Theorem List for Intuitionistic Logic Explorer - 7701-7800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | apsqgt0 7701 |
The square of a real number apart from zero is positive. (Contributed by
Jim Kingdon, 7-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴 · 𝐴)) |
|
Theorem | cru 7702 |
The representation of complex numbers in terms of real and imaginary parts
is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM,
9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) = (𝐶 + (i · 𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
|
Theorem | apreim 7703 |
Complex apartness in terms of real and imaginary parts. (Contributed by
Jim Kingdon, 12-Feb-2020.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) # (𝐶 + (i · 𝐷)) ↔ (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) |
|
Theorem | mulreim 7704 |
Complex multiplication in terms of real and imaginary parts. (Contributed
by Jim Kingdon, 23-Feb-2020.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) · (𝐶 + (i · 𝐷))) = (((𝐴 · 𝐶) + -(𝐵 · 𝐷)) + (i · ((𝐶 · 𝐵) + (𝐷 · 𝐴))))) |
|
Theorem | apirr 7705 |
Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
|
⊢ (𝐴 ∈ ℂ → ¬ 𝐴 # 𝐴) |
|
Theorem | apsym 7706 |
Apartness is symmetric. This theorem for real numbers is part of
Definition 11.2.7(v) of [HoTT], p.
(varies). (Contributed by Jim
Kingdon, 16-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ 𝐵 # 𝐴)) |
|
Theorem | apcotr 7707 |
Apartness is cotransitive. (Contributed by Jim Kingdon,
16-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶))) |
|
Theorem | apadd1 7708 |
Addition respects apartness. Analogue of addcan 7288 for apartness.
(Contributed by Jim Kingdon, 13-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶))) |
|
Theorem | apadd2 7709 |
Addition respects apartness. (Contributed by Jim Kingdon,
16-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐶 + 𝐴) # (𝐶 + 𝐵))) |
|
Theorem | addext 7710 |
Strong extensionality for addition. Given excluded middle, apartness
would be equivalent to negated equality and this would follow readily (for
all operations) from oveq12 5541. For us, it is proved a different way.
(Contributed by Jim Kingdon, 15-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) |
|
Theorem | apneg 7711 |
Negation respects apartness. (Contributed by Jim Kingdon,
14-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵)) |
|
Theorem | mulext1 7712 |
Left extensionality for complex multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵)) |
|
Theorem | mulext2 7713 |
Right extensionality for complex multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵)) |
|
Theorem | mulext 7714 |
Strong extensionality for multiplication. Given excluded middle,
apartness would be equivalent to negated equality and this would follow
readily (for all operations) from oveq12 5541. For us, it is proved a
different way. (Contributed by Jim Kingdon, 23-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) # (𝐶 · 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) |
|
Theorem | mulap0r 7715 |
A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed
by Jim Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 · 𝐵) # 0) → (𝐴 # 0 ∧ 𝐵 # 0)) |
|
Theorem | msqge0 7716 |
A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by
NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴)) |
|
Theorem | msqge0i 7717 |
A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof
shortened by Andrew Salmon, 19-Nov-2011.)
|
⊢ 𝐴 ∈ ℝ
⇒ ⊢ 0 ≤ (𝐴 · 𝐴) |
|
Theorem | msqge0d 7718 |
A square is nonnegative. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ)
⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐴)) |
|
Theorem | mulge0 7719 |
The product of two nonnegative numbers is nonnegative. (Contributed by
NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 · 𝐵)) |
|
Theorem | mulge0i 7720 |
The product of two nonnegative numbers is nonnegative. (Contributed by
NM, 30-Jul-1999.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 · 𝐵)) |
|
Theorem | mulge0d 7721 |
The product of two nonnegative numbers is nonnegative. (Contributed by
Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴)
& ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) |
|
Theorem | apti 7722 |
Complex apartness is tight. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵)) |
|
Theorem | apne 7723 |
Apartness implies negated equality. We cannot in general prove the
converse, which is the whole point of having separate notations for
apartness and negated equality. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 → 𝐴 ≠ 𝐵)) |
|
Theorem | leltap 7724 |
'<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon,
13-Aug-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴)) |
|
Theorem | gt0ap0 7725 |
Positive implies apart from zero. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0) |
|
Theorem | gt0ap0i 7726 |
Positive means apart from zero (useful for ordering theorems involving
division). (Contributed by Jim Kingdon, 27-Feb-2020.)
