	caucvgprprlemmu 6885	Lemma for caucvgprpr 6902. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → ∃𝑡 ∈ Q 𝑡 ∈ (2^nd ‘𝐿))

29-Dec-2020	caucvgprprlemml 6884	Lemma for caucvgprpr 6902. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (1^st ‘𝐿))

21-Dec-2020	caucvgprprlemloc 6893	Lemma for caucvgprpr 6902. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑡 ∈ Q (𝑠 <_Q 𝑡 → (𝑠 ∈ (1^st ‘𝐿) ∨ 𝑡 ∈ (2^nd ‘𝐿))))

21-Dec-2020	caucvgprprlemdisj 6892	Lemma for caucvgprpr 6902. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1^st ‘𝐿) ∧ 𝑠 ∈ (2^nd ‘𝐿)))

21-Dec-2020	caucvgprprlemrnd 6891	Lemma for caucvgprpr 6902. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1^st ‘𝐿) ↔ ∃𝑡 ∈ Q (𝑠 <_Q 𝑡 ∧ 𝑡 ∈ (1^st ‘𝐿))) ∧ ∀𝑡 ∈ Q (𝑡 ∈ (2^nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <_Q 𝑡 ∧ 𝑠 ∈ (2^nd ‘𝐿)))))

21-Dec-2020	caucvgprprlemupu 6890	Lemma for caucvgprpr 6902. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 <_Q 𝑡 ∧ 𝑠 ∈ (2^nd ‘𝐿)) → 𝑡 ∈ (2^nd ‘𝐿))

21-Dec-2020	caucvgprprlemopu 6889	Lemma for caucvgprpr 6902. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑡 ∈ (2^nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <_Q 𝑡 ∧ 𝑠 ∈ (2^nd ‘𝐿)))

21-Dec-2020	caucvgprprlemlol 6888	Lemma for caucvgprpr 6902. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 <_Q 𝑡 ∧ 𝑡 ∈ (1^st ‘𝐿)) → 𝑠 ∈ (1^st ‘𝐿))

21-Dec-2020	caucvgprprlemopl 6887	Lemma for caucvgprpr 6902. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) → ∃𝑡 ∈ Q (𝑠 <_Q 𝑡 ∧ 𝑡 ∈ (1^st ‘𝐿)))

21-Dec-2020	caucvgprprlemm 6886	Lemma for caucvgprpr 6902. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1^st ‘𝐿) ∧ ∃𝑡 ∈ Q 𝑡 ∈ (2^nd ‘𝐿)))

29-Nov-2020	nqpru 6742	Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by <_P. (Contributed by Jim Kingdon, 29-Nov-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2^nd ‘𝐵) ↔ 𝐵<_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩))

28-Nov-2020	caucvgprprlemk 6873	Lemma for caucvgprpr 6902. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)
⊢ (𝜑 → 𝐽 <_N 𝐾) & ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝑄) ⇒ ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝑄)

25-Nov-2020	caucvgprprlem2 6900	Lemma for caucvgprpr 6902. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ & ⊢ (𝜑 → 𝑄 ∈ P) & ⊢ (𝜑 → 𝐽 <_N 𝐾) & ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝑄) ⇒ ⊢ (𝜑 → 𝐿<_P ((𝐹‘𝐾) +_P 𝑄))

25-Nov-2020	caucvgprprlem1 6899	Lemma for caucvgprpr 6902. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ & ⊢ (𝜑 → 𝑄 ∈ P) & ⊢ (𝜑 → 𝐽 <_N 𝐾) & ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝑄) ⇒ ⊢ (𝜑 → (𝐹‘𝐾)<_P (𝐿 +_P 𝑄))

25-Nov-2020	archrecpr 6854	Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)
⊢ (𝐴 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩<_P 𝐴)

21-Nov-2020	caucvgprprlemlim 6901	Lemma for caucvgprpr 6902. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → ∀𝑥 ∈ P ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <_N 𝑘 → ((𝐹‘𝑘)<_P (𝐿 +_P 𝑥) ∧ 𝐿<_P ((𝐹‘𝑘) +_P 𝑥))))

21-Nov-2020	caucvgprprlemcl 6894	Lemma for caucvgprpr 6902. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <_Q (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ))}, {𝑞 ∣ (𝑙 +_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )) <_Q 𝑞}⟩<_P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +_P ⟨{𝑝 ∣ 𝑝 <_Q (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q )}, {𝑞 ∣ (_Q‘[⟨𝑟, 1_𝑜⟩] ~_Q ) <_Q 𝑞}⟩)<_P ⟨{𝑝 ∣ 𝑝 <_Q 𝑢}, {𝑞 ∣ 𝑢 <_Q 𝑞}⟩}⟩ ⇒ ⊢ (𝜑 → 𝐿 ∈ P)

17-Nov-2020	pm3.2 137	Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) (Proof shortened by Jia Ming, 17-Nov-2020.)
⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))

14-Nov-2020	caucvgprpr 6902	A Cauchy sequence of positive reals with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a given value 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). This is similar to caucvgpr 6872 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 6852) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.)
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛)<_P ((𝐹‘𝑘) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩) ∧ (𝐹‘𝑘)<_P ((𝐹‘𝑛) +_P ⟨{𝑙 ∣ 𝑙 <_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )}, {𝑢 ∣ (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ) <_Q 𝑢}⟩)))) & ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<_P (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ P ∀𝑥 ∈ P ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <_N 𝑘 → ((𝐹‘𝑘)<_P (𝑦 +_P 𝑥) ∧ 𝑦<_P ((𝐹‘𝑘) +_P 𝑥))))

27-Oct-2020	bj-omssonALT 10758	Alternate proof of bj-omsson 10757. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ⊆ On

27-Oct-2020	bj-omsson 10757	Constructive proof of omsson 4353. See also bj-omssonALT 10758. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
⊢ ω ⊆ On

27-Oct-2020	bj-nnelon 10754	A natural number is an ordinal. Constructive proof of nnon 4350. Can also be proved from bj-omssonALT 10758. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)

27-Oct-2020	bj-nnord 10753	A natural number is an ordinal. Constructive proof of nnord 4352. Can also be proved from bj-nnelon 10754 if the latter is proved from bj-omssonALT 10758. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Ord 𝐴)

27-Oct-2020	bj-axempty2 10685	Axiom of the empty set from bounded separation, alternate version to bj-axempty 10684. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

25-Oct-2020	bj-indind 10727	If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴 ∩ 𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.)
⊢ ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → Ind (𝐴 ∩ 𝐵))

25-Oct-2020	bj-axempty 10684	Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a non-empty universe. See axnul 3903. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥

