cvmliftlem10 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem10		Structured version Visualization version GIF version

Theorem cvmliftlem10 31276

Description: Lemma for cvmlift 31281. The function 𝐾 is going to be our complete lifted path, formed by unioning together all the 𝑄 functions (each of which is defined on one segment [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁] of the interval). Here we prove by induction that 𝐾 is a continuous function and a lift of 𝐺 by applying cvmliftlem6 31272, cvmliftlem7 31273 (to show it is a function and a lift), cvmliftlem8 31274 (to show it is continuous), and cvmliftlem9 31275 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 21098 gives that 𝐾 is well-defined and continuous). (Contributed by Mario Carneiro, 14-Feb-2015.)

**Hypotheses**
Ref	Expression
cvmliftlem.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
cvmliftlem.b	⊢ 𝐵 = ∪ 𝐶
cvmliftlem.x	⊢ 𝑋 = ∪ 𝐽
cvmliftlem.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftlem.g	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmliftlem.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
cvmliftlem.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
cvmliftlem.t	⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
cvmliftlem.a	⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1^st ‘(𝑇‘𝑘)))
cvmliftlem.l	⊢ 𝐿 = (topGen‘ran (,))
cvmliftlem.q	⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
cvmliftlem.k	⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)
cvmliftlem10.1	⊢ (𝜒 ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))

**Assertion**
Ref	Expression
cvmliftlem10	⊢ (𝜑 → (𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))

Distinct variable groups: 𝑣,𝑏,𝑧,𝐵 𝑗,𝑏,𝑘,𝑚,𝑛,𝑠,𝑢,𝑥,𝐹,𝑣,𝑧 𝑛,𝐿,𝑧 𝑃,𝑏,𝑘,𝑚,𝑛,𝑢,𝑣,𝑥,𝑧 𝐶,𝑏,𝑗,𝑘,𝑛,𝑠,𝑢,𝑣,𝑧 𝜑,𝑗,𝑛,𝑠,𝑥,𝑧 𝑁,𝑏,𝑘,𝑚,𝑛,𝑢,𝑣,𝑥,𝑧 𝑆,𝑏,𝑗,𝑘,𝑛,𝑠,𝑢,𝑣,𝑥,𝑧 𝑗,𝑋 𝐺,𝑏,𝑗,𝑘,𝑚,𝑛,𝑠,𝑢,𝑣,𝑥,𝑧 𝑇,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧 𝐽,𝑏,𝑗,𝑘,𝑛,𝑠,𝑢,𝑣,𝑥,𝑧 𝑄,𝑏,𝑘,𝑚,𝑛,𝑢,𝑣,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑣,𝑢,𝑘,𝑚,𝑏) 𝜒(𝑥,𝑧,𝑣,𝑢,𝑗,𝑘,𝑚,𝑛,𝑠,𝑏) 𝐵(𝑥,𝑢,𝑗,𝑘,𝑚,𝑛,𝑠) 𝐶(𝑥,𝑚) 𝑃(𝑗,𝑠) 𝑄(𝑗,𝑠) 𝑆(𝑚) 𝑇(𝑛) 𝐽(𝑚) 𝐾(𝑥,𝑧,𝑣,𝑢,𝑗,𝑘,𝑚,𝑛,𝑠,𝑏) 𝐿(𝑥,𝑣,𝑢,𝑗,𝑘,𝑚,𝑠,𝑏) 𝑁(𝑗,𝑠) 𝑋(𝑥,𝑧,𝑣,𝑢,𝑘,𝑚,𝑛,𝑠,𝑏)

Proof of Theorem cvmliftlem10 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftlem.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		nnuz 11723	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	syl6eleq 2711	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		eluzfz2 12349	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
6		eleq1 2689	. . . . . 6 ⊢ (𝑦 = 1 → (𝑦 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
7		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (1...𝑦) = (1...1))
8		1z 11407	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
9		fzsn 12383	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → (1...1) = {1})
10	8, 9	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...1) = {1}
11	7, 10	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (1...𝑦) = {1})
12	11	iuneq1d 4545	. . . . . . . . 9 ⊢ (𝑦 = 1 → ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) = ∪ 𝑘 ∈ {1} (𝑄‘𝑘))
13		1ex 10035	. . . . . . . . . 10 ⊢ 1 ∈ V
14		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (𝑄‘𝑘) = (𝑄‘1))
15	13, 14	iunxsn 4603	. . . . . . . . 9 ⊢ ∪ 𝑘 ∈ {1} (𝑄‘𝑘) = (𝑄‘1)
16	12, 15	syl6eq 2672	. . . . . . . 8 ⊢ (𝑦 = 1 → ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) = (𝑄‘1))
17		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (𝑦 / 𝑁) = (1 / 𝑁))
18	17	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (0[,](𝑦 / 𝑁)) = (0[,](1 / 𝑁)))
19	18	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑦 = 1 → (𝐿 ↾_t (0[,](𝑦 / 𝑁))) = (𝐿 ↾_t (0[,](1 / 𝑁))))
20	19	oveq1d 6665	. . . . . . . 8 ⊢ (𝑦 = 1 → ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶))
21	16, 20	eleq12d 2695	. . . . . . 7 ⊢ (𝑦 = 1 → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ (𝑄‘1) ∈ ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶)))
22	16	coeq2d 5284	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐹 ∘ (𝑄‘1)))
23	18	reseq2d 5396	. . . . . . . 8 ⊢ (𝑦 = 1 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](1 / 𝑁))))