|
⊢ 𝐴 ∈ ℝ
⇒ ⊢ (0 < 𝐴 → 𝐴 # 0) |
|
Theorem | gt0ap0ii 7727 |
Positive implies apart from zero. (Contributed by Jim Kingdon,
27-Feb-2020.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 𝐴 # 0 |
|
Theorem | gt0ap0d 7728 |
Positive implies apart from zero. Because of the way we define #,
𝐴 must be an element of ℝ, not just ℝ*. (Contributed by
Jim Kingdon, 27-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) ⇒ ⊢ (𝜑 → 𝐴 # 0) |
|
Theorem | negap0 7729 |
A number is apart from zero iff its negative is apart from zero.
(Contributed by Jim Kingdon, 27-Feb-2020.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ -𝐴 # 0)) |
|
Theorem | ltleap 7730 |
Less than in terms of non-strict order and apartness. (Contributed by Jim
Kingdon, 28-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵))) |
|
Theorem | ltap 7731 |
'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴) |
|
Theorem | gtapii 7732 |
'Greater than' implies apart. (Contributed by Jim Kingdon,
12-Aug-2021.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐵 # 𝐴 |
|
Theorem | ltapii 7733 |
'Less than' implies apart. (Contributed by Jim Kingdon,
12-Aug-2021.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 # 𝐵 |
|
Theorem | ltapi 7734 |
'Less than' implies apart. (Contributed by Jim Kingdon,
12-Aug-2021.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 # 𝐴) |
|
Theorem | 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) |
|
Theorem | 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) |
|
Theorem | 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)) |
|
Theorem | 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)) |
|
Theorem | 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) |
|
Theorem | 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) |
|
3.3.7 Reciprocals
|
|
Theorem | recextlem1 7741 |
Lemma for recexap 7743. (Contributed by Eric Schmidt, 23-May-2007.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵))) |
|
Theorem | recexaplem2 7742 |
Lemma for recexap 7743. (Contributed by Jim Kingdon, 20-Feb-2020.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0) |
|
Theorem | recexap 7743* |
Existence of reciprocal of nonzero complex number. (Contributed by Jim
Kingdon, 20-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1) |
|
Theorem | mulap0 7744 |
The product of two numbers apart from zero is apart from zero. Lemma
2.15 of [Geuvers], p. 6. (Contributed
by Jim Kingdon, 22-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 · 𝐵) # 0) |
|
Theorem | mulap0b 7745 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0)) |
|
Theorem | mulap0i 7746 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 23-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐴 # 0 & ⊢ 𝐵 # 0
⇒ ⊢ (𝐴 · 𝐵) # 0 |
|
Theorem | mulap0bd 7747 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0)) |
|
Theorem | mulap0d 7748 |
The product of two numbers apart from zero is apart from zero.
(Contributed by Jim Kingdon, 23-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) # 0) |
|
Theorem | mulap0bad 7749 |
A factor of a complex number apart from zero is apart from zero.
Partial converse of mulap0d 7748 and consequence of mulap0bd 7747.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) # 0) ⇒ ⊢ (𝜑 → 𝐴 # 0) |
|
Theorem | mulap0bbd 7750 |
A factor of a complex number apart from zero is apart from zero.
Partial converse of mulap0d 7748 and consequence of mulap0bd 7747.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) # 0) ⇒ ⊢ (𝜑 → 𝐵 # 0) |
|
Theorem | mulcanapd 7751 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | mulcanap2d 7752 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | mulcanapad 7753 |
Cancellation of a nonzero factor on the left in an equation. One-way
deduction form of mulcanapd 7751. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | mulcanap2ad 7754 |
Cancellation of a nonzero factor on the right in an equation. One-way
deduction form of mulcanap2d 7752. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) |
|
Theorem | mulcanap 7755 |
Cancellation law for multiplication (full theorem form). (Contributed by
Jim Kingdon, 21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | mulcanap2 7756 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | mulcanapi 7757 |
Cancellation law for multiplication. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 # 0
⇒ ⊢ ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵) |
|
Theorem | muleqadd 7758 |
Property of numbers whose product equals their sum. Equation 5 of
[Kreyszig] p. 12. (Contributed by NM,
13-Nov-2006.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 + 𝐵) ↔ ((𝐴 − 1) · (𝐵 − 1)) = 1)) |
|
Theorem | receuap 7759* |
Existential uniqueness of reciprocals. (Contributed by Jim Kingdon,
21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) |
|
3.3.8 Division
|
|
Syntax | cdiv 7760 |
Extend class notation to include division.