25-Oct-2020	bj-axemptylem 10683	Lemma for bj-axempty 10684 and bj-axempty2 10685. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)

23-Oct-2020	caucvgprlemnkj 6856	Lemma for caucvgpr 6872. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → ¬ ((𝑆 +_Q (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )) <_Q 𝑆))

20-Oct-2020	caucvgprlemupu 6862	Lemma for caucvgpr 6872. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 <_Q 𝑟 ∧ 𝑠 ∈ (2^nd ‘𝐿)) → 𝑟 ∈ (2^nd ‘𝐿))

20-Oct-2020	caucvgprlemopu 6861	Lemma for caucvgpr 6872. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑟 ∈ (2^nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑠 ∈ (2^nd ‘𝐿)))

20-Oct-2020	caucvgprlemlol 6860	Lemma for caucvgpr 6872. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿)) → 𝑠 ∈ (1^st ‘𝐿))

20-Oct-2020	caucvgprlemopl 6859	Lemma for caucvgpr 6872. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿)))

18-Oct-2020	caucvgprlemnbj 6857	Lemma for caucvgpr 6872. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → 𝐵 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) ⇒ ⊢ (𝜑 → ¬ (((𝐹‘𝐵) +_Q (_Q‘[⟨𝐵, 1_𝑜⟩] ~_Q )) +_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝐽))

11-Oct-2020	elexd 2612	If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ∈ V)

9-Oct-2020	caucvgprlemladdfu 6867	Lemma for caucvgpr 6872. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)) ⊆ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) +_Q 𝑆) <_Q 𝑢})

9-Oct-2020	caucvgprlemk 6855	Lemma for caucvgpr 6872. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)
⊢ (𝜑 → 𝐽 <_N 𝐾) & ⊢ (𝜑 → (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑄) ⇒ ⊢ (𝜑 → (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ) <_Q 𝑄)

8-Oct-2020	caucvgprlemladdrl 6868	Lemma for caucvgpr 6872. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q ((𝐹‘𝑗) +_Q 𝑆)} ⊆ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)))

3-Oct-2020	caucvgprlem2 6870	Lemma for caucvgpr 6872. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝐽 <_N 𝐾) & ⊢ (𝜑 → (*_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑄) ⇒ ⊢ (𝜑 → 𝐿<_P ⟨{𝑙 ∣ 𝑙 <_Q ((𝐹‘𝐾) +_Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +_Q 𝑄) <_Q 𝑢}⟩)

3-Oct-2020	caucvgprlem1 6869	Lemma for caucvgpr 6872. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝐽 <_N 𝐾) & ⊢ (𝜑 → (*_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑄) ⇒ ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <_Q (𝐹‘𝐾)}, {𝑢 ∣ (𝐹‘𝐾) <_Q 𝑢}⟩<_P (𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑄}, {𝑢 ∣ 𝑄 <_Q 𝑢}⟩))

3-Oct-2020	ltnnnq 6613	Ordering of positive integers via <_N or <_Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_N 𝐵 ↔ [⟨𝐴, 1_𝑜⟩] ~_Q <_Q [⟨𝐵, 1_𝑜⟩] ~_Q ))

1-Oct-2020	caucvgprlemlim 6871	Lemma for caucvgpr 6872. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <_N 𝑘 → (⟨{𝑙 ∣ 𝑙 <_Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <_Q 𝑢}⟩<_P (𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑥}, {𝑢 ∣ 𝑥 <_Q 𝑢}⟩) ∧ 𝐿<_P ⟨{𝑙 ∣ 𝑙 <_Q ((𝐹‘𝑘) +_Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +_Q 𝑥) <_Q 𝑢}⟩)))

27-Sep-2020	caucvgprlemloc 6865	Lemma for caucvgpr 6872. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <_Q 𝑟 → (𝑠 ∈ (1^st ‘𝐿) ∨ 𝑟 ∈ (2^nd ‘𝐿))))

27-Sep-2020	caucvgprlemdisj 6864	Lemma for caucvgpr 6872. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1^st ‘𝐿) ∧ 𝑠 ∈ (2^nd ‘𝐿)))

27-Sep-2020	caucvgprlemrnd 6863	Lemma for caucvgpr 6872. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1^st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑠 ∈ (2^nd ‘𝐿)))))

27-Sep-2020	caucvgprlemm 6858	Lemma for caucvgpr 6872. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1^st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐿)))

27-Sep-2020	archrecnq 6853	Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)
⊢ (𝐴 ∈ Q → ∃𝑗 ∈ N (*_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q ) <_Q 𝐴)

26-Sep-2020	caucvgprlemcl 6866	Lemma for caucvgpr 6872. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +_Q (_Q‘[⟨𝑗, 1_𝑜⟩] ~_Q )) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → 𝐿 ∈ P)

16-Sep-2020	lcmass 10467	Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = (𝑁 lcm (𝑀 lcm 𝑃)))

16-Sep-2020	lcmgcdlem 10459	Lemma for lcmgcd 10460 and lcmdvds 10461. Prove them for positive 𝑀, 𝑁, and 𝐾. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)) ∧ ((𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾)) → (𝑀 lcm 𝑁) ∥ 𝐾)))

16-Sep-2020	lcmledvds 10452	A positive integer which both operands of the lcm operator divide bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾) → (𝑀 lcm 𝑁) ≤ 𝐾))

16-Sep-2020	lcmcllem 10449	Lemma for lcmn0cl 10450 and dvdslcm 10451. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)})

16-Sep-2020	lcmn0val 10448	The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))

16-Sep-2020	lcm0val 10447	The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 10446 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0)

16-Sep-2020	lcmcom 10446	The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀))

16-Sep-2020	lcmval 10445	Value of the lcm operator. (𝑀 lcm 𝑁) is the least common multiple of 𝑀 and 𝑁. If either 𝑀 or 𝑁 is 0, the result is defined conventionally as 0. Contrast with df-gcd 10339 and gcdval 10351. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )))

16-Sep-2020	df-lcm 10443	Define the lcm operator. For example, (6 lcm 9) = 18. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))

4-Sep-2020	dfinfre 8034	The infimum of a set of reals 𝐴. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ (𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})

4-Sep-2020	lbinfle 8028	If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)

4-Sep-2020	lbinfcl 8027	If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → inf(𝑆, ℝ, < ) ∈ 𝑆)

4-Sep-2020	lbinf 8026	If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝑆 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → inf(𝑆, ℝ, < ) = (℩𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦))

4-Sep-2020	inf00 6444	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ inf(𝐵, ∅, 𝑅) = ∅