24	22, 23	eqeq12d 2637	. . . . . . 7 ⊢ (𝑦 = 1 → ((𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁)))))
25	21, 24	anbi12d 747	. . . . . 6 ⊢ (𝑦 = 1 → ((∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))) ↔ ((𝑄‘1) ∈ ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))))
26	6, 25	imbi12d 334	. . . . 5 ⊢ (𝑦 = 1 → ((𝑦 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))))) ↔ (1 ∈ (1...𝑁) → ((𝑄‘1) ∈ ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁)))))))
27	26	imbi2d 330	. . . 4 ⊢ (𝑦 = 1 → ((𝜑 → (𝑦 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))))) ↔ (𝜑 → (1 ∈ (1...𝑁) → ((𝑄‘1) ∈ ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))))))
28		eleq1 2689	. . . . . 6 ⊢ (𝑦 = 𝑛 → (𝑦 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
29		oveq2 6658	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (1...𝑦) = (1...𝑛))
30	29	iuneq1d 4545	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) = ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘))
31		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
32	31	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (0[,](𝑦 / 𝑁)) = (0[,](𝑛 / 𝑁)))
33	32	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝐿 ↾_t (0[,](𝑦 / 𝑁))) = (𝐿 ↾_t (0[,](𝑛 / 𝑁))))
34	33	oveq1d 6665	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶))
35	30, 34	eleq12d 2695	. . . . . . 7 ⊢ (𝑦 = 𝑛 → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶)))
36	30	coeq2d 5284	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)))
37	32	reseq2d 5396	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))
38	36, 37	eqeq12d 2637	. . . . . . 7 ⊢ (𝑦 = 𝑛 → ((𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
39	35, 38	anbi12d 747	. . . . . 6 ⊢ (𝑦 = 𝑛 → ((∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))) ↔ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))
40	28, 39	imbi12d 334	. . . . 5 ⊢ (𝑦 = 𝑛 → ((𝑦 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))))) ↔ (𝑛 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))))
41	40	imbi2d 330	. . . 4 ⊢ (𝑦 = 𝑛 → ((𝜑 → (𝑦 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))))) ↔ (𝜑 → (𝑛 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))))
42		eleq1 2689	. . . . . 6 ⊢ (𝑦 = (𝑛 + 1) → (𝑦 ∈ (1...𝑁) ↔ (𝑛 + 1) ∈ (1...𝑁)))
43		oveq2 6658	. . . . . . . . 9 ⊢ (𝑦 = (𝑛 + 1) → (1...𝑦) = (1...(𝑛 + 1)))
44	43	iuneq1d 4545	. . . . . . . 8 ⊢ (𝑦 = (𝑛 + 1) → ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) = ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘))
45		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
46	45	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑦 = (𝑛 + 1) → (0[,](𝑦 / 𝑁)) = (0[,]((𝑛 + 1) / 𝑁)))
47	46	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑦 = (𝑛 + 1) → (𝐿 ↾_t (0[,](𝑦 / 𝑁))) = (𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))))
48	47	oveq1d 6665	. . . . . . . 8 ⊢ (𝑦 = (𝑛 + 1) → ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
49	44, 48	eleq12d 2695	. . . . . . 7 ⊢ (𝑦 = (𝑛 + 1) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶)))
50	44	coeq2d 5284	. . . . . . . 8 ⊢ (𝑦 = (𝑛 + 1) → (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)))
51	46	reseq2d 5396	. . . . . . . 8 ⊢ (𝑦 = (𝑛 + 1) → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))
52	50, 51	eqeq12d 2637	. . . . . . 7 ⊢ (𝑦 = (𝑛 + 1) → ((𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))
53	49, 52	anbi12d 747	. . . . . 6 ⊢ (𝑦 = (𝑛 + 1) → ((∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))) ↔ (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))
54	42, 53	imbi12d 334	. . . . 5 ⊢ (𝑦 = (𝑛 + 1) → ((𝑦 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))))) ↔ ((𝑛 + 1) ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))))
55	54	imbi2d 330	. . . 4 ⊢ (𝑦 = (𝑛 + 1) → ((𝜑 → (𝑦 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))))) ↔ (𝜑 → ((𝑛 + 1) ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))))
56		eleq1 2689	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑦 ∈ (1...𝑁) ↔ 𝑁 ∈ (1...𝑁)))
57		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑦 = 𝑁 → (1...𝑦) = (1...𝑁))
58	57	iuneq1d 4545	. . . . . . . . 9 ⊢ (𝑦 = 𝑁 → ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘))