|
class / |
|
Definition | df-div 7761* |
Define division. Theorem divmulap 7763 relates it to multiplication, and
divclap 7766 and redivclap 7819 prove its closure laws. (Contributed by NM,
2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.)
(New usage is discouraged.)
|
⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦
(℩𝑧 ∈
ℂ (𝑦 · 𝑧) = 𝑥)) |
|
Theorem | divvalap 7762* |
Value of division: the (unique) element 𝑥 such that
(𝐵
· 𝑥) = 𝐴. This is meaningful
only when 𝐵 is apart from
zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
|
Theorem | divmulap 7763 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴)) |
|
Theorem | divmulap2 7764 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵))) |
|
Theorem | divmulap3 7765 |
Relationship between division and multiplication. (Contributed by Jim
Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶))) |
|
Theorem | divclap 7766 |
Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ) |
|
Theorem | recclap 7767 |
Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ) |
|
Theorem | divcanap2 7768 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
|
Theorem | divcanap1 7769 |
A cancellation law for division. (Contributed by Jim Kingdon,
22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
|
Theorem | diveqap0 7770 |
A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon,
22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0)) |
|
Theorem | divap0b 7771 |
The ratio of numbers apart from zero is apart from zero. (Contributed by
Jim Kingdon, 22-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0)) |
|
Theorem | divap0 7772 |
The ratio of numbers apart from zero is apart from zero. (Contributed by
Jim Kingdon, 22-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / 𝐵) # 0) |
|
Theorem | recap0 7773 |
The reciprocal of a number apart from zero is apart from zero.
(Contributed by Jim Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) # 0) |
|
Theorem | recidap 7774 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1) |
|
Theorem | recidap2 7775 |
Multiplication of a number and its reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1) |
|
Theorem | divrecap 7776 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 24-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
|
Theorem | divrecap2 7777 |
Relationship between division and reciprocal. (Contributed by Jim
Kingdon, 25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
|
Theorem | divassap 7778 |
An associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) |
|
Theorem | div23ap 7779 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) |
|
Theorem | div32ap 7780 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵))) |
|
Theorem | div13ap 7781 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴)) |
|
Theorem | div12ap 7782 |
A commutative/associative law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶))) |
|
Theorem | 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))) |
|
Theorem | 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))) |
|
Theorem | divdirap 7785 |
Distribution of division over addition. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))) |
|
Theorem | divcanap3 7786 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴) |
|
Theorem | divcanap4 7787 |
A cancellation law for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) |
|
Theorem | div11ap 7788 |
One-to-one relationship for division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | dividap 7789 |
A number divided by itself is one. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1) |
|
Theorem | div0ap 7790 |
Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0) |
|
Theorem | div1 7791 |
A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof
shortened by Mario Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) |
|
Theorem | 1div1e1 7792 |
1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler,
7-Dec-2018.)
|
⊢ (1 / 1) = 1 |
|
Theorem | diveqap1 7793 |
Equality in terms of unit ratio. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) |
|
Theorem | divnegap 7794 |
Move negative sign inside of a division. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) |
|
Theorem | 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))) |
|
Theorem | divsubdirap 7796 |
Distribution of division over subtraction. (Contributed by NM,
4-Mar-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 − 𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶))) |
|
Theorem | recrecap 7797 |
A number is equal to the reciprocal of its reciprocal. (Contributed by
Jim Kingdon, 25-Feb-2020.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (1 / 𝐴)) = 𝐴) |
|
Theorem | rec11ap 7798 |
Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵)) |
|
Theorem | rec11rap 7799 |
Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴)) |
|
Theorem | divmuldivap 7800 |
Multiplication of two ratios. (Contributed by Jim Kingdon,
25-Feb-2020.)
|
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 · 𝐵) / (𝐶 · 𝐷))) |