4-Sep-2020	sup00 6416	The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ sup(𝐵, ∅, 𝑅) = ∅

2-Sep-2020	nfinf 6430	Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅)

2-Sep-2020	infeq123d 6429	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝐴 = 𝐷) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ (𝜑 → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

2-Sep-2020	infeq3 6428	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))

2-Sep-2020	infeq2 6427	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))

2-Sep-2020	infeq1i 6426	Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ 𝐵 = 𝐶 ⇒ ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)

2-Sep-2020	infeq1d 6425	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

2-Sep-2020	infeq1 6424	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

2-Sep-2020	df-inf 6398	Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ^◡𝑅)

2-Sep-2020	impel 274	An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜃) → 𝜒)

1-Sep-2020	2a1 25	A double form of ax-1 5. Its associated inference is 2a1i 27. Its associated deduction is 2a1d 23. (Contributed by BJ, 10-Aug-2020.) (Proof shortened by Wolf Lammen, 1-Sep-2020.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜑)))

27-Aug-2020	3lcm2e6 10539	The least common multiple of three and two is six. The operands are unequal primes and thus coprime, so the result is (the absolute value of) their product. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 27-Aug-2020.)
⊢ (3 lcm 2) = 6

27-Aug-2020	6lcm4e12 10469	The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.)
⊢ (6 lcm 4) = ;12

27-Aug-2020	3lcm2e6woprm 10468	The least common multiple of three and two is six. This proof does not use the property of 2 and 3 being prime. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.)
⊢ (3 lcm 2) = 6

27-Aug-2020	lcmgcdnn 10464	The product of two positive integers' least common multiple and greatest common divisor is the product of the two integers. (Contributed by AV, 27-Aug-2020.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 · 𝑁))

27-Aug-2020	6gcd4e2 10384	The greatest common divisor of six and four is two. To calculate this gcd, a simple form of Euclid's algorithm is used: (6 gcd 4) = ((4 + 2) gcd 4) = (2 gcd 4) and (2 gcd 4) = (2 gcd (2 + 2)) = (2 gcd 2) = 2. (Contributed by AV, 27-Aug-2020.)
⊢ (6 gcd 4) = 2

26-Aug-2020	sqrt0rlem 9889	Lemma for sqrt0 9890. (Contributed by Jim Kingdon, 26-Aug-2020.)
⊢ ((𝐴 ∈ ℝ ∧ ((𝐴↑2) = 0 ∧ 0 ≤ 𝐴)) ↔ 𝐴 = 0)

23-Aug-2020	lcm1 10463	The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.)
⊢ (𝑀 ∈ ℤ → (𝑀 lcm 1) = (abs‘𝑀))

23-Aug-2020	sqrtrval 9886	Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)
⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))

23-Aug-2020	df-rsqrt 9884	Define a function whose value is the square root of a nonnegative real number. Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root. (Contributed by Jim Kingdon, 23-Aug-2020.)
⊢ √ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦)))

19-Aug-2020	addnqpr 6751	Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <_Q (𝐴 +_Q 𝐵)}, {𝑢 ∣ (𝐴 +_Q 𝐵) <_Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩))

19-Aug-2020	addnqprlemfu 6750	Lemma for addnqpr 6751. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2^nd ‘⟨{𝑙 ∣ 𝑙 <_Q (𝐴 +_Q 𝐵)}, {𝑢 ∣ (𝐴 +_Q 𝐵) <_Q 𝑢}⟩) ⊆ (2^nd ‘(⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)))

19-Aug-2020	addnqprlemfl 6749	Lemma for addnqpr 6751. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1^st ‘⟨{𝑙 ∣ 𝑙 <_Q (𝐴 +_Q 𝐵)}, {𝑢 ∣ (𝐴 +_Q 𝐵) <_Q 𝑢}⟩) ⊆ (1^st ‘(⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)))

19-Aug-2020	addnqprlemru 6748	Lemma for addnqpr 6751. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2^nd ‘(⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)) ⊆ (2^nd ‘⟨{𝑙 ∣ 𝑙 <_Q (𝐴 +_Q 𝐵)}, {𝑢 ∣ (𝐴 +_Q 𝐵) <_Q 𝑢}⟩))

19-Aug-2020	addnqprlemrl 6747	Lemma for addnqpr 6751. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1^st ‘(⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝐵}, {𝑢 ∣ 𝐵 <_Q 𝑢}⟩)) ⊆ (1^st ‘⟨{𝑙 ∣ 𝑙 <_Q (𝐴 +_Q 𝐵)}, {𝑢 ∣ (𝐴 +_Q 𝐵) <_Q 𝑢}⟩))

15-Aug-2020	prmdvdsfz 10520	Each integer greater than 1 and less then or equal to a fixed number is divisible by a prime less then or equal to this fixed number. (Contributed by AV, 15-Aug-2020.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∃𝑝 ∈ ℙ (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼))

15-Aug-2020	caucvgprlemcanl 6834	Lemma for cauappcvgprlemladdrl 6847. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
⊢ (𝜑 → 𝐿 ∈ P) & ⊢ (𝜑 → 𝑆 ∈ Q) & ⊢ (𝜑 → 𝑅 ∈ Q) & ⊢ (𝜑 → 𝑄 ∈ Q) ⇒ ⊢ (𝜑 → ((𝑅 +_Q 𝑄) ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑆 +_Q 𝑄)}, {𝑢 ∣ (𝑆 +_Q 𝑄) <_Q 𝑢}⟩)) ↔ 𝑅 ∈ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩))))

10-Aug-2020	2a1d 23	Deduction introducing two antecedents. Two applications of a1d 22. Deduction associated with 2a1 25 and 2a1i 27. (Contributed by BJ, 10-Aug-2020.)
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜓)))

9-Aug-2020	ncoprmgcdgt1b 10472	Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is greater than 1. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑖 ∈ (ℤ_≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) ↔ 1 < (𝐴 gcd 𝐵)))

9-Aug-2020	ncoprmgcdne1b 10471	Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑖 ∈ (ℤ_≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) ↔ (𝐴 gcd 𝐵) ≠ 1))

9-Aug-2020	coprmgcdb 10470	Two positive integers are coprime, i.e. the only positive integer that divides both of them is 1, iff their greatest common divisor is 1. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∀𝑖 ∈ ℕ ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐵) → 𝑖 = 1) ↔ (𝐴 gcd 𝐵) = 1))

9-Aug-2020	nndvdslegcd 10357	A positive integer which divides both positive operands of the gcd operator is bounded by it. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁) → 𝐾 ≤ (𝑀 gcd 𝑁)))