59		cvmliftlem.k	. . . . . . . . 9 ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)
60	58, 59	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) = 𝐾)
61		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑁 → (𝑦 / 𝑁) = (𝑁 / 𝑁))
62	61	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑦 = 𝑁 → (0[,](𝑦 / 𝑁)) = (0[,](𝑁 / 𝑁)))
63	62	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑦 = 𝑁 → (𝐿 ↾_t (0[,](𝑦 / 𝑁))) = (𝐿 ↾_t (0[,](𝑁 / 𝑁))))
64	63	oveq1d 6665	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶))
65	60, 64	eleq12d 2695	. . . . . . 7 ⊢ (𝑦 = 𝑁 → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ 𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶)))
66	60	coeq2d 5284	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐹 ∘ 𝐾))
67	62	reseq2d 5396	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))
68	66, 67	eqeq12d 2637	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))
69	65, 68	anbi12d 747	. . . . . 6 ⊢ (𝑦 = 𝑁 → ((∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))) ↔ (𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))))
70	56, 69	imbi12d 334	. . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑦 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))))) ↔ (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))))
71	70	imbi2d 330	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝜑 → (𝑦 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑦)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))))) ↔ (𝜑 → (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))))))
72		eluzfz1 12348	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
73	3, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ (1...𝑁))
74		cvmliftlem.1	. . . . . . . . 9 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
75		cvmliftlem.b	. . . . . . . . 9 ⊢ 𝐵 = ∪ 𝐶
76		cvmliftlem.x	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
77		cvmliftlem.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
78		cvmliftlem.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
79		cvmliftlem.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
80		cvmliftlem.e	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
81		cvmliftlem.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
82		cvmliftlem.a	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1^st ‘(𝑇‘𝑘)))
83		cvmliftlem.l	. . . . . . . . 9 ⊢ 𝐿 = (topGen‘ran (,))
84		cvmliftlem.q	. . . . . . . . 9 ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
85		eqid 2622	. . . . . . . . 9 ⊢ (((1 − 1) / 𝑁)[,](1 / 𝑁)) = (((1 − 1) / 𝑁)[,](1 / 𝑁))
86	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85	cvmliftlem8 31274	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 ∈ (1...𝑁)) → (𝑄‘1) ∈ ((𝐿 ↾_t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶))
87	73, 86	mpdan 702	. . . . . . 7 ⊢ (𝜑 → (𝑄‘1) ∈ ((𝐿 ↾_t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶))
88		1m1e0 11089	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
89	88	oveq1i 6660	. . . . . . . . . . 11 ⊢ ((1 − 1) / 𝑁) = (0 / 𝑁)
90	1	nncnd 11036	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
91	1	nnne0d 11065	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ≠ 0)
92	90, 91	div0d 10800	. . . . . . . . . . 11 ⊢ (𝜑 → (0 / 𝑁) = 0)
93	89, 92	syl5eq 2668	. . . . . . . . . 10 ⊢ (𝜑 → ((1 − 1) / 𝑁) = 0)
94	93	oveq1d 6665	. . . . . . . . 9 ⊢ (𝜑 → (((1 − 1) / 𝑁)[,](1 / 𝑁)) = (0[,](1 / 𝑁)))
95	94	oveq2d 6666	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ↾_t (((1 − 1) / 𝑁)[,](1 / 𝑁))) = (𝐿 ↾_t (0[,](1 / 𝑁))))
96	95	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → ((𝐿 ↾_t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶) = ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶))
97	87, 96	eleqtrd 2703	. . . . . 6 ⊢ (𝜑 → (𝑄‘1) ∈ ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶))
98		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ∈ (1...𝑁)) → 1 ∈ (1...𝑁))
99	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85	cvmliftlem7 31273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ∈ (1...𝑁)) → ((𝑄‘(1 − 1))‘((1 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((1 − 1) / 𝑁))}))