9-Aug-2020	negfi 10110	The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → {𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴} ∈ Fin)

9-Aug-2020	negf1o 7486	Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥) ⇒ ⊢ (𝐴 ⊆ ℝ → 𝐹:𝐴–1-1-onto→{𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴})

4-Aug-2020	cauappcvgprlemupu 6839	Lemma for cauappcvgpr 6852. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 <_Q 𝑟 ∧ 𝑠 ∈ (2^nd ‘𝐿)) → 𝑟 ∈ (2^nd ‘𝐿))

4-Aug-2020	cauappcvgprlemopu 6838	Lemma for cauappcvgpr 6852. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑟 ∈ (2^nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑠 ∈ (2^nd ‘𝐿)))

4-Aug-2020	cauappcvgprlemlol 6837	Lemma for cauappcvgpr 6852. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿)) → 𝑠 ∈ (1^st ‘𝐿))

4-Aug-2020	cauappcvgprlemopl 6836	Lemma for cauappcvgpr 6852. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ ((𝜑 ∧ 𝑠 ∈ (1^st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿)))

22-Jul-2020	prmex 10495	The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.)
⊢ ℙ ∈ V

22-Jul-2020	prmssnn 10494	The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.)
⊢ ℙ ⊆ ℕ

18-Jul-2020	cauappcvgprlemloc 6842	Lemma for cauappcvgpr 6852. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <_Q 𝑟 → (𝑠 ∈ (1^st ‘𝐿) ∨ 𝑟 ∈ (2^nd ‘𝐿))))

18-Jul-2020	cauappcvgprlemdisj 6841	Lemma for cauappcvgpr 6852. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1^st ‘𝐿) ∧ 𝑠 ∈ (2^nd ‘𝐿)))

18-Jul-2020	cauappcvgprlemrnd 6840	Lemma for cauappcvgpr 6852. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1^st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <_Q 𝑟 ∧ 𝑠 ∈ (2^nd ‘𝐿)))))

18-Jul-2020	cauappcvgprlemm 6835	Lemma for cauappcvgpr 6852. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1^st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐿)))

11-Jul-2020	cauappcvgprlemladdrl 6847	Lemma for cauappcvgprlemladd 6848. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (1^st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q ((𝐹‘𝑞) +_Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +_Q 𝑞) +_Q 𝑆) <_Q 𝑢}⟩) ⊆ (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)))

11-Jul-2020	cauappcvgprlemladdru 6846	Lemma for cauappcvgprlemladd 6848. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (2^nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q ((𝐹‘𝑞) +_Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +_Q 𝑞) +_Q 𝑆) <_Q 𝑢}⟩) ⊆ (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)))

11-Jul-2020	cauappcvgprlemladdfl 6845	Lemma for cauappcvgprlemladd 6848. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (1^st ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)) ⊆ (1^st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q ((𝐹‘𝑞) +_Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +_Q 𝑞) +_Q 𝑆) <_Q 𝑢}⟩))

11-Jul-2020	cauappcvgprlemladdfu 6844	Lemma for cauappcvgprlemladd 6848. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (2^nd ‘(𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩)) ⊆ (2^nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q ((𝐹‘𝑞) +_Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +_Q 𝑞) +_Q 𝑆) <_Q 𝑢}⟩))

8-Jul-2020	nqprl 6741	Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <_P. (Contributed by Jim Kingdon, 8-Jul-2020.)
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (1^st ‘𝐵) ↔ ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩<_P 𝐵))

5-Jul-2020	addmodid 9374	The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴)

1-Jul-2020	gcdzeq 10411	A positive integer 𝐴 is equal to its gcd with an integer 𝐵 if and only if 𝐴 divides 𝐵. Generalization of gcdeq 10412. (Contributed by AV, 1-Jul-2020.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) = 𝐴 ↔ 𝐴 ∥ 𝐵))

29-Jun-2020	rspcda 2706	Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.)
⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → 𝜒)

26-Jun-2020	zeo5 10288	An integer is either even or odd, version of zeo3 10267 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.)
⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ 2 ∥ (𝑁 + 1)))

25-Jun-2020	even2n 10273	An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.)
⊢ (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)

24-Jun-2020	nqprlu 6737	The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ ∈ P)

23-Jun-2020	cauappcvgprlem2 6850	Lemma for cauappcvgpr 6852. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝑅 ∈ Q) ⇒ ⊢ (𝜑 → 𝐿<_P ⟨{𝑙 ∣ 𝑙 <_Q ((𝐹‘𝑄) +_Q (𝑄 +_Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +_Q (𝑄 +_Q 𝑅)) <_Q 𝑢}⟩)

23-Jun-2020	cauappcvgprlem1 6849	Lemma for cauappcvgpr 6852. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝑅 ∈ Q) ⇒ ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <_Q (𝐹‘𝑄)}, {𝑢 ∣ (𝐹‘𝑄) <_Q 𝑢}⟩<_P (𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑄 +_Q 𝑅)}, {𝑢 ∣ (𝑄 +_Q 𝑅) <_Q 𝑢}⟩))

23-Jun-2020	cauappcvgprlemladd 6848	Lemma for cauappcvgpr 6852. This takes 𝐿 and offsets it by the positive fraction 𝑆. (Contributed by Jim Kingdon, 23-Jun-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑆}, {𝑢 ∣ 𝑆 <_Q 𝑢}⟩) = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q ((𝐹‘𝑞) +_Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +_Q 𝑞) +_Q 𝑆) <_Q 𝑢}⟩)

21-Jun-2020	rspcdva 2707	Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝜒)

20-Jun-2020	cauappcvgprlemlim 6851	Lemma for cauappcvgpr 6852. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <_Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <_Q 𝑢}⟩<_P (𝐿 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑞 +_Q 𝑟)}, {𝑢 ∣ (𝑞 +_Q 𝑟) <_Q 𝑢}⟩) ∧ 𝐿<_P ⟨{𝑙 ∣ 𝑙 <_Q ((𝐹‘𝑞) +_Q (𝑞 +_Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +_Q (𝑞 +_Q 𝑟)) <_Q 𝑢}⟩))

20-Jun-2020	cauappcvgprlemcl 6843	Lemma for cauappcvgpr 6852. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) & ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +_Q 𝑞) <_Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +_Q 𝑞) <_Q 𝑢}⟩ ⇒ ⊢ (𝜑 → 𝐿 ∈ P)

19-Jun-2020	cauappcvgpr 6852	A Cauchy approximation has a limit. A Cauchy approximation, here 𝐹, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of 𝐹 is Q rather than P. We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). This proof (including its lemmas) is similar to the proofs of caucvgpr 6872 and caucvgprpr 6902 but is somewhat simpler, so reading this one first may help understanding the other two. (Contributed by Jim Kingdon, 19-Jun-2020.)
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <_Q ((𝐹‘𝑞) +_Q (𝑝 +_Q 𝑞)) ∧ (𝐹‘𝑞) <_Q ((𝐹‘𝑝) +_Q (𝑝 +_Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <_Q (𝐹‘𝑝)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ P ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <_Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <_Q 𝑢}⟩<_P (𝑦 +_P ⟨{𝑙 ∣ 𝑙 <_Q (𝑞 +_Q 𝑟)}, {𝑢 ∣ (𝑞 +_Q 𝑟) <_Q 𝑢}⟩) ∧ 𝑦<_P ⟨{𝑙 ∣ 𝑙 <_Q ((𝐹‘𝑞) +_Q (𝑞 +_Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +_Q (𝑞 +_Q 𝑟)) <_Q 𝑢}⟩))

18-Jun-2020	caucvgpr 6872	A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). This proof (including its lemmas) is similar to the proofs of cauappcvgpr 6852 and caucvgprpr 6902. Reading cauappcvgpr 6852 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-2020.)
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <_N 𝑘 → ((𝐹‘𝑛) <_Q ((𝐹‘𝑘) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q )) ∧ (𝐹‘𝑘) <_Q ((𝐹‘𝑛) +_Q (_Q‘[⟨𝑛, 1_𝑜⟩] ~_Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <_Q (𝐹‘𝑗)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ P ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <_N 𝑘 → (⟨{𝑙 ∣ 𝑙 <_Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <_Q 𝑢}⟩<_P (𝑦 +_P ⟨{𝑙 ∣ 𝑙 <_Q 𝑥}, {𝑢 ∣ 𝑥 <_Q 𝑢}⟩) ∧ 𝑦<_P ⟨{𝑙 ∣ 𝑙 <_Q ((𝐹‘𝑘) +_Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +_Q 𝑥) <_Q 𝑢}⟩)))

15-Jun-2020	imdivapd 9862	Imaginary part of a division. Related to remul2 9760. (Contributed by Jim Kingdon, 15-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (ℑ‘(𝐵 / 𝐴)) = ((ℑ‘𝐵) / 𝐴))

15-Jun-2020	redivapd 9861	Real part of a division. Related to remul2 9760. (Contributed by Jim Kingdon, 15-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (ℜ‘(𝐵 / 𝐴)) = ((ℜ‘𝐵) / 𝐴))

15-Jun-2020	cjdivapd 9855	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))

15-Jun-2020	riotaexg 5492	Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V)

14-Jun-2020	cjdivapi 9822	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))

14-Jun-2020	cjdivap 9796	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))

14-Jun-2020	cjap0 9794	A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ (∗‘𝐴) # 0))

14-Jun-2020	cjap 9793	Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) # (∗‘𝐵) ↔ 𝐴 # 𝐵))

14-Jun-2020	imdivap 9768	Imaginary part of a division. Related to immul2 9767. (Contributed by Jim Kingdon, 14-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))

14-Jun-2020	redivap 9761	Real part of a division. Related to remul2 9760. (Contributed by Jim Kingdon, 14-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))

14-Jun-2020	mulreap 9751	A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 ∈ ℝ ↔ (𝐵 · 𝐴) ∈ ℝ))

13-Jun-2020	sqgt0apd 9633	The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → 0 < (𝐴↑2))

13-Jun-2020	reexpclzapd 9630	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ)

13-Jun-2020	expdivapd 9619	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁)))

13-Jun-2020	sqdivapd 9618	Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))

12-Jun-2020	expsubapd 9616	Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁)))

12-Jun-2020	expm1apd 9615	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴))

12-Jun-2020	expp1zapd 9614	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))

12-Jun-2020	exprecapd 9613	Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))

12-Jun-2020	expnegapd 9612	Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))

12-Jun-2020	expap0d 9611	Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) # 0)

12-Jun-2020	expclzapd 9610	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ)

12-Jun-2020	sqrecapd 9609	Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴)↑2) = (1 / (𝐴↑2)))

12-Jun-2020	sqgt0api 9561	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 # 0 → 0 < (𝐴↑2))

12-Jun-2020	sqdivapi 9559	Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))

11-Jun-2020	sqgt0ap 9544	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴↑2))

11-Jun-2020	sqdivap 9540	Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))

11-Jun-2020	expdivap 9527	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℕ₀) → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁)))

11-Jun-2020	expm1ap 9526	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴))

11-Jun-2020	expp1zap 9525	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))

11-Jun-2020	expsubap 9524	Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁)))

11-Jun-2020	expmulzap 9522	Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴↑𝑀)↑𝑁))

10-Jun-2020	expaddzap 9520	Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁)))

10-Jun-2020	expaddzaplem 9519	Lemma for expaddzap 9520. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ₀) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁)))

10-Jun-2020	exprecap 9517	Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))

10-Jun-2020	mulexpzap 9516	Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))

10-Jun-2020	expap0i 9508	Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) # 0)

10-Jun-2020	expap0 9506	Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 9507 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) # 0 ↔ 𝐴 # 0))

10-Jun-2020	mvllmulapd 7921	Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → (𝐴 · 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = (𝐶 / 𝐴))

9-Jun-2020	expclzap 9501	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ)

9-Jun-2020	expclzaplem 9500	Closure law for integer exponentiation. Lemma for expclzap 9501 and expap0i 9508. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})

9-Jun-2020	reexpclzap 9496	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ)

9-Jun-2020	neg1ap0 8148	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ -1 # 0

8-Jun-2020	expcl2lemap 9488	Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ 𝐹 ⊆ ℂ & ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) & ⊢ 1 ∈ 𝐹 & ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 # 0) → (1 / 𝑥) ∈ 𝐹) ⇒ ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐴 # 0 ∧ 𝐵 ∈ ℤ) → (𝐴↑𝐵) ∈ 𝐹)

8-Jun-2020	expn1ap0 9486	A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴↑-1) = (1 / 𝐴))

8-Jun-2020	expineg2 9485	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ₀)) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁)))

8-Jun-2020	expnegap0 9484	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ₀) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))

8-Jun-2020	expinnval 9479	Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁))

7-Jun-2020	zob 10291	Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.)
⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 − 1) / 2) ∈ ℤ))

7-Jun-2020	expival 9478	Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))

7-Jun-2020	expivallem 9477	Lemma for expival 9478. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) # 0)

7-Jun-2020	df-iexp 9476	Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 9480 and expp1 9483 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that 0↑0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case 𝑥 = 0, 𝑦 < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.)
⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))))

7-Jun-2020	halfge0 8247	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
⊢ 0 ≤ (1 / 2)

4-Jun-2020	nn0ob 10308	Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)
⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) / 2) ∈ ℕ₀ ↔ ((𝑁 − 1) / 2) ∈ ℕ₀))

4-Jun-2020	iseqfveq 9450	Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))

3-Jun-2020	iseqfeq2 9449	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘𝑀)) & ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾)) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ_≥‘𝐾)) = seq𝐾( + , 𝐺, 𝑆))

3-Jun-2020	iseqfveq2 9448	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘𝑀)) & ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾)) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝐾)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))

2-Jun-2020	nn0o 10307	An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)
⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ₀)

2-Jun-2020	nno 10306	An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)
⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ)

2-Jun-2020	nn0o1gt2 10305	An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)
⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 = 1 ∨ 2 < 𝑁))

2-Jun-2020	3halfnz 8444	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
⊢ ¬ (3 / 2) ∈ ℤ

1-Jun-2020	iseqcl 9443	Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆)

1-Jun-2020	fzdcel 9059	Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀...𝑁))

1-Jun-2020	fztri3or 9058	Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑀 ∨ 𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾))

1-Jun-2020	zdclt 8425	Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)

31-May-2020	iseqp1 9445	Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1))))

31-May-2020	iseq1 9442	Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀))

31-May-2020	iseqovex 9439	Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ_≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)

31-May-2020	frecuzrdgcl 9415	Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 9401 for the description of 𝐺 as the mapping from ω to (ℤ_≥‘𝐶). (Contributed by Jim Kingdon, 31-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) & ⊢ (𝜑 → 𝑇 = ran 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ_≥‘𝐶)) → (𝑇‘𝐵) ∈ 𝑆)

30-May-2020	nn0enne 10302	A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ₀ ↔ (𝑁 / 2) ∈ ℕ))

30-May-2020	iseqfn 9441	The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ_≥‘𝑀))

30-May-2020	iseqval 9440	Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ_≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = ran 𝑅)

30-May-2020	nfiseq 9438	Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ Ⅎ𝑥𝑀 & ⊢ Ⅎ𝑥 + & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝑆 ⇒ ⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆)

30-May-2020	iseqeq4 9437	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝑆 = 𝑇 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))

30-May-2020	iseqeq3 9436	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))

30-May-2020	iseqeq2 9435	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ ( + = 𝑄 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀(𝑄, 𝐹, 𝑆))

30-May-2020	iseqeq1 9434	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆))

30-May-2020	nffrec 6005	Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥frec(𝐹, 𝐴)

30-May-2020	freceq2 6003	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐴 = 𝐵 → frec(𝐹, 𝐴) = frec(𝐹, 𝐵))

30-May-2020	freceq1 6002	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))

29-May-2020	df-iseq 9432	Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers ℕ or some other upper integer set) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by iseq1 9442 and iseqp1 9445. Typically, those are the main theorems that would be used in practice. The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence 𝐹 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq1( + , 𝐹, ℚ) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹, ℚ)‘1) = 1, (seq1( + , 𝐹, ℚ)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹, ℚ) transforms a sequence 𝐹 into an infinite series. Internally, the frec function generates as its values a set of ordered pairs starting at ⟨𝑀, (𝐹‘𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain. (Contributed by Jim Kingdon, 29-May-2020.)
⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ_≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)

28-May-2020	frecuzrdgsuc 9417	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9401 for the description of 𝐺 as the mapping from ω to (ℤ_≥‘𝐶). (Contributed by Jim Kingdon, 28-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) & ⊢ (𝜑 → 𝑇 = ran 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ_≥‘𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇‘𝐵)))

27-May-2020	frecuzrdg0 9416	Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 9401 for the description of 𝐺 as the mapping from ω to (ℤ_≥‘𝐶). (Contributed by Jim Kingdon, 27-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) & ⊢ (𝜑 → 𝑇 = ran 𝑅) ⇒ ⊢ (𝜑 → (𝑇‘𝐶) = 𝐴)

27-May-2020	frecuzrdgrrn 9410	The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. (Contributed by Jim Kingdon, 27-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ⇒ ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → (𝑅‘𝐷) ∈ ((ℤ_≥‘𝐶) × 𝑆))

27-May-2020	dffun5r 4934	A way of proving a relation is a function, analogous to mo2r 1993. (Contributed by Jim Kingdon, 27-May-2020.)
⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) → Fun 𝐴)

26-May-2020	frecuzrdgfn 9414	The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9401 for the description of 𝐺 as the mapping from ω to (ℤ_≥‘𝐶). (Contributed by Jim Kingdon, 26-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) & ⊢ (𝜑 → 𝑇 = ran 𝑅) ⇒ ⊢ (𝜑 → 𝑇 Fn (ℤ_≥‘𝐶))

26-May-2020	frecuzrdglem 9413	A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) & ⊢ (𝜑 → 𝐵 ∈ (ℤ_≥‘𝐶)) ⇒ ⊢ (𝜑 → ⟨𝐵, (2^nd ‘(𝑅‘(^◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)

26-May-2020	frecuzrdgrom 9412	The function 𝑅 (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 26-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ⇒ ⊢ (𝜑 → 𝑅 Fn ω)

25-May-2020	divlt1lt 8801	A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((𝐴 / 𝐵) < 1 ↔ 𝐴 < 𝐵))

25-May-2020	freccl 6016	Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)
⊢ (𝜑 → ∀𝑧(𝐹‘𝑧) ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ ω) ⇒ ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

24-May-2020	mod2eq1n2dvds 10279	An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.)
⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁))

24-May-2020	frec2uzrdg 9411	A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ₀) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. This lemma shows that evaluating 𝑅 at an element of ω gives an ordered pair whose first element is the index (translated from ω to (ℤ_≥‘𝐶)). See comment in frec2uz0d 9401 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ_≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) & ⊢ 𝑅 = frec((𝑥 ∈ (ℤ_≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) & ⊢ (𝜑 → 𝐵 ∈ ω) ⇒ ⊢ (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2^nd ‘(𝑅‘𝐵))⟩)

24-May-2020	eluz2gt1 8689	An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 < 𝑁)

24-May-2020	xp1d2m1eqxm1d2 8283	A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
⊢ (𝑋 ∈ ℂ → (((𝑋 + 1) / 2) − 1) = ((𝑋 − 1) / 2))

21-May-2020	fzofig 9424	Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin)

21-May-2020	fzfigd 9423	Deduction form of fzfig 9422. (Contributed by Jim Kingdon, 21-May-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)

19-May-2020	fzfig 9422	A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin)

19-May-2020	frechashgf1o 9421	𝐺 maps ω one-to-one onto ℕ₀. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ 𝐺:ω–1-1-onto→ℕ₀

19-May-2020	ssfiexmid 6361	If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑)

19-May-2020	enm 6317	A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ 𝐵)

18-May-2020	frecfzen2 9420	The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑀...𝑁) ≈ (^◡𝐺‘((𝑁 + 1) − 𝑀)))

18-May-2020	frecfzennn 9419	The cardinality of a finite set of sequential integers. (See frec2uz0d 9401 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝑁 ∈ ℕ₀ → (1...𝑁) ≈ (^◡𝐺‘𝑁))

17-May-2020	frec2uzisod 9409	𝐺 (see frec2uz0d 9401) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → 𝐺 Isom E , < (ω, (ℤ_≥‘𝐶)))

17-May-2020	frec2uzf1od 9408	𝐺 (see frec2uz0d 9401) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ_≥‘𝐶))

17-May-2020	frec2uzrand 9407	Range of 𝐺 (see frec2uz0d 9401). (Contributed by Jim Kingdon, 17-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → ran 𝐺 = (ℤ_≥‘𝐶))

16-May-2020	frec2uzlt2d 9406	The mapping 𝐺 (see frec2uz0d 9401) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ∈ ω) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵)))

16-May-2020	frec2uzltd 9405	Less-than relation for 𝐺 (see frec2uz0d 9401). (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ∈ ω) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) < (𝐺‘𝐵)))

16-May-2020	frec2uzuzd 9404	The value 𝐺 (see frec2uz0d 9401) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝐴 ∈ ω) ⇒ ⊢ (𝜑 → (𝐺‘𝐴) ∈ (ℤ_≥‘𝐶))

16-May-2020	frec2uzsucd 9403	The value of 𝐺 (see frec2uz0d 9401) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝐴 ∈ ω) ⇒ ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1))

16-May-2020	frec2uzzd 9402	The value of 𝐺 (see frec2uz0d 9401) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) & ⊢ (𝜑 → 𝐴 ∈ ω) ⇒ ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ)

16-May-2020	frec2uz0d 9401	The mapping 𝐺 is a one-to-one mapping from ω onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers ℕ₀ or 1 for the upper integers ℕ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → (𝐺‘∅) = 𝐶)

15-May-2020	nntri3 6098	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))

14-May-2020	rdgifnon2 5990	The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)

14-May-2020	rdgtfr 5984	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)
⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))

13-May-2020	frecfnom 6009	The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → frec(𝐹, 𝐴) Fn ω)

13-May-2020	frecabex 6007	The class abstraction from df-frec 6001 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑦(𝐹‘𝑦) ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) ⇒ ⊢ (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)

8-May-2020	tfr0 5960	Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
⊢ 𝐹 = recs(𝐺) ⇒ ⊢ ((𝐺‘∅) ∈ 𝑉 → (𝐹‘∅) = (𝐺‘∅))

7-May-2020	frec0g 6006	The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

3-May-2020	dcned 2251	Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
⊢ (𝜑 → DECID 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵)

2-May-2020	ax-arch 7095	Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by theorem axarch 7057. This axiom should not be used directly; instead use arch 8285 (which is the same, but stated in terms of ℕ and <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <_ℝ 𝑛)

30-Apr-2020	ltexnqi 6599	Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.)
⊢ (𝐴 <_Q 𝐵 → ∃𝑥 ∈ Q (𝐴 +_Q 𝑥) = 𝐵)

26-Apr-2020	pitonnlem1p1 7014	Lemma for pitonn 7016. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ P → [⟨(𝐴 +_P (1_P +_P 1_P)), (1_P +_P 1_P)⟩] ~_R = [⟨(𝐴 +_P 1_P), 1_P⟩] ~_R )

26-Apr-2020	addnqpr1 6752	Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 6751. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q (𝐴 +_Q 1_Q)}, {𝑢 ∣ (𝐴 +_Q 1_Q) <_Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <_Q 𝐴}, {𝑢 ∣ 𝐴 <_Q 𝑢}⟩ +_P 1_P))

26-Apr-2020	addpinq1 6654	Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ N → [⟨(𝐴 +_N 1_𝑜), 1_𝑜⟩] ~_Q = ([⟨𝐴, 1_𝑜⟩] ~_Q +_Q 1_Q))

26-Apr-2020	nnnq 6612	The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
⊢ (𝐴 ∈ N → [⟨𝐴, 1_𝑜⟩] ~_Q ∈ Q)

24-Apr-2020	pitonnlem2 7015	Lemma for pitonn 7016. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)
⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨(𝐾 +_N 1_𝑜), 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨(𝐾 +_N 1_𝑜), 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩)

24-Apr-2020	pitonnlem1 7013	Lemma for pitonn 7016. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)
⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨1_𝑜, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨1_𝑜, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ = 1

23-Apr-2020	archsr 6958	For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑥, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑥, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R is the embedding of the positive integer 𝑥 into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
⊢ (𝐴 ∈ R → ∃𝑥 ∈ N 𝐴 <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑥, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑥, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )

23-Apr-2020	nnprlu 6743	The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
⊢ (𝐴 ∈ N → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐴, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐴, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ∈ P)

22-Apr-2020	axarch 7057	Archimedean axiom. The Archimedean property is more naturally stated once we have defined ℕ. Unless we find another way to state it, we'll just use the right hand side of dfnn2 8041 in stating what we mean by "natural number" in the context of this axiom. This construction-dependent theorem should not be referenced directly; instead, use ax-arch 7095. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <_ℝ 𝑛)

22-Apr-2020	pitonn 7016	Mapping from N to ℕ. (Contributed by Jim Kingdon, 22-Apr-2020.)
⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑁, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑁, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})

22-Apr-2020	archpr 6833	For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer 𝑥 is embedded into the reals as described at nnprlu 6743. (Contributed by Jim Kingdon, 22-Apr-2020.)
⊢ (𝐴 ∈ P → ∃𝑥 ∈ N 𝐴<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝑥, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝑥, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩)

20-Apr-2020	fzo0m 9200	A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.)
⊢ (∃𝑥 𝑥 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)

20-Apr-2020	fzom 9173	A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
⊢ (∃𝑥 𝑥 ∈ (𝑀..^𝑁) ↔ 𝑀 ∈ (𝑀..^𝑁))

18-Apr-2020	eluzdc 8697	Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ_≥‘𝑀))

17-Apr-2020	zlelttric 8396	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴))

16-Apr-2020	fznlem 9060	A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → (𝑀...𝑁) = ∅))

15-Apr-2020	fzm 9057	Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ_≥‘𝑀))

15-Apr-2020	xpdom3m 6331	A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ∃𝑥 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 × 𝐵))

13-Apr-2020	snfig 6314	A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin)

13-Apr-2020	en1bg 6303	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ 1_𝑜 ↔ 𝐴 = {∪ 𝐴}))

10-Apr-2020	negm 8700	The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴})

8-Apr-2020	zleloe 8398	Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))

7-Apr-2020	zdcle 8424	Integer ≤ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 ≤ 𝐵)

5-Apr-2020	faccld 9663	Closure of the factorial function, deduction version of faccl 9662. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)

5-Apr-2020	divge1 8800	The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 1 ≤ (𝐵 / 𝐴))

4-Apr-2020	ioorebasg 8998	Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ ran (,))

30-Mar-2020	icc0r 8949	An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 → (𝐴[,]𝐵) = ∅))

30-Mar-2020	ubioog 8937	An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ¬ 𝐵 ∈ (𝐴(,)𝐵))

30-Mar-2020	lbioog 8936	An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ¬ 𝐴 ∈ (𝐴(,)𝐵))

29-Mar-2020	iooidg 8932	An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
⊢ (𝐴 ∈ ℝ^* → (𝐴(,)𝐴) = ∅)

27-Mar-2020	zletric 8395	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))

26-Mar-2020	4z 8381	4 is an integer. (Contributed by BJ, 26-Mar-2020.)
⊢ 4 ∈ ℤ

25-Mar-2020	elfzmlbm 9142	Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)))

25-Mar-2020	elfz0add 9134	An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ₀) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵))))

25-Mar-2020	2eluzge0 8663	2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ 2 ∈ (ℤ_≥‘0)

23-Mar-2020	rpnegap 8766	Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (𝐴 ∈ ℝ⁺ ⊻ -𝐴 ∈ ℝ⁺))

23-Mar-2020	reapltxor 7689	Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ⊻ 𝐵 < 𝐴)))

22-Mar-2020	rpcnap0 8754	A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ 𝐴 # 0))

22-Mar-2020	rpreap0 8752	A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 𝐴 # 0))

22-Mar-2020	rpap0 8750	A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
⊢ (𝐴 ∈ ℝ⁺ → 𝐴 # 0)

20-Mar-2020	qapne 8724	Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 # 𝐵 ↔ 𝐴 ≠ 𝐵))

20-Mar-2020	divap1d 7888	If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐴 # 𝐵) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) # 1)

20-Mar-2020	apmul1 7876	Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))

19-Mar-2020	divfnzn 8706	Division restricted to ℤ × ℕ is a function. Given excluded middle, it would be easy to prove this for ℂ × (ℂ ∖ {0}). The key difference is that an element of ℕ is apart from zero, whereas being an element of ℂ ∖ {0} implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ)

19-Mar-2020	div2negapd 7892	Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵))

19-Mar-2020	divneg2apd 7891	Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵))

19-Mar-2020	divnegapd 7890	Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))

19-Mar-2020	divap0bd 7889	A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0))

19-Mar-2020	diveqap0ad 7887	A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7770. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))

19-Mar-2020	diveqap1ad 7886	The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7793. Generalization of diveqap1d 7885. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))

19-Mar-2020	diveqap1d 7885	Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → (𝐴 / 𝐵) = 1) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵)

19-Mar-2020	diveqap0d 7884	If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → (𝐴 / 𝐵) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0)

15-Mar-2020	nneoor 8449	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))

14-Mar-2020	zltlen 8426	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7730 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)))

14-Mar-2020	zdceq 8423	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)

14-Mar-2020	zapne 8422	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ 𝑀 ≠ 𝑁))

14-Mar-2020	zltnle 8397	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

14-Mar-2020	ztri3or 8394	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))

14-Mar-2020	ztri3or0 8393	Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))

14-Mar-2020	zaddcllemneg 8390	Lemma for zaddcl 8391. Special case in which -𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)

14-Mar-2020	zaddcllempos 8388	Lemma for zaddcl 8391. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)

14-Mar-2020	dcne 2256	Decidable equality expressed in terms of ≠. Basically the same as df-dc 776. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵))

9-Mar-2020	dvdsmultr1d 10234	Natural deduction form of dvdsmultr1 10233. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∥ 𝑀) ⇒ ⊢ (𝜑 → 𝐾 ∥ (𝑀 · 𝑁))

9-Mar-2020	dvds2subd 10231	Natural deduction form of dvds2sub 10230. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∥ 𝑀) & ⊢ (𝜑 → 𝐾 ∥ 𝑁) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐾 ∥ (𝑀 − 𝑁))

9-Mar-2020	2muliap0 8255	2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ (2 · i) # 0

9-Mar-2020	iap0 8254	The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ i # 0

9-Mar-2020	2ap0 8132	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 2 # 0

9-Mar-2020	1ne0 8107	1 ≠ 0. See aso 1ap0 7690. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 1 ≠ 0

9-Mar-2020	redivclapi 7867	Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐵 # 0 ⇒ ⊢ (𝐴 / 𝐵) ∈ ℝ

9-Mar-2020	redivclapzi 7866	Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ)

9-Mar-2020	rerecclapi 7865	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 # 0 ⇒ ⊢ (1 / 𝐴) ∈ ℝ

9-Mar-2020	rerecclapzi 7864	Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℝ)

9-Mar-2020	divdivdivapi 7863	Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐷 # 0 & ⊢ 𝐶 # 0 ⇒ ⊢ ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))

9-Mar-2020	divadddivapi 7862	Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐷 # 0 ⇒ ⊢ ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷))

9-Mar-2020	divmul13api 7861	Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐷 # 0 ⇒ ⊢ ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐶 / 𝐵) · (𝐴 / 𝐷))

9-Mar-2020	divmuldivapi 7860	Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ & ⊢ 𝐵 # 0 & ⊢ 𝐷 # 0 ⇒ ⊢ ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))

9-Mar-2020	div11api 7859	One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵)

9-Mar-2020	div23api 7858	A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)

9-Mar-2020	divdirapi 7857	Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))

9-Mar-2020	divassapi 7856	An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0 ⇒ ⊢ ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))

8-Mar-2020	nnap0 8068	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
⊢ (𝐴 ∈ ℕ → 𝐴 # 0)

8-Mar-2020	divdivap2d 7909	Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))

8-Mar-2020	divdivap1d 7908	Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))

8-Mar-2020	dmdcanap2d 7907	Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐶)) = (𝐴 / 𝐶))

8-Mar-2020	dmdcanapd 7906	Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐵 / 𝐶) · (𝐴 / 𝐵)) = (𝐴 / 𝐶))

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