100	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85, 98, 99	cvmliftlem6 31272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ∈ (1...𝑁)) → ((𝑄‘1):(((1 − 1) / 𝑁)[,](1 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁)))))
101	73, 100	mpdan 702	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘1):(((1 − 1) / 𝑁)[,](1 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁)))))
102	101	simprd 479	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁))))
103	94	reseq2d 5396	. . . . . . 7 ⊢ (𝜑 → (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁))) = (𝐺 ↾ (0[,](1 / 𝑁))))
104	102, 103	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))
105	97, 104	jca 554	. . . . 5 ⊢ (𝜑 → ((𝑄‘1) ∈ ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁)))))
106	105	a1d 25	. . . 4 ⊢ (𝜑 → (1 ∈ (1...𝑁) → ((𝑄‘1) ∈ ((𝐿 ↾_t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))))
107		elnnuz 11724	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
108	107	biimpi 206	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘1))
109	108	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
110		peano2fzr 12354	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘1) ∧ (𝑛 + 1) ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
111	110	ex 450	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘1) → ((𝑛 + 1) ∈ (1...𝑁) → 𝑛 ∈ (1...𝑁)))
112	109, 111	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) ∈ (1...𝑁) → 𝑛 ∈ (1...𝑁)))
113	112	imim1d 82	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))) → ((𝑛 + 1) ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))))
114		cvmliftlem10.1	. . . . . . . . . . 11 ⊢ (𝜒 ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))
115		eqid 2622	. . . . . . . . . . . . 13 ⊢ ∪ (𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) = ∪ (𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁)))
116		0re 10040	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
117	114	simplbi 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)))
118	117	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)))
119	118	simprd 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 + 1) ∈ (1...𝑁))
120		elfznn 12370	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 + 1) ∈ (1...𝑁) → (𝑛 + 1) ∈ ℕ)
121	119, 120	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 + 1) ∈ ℕ)
122	121	nnred 11035	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 + 1) ∈ ℝ)
123	1	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → 𝑁 ∈ ℕ)
124	122, 123	nndivred 11069	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
125		iccssre 12255	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ) → (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ)
126	116, 124, 125	sylancr 695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ)
127	117	simpld 475	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝑛 ∈ ℕ)
128	127	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → 𝑛 ∈ ℕ)
129	128	nnred 11035	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → 𝑛 ∈ ℝ)
130	129, 123	nndivred 11069	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) ∈ ℝ)
131		icccld 22570	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ (𝑛 / 𝑁) ∈ ℝ) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
132	116, 130, 131	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
133	83	fveq2i 6194	. . . . . . . . . . . . . . 15 ⊢ (Clsd‘𝐿) = (Clsd‘(topGen‘ran (,)))
134	132, 133	syl6eleqr 2712	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘𝐿))
135		ssun1 3776	. . . . . . . . . . . . . . 15 ⊢ (0[,](𝑛 / 𝑁)) ⊆ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
136	116	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → 0 ∈ ℝ)
137	128	nnnn0d 11351	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → 𝑛 ∈ ℕ₀)
138	137	nn0ge0d 11354	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → 0 ≤ 𝑛)
139	123	nnred 11035	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → 𝑁 ∈ ℝ)
140	123	nngt0d 11064	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → 0 < 𝑁)
141		divge0 10892	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ (𝑛 / 𝑁))
142	129, 138, 139, 140, 141	syl22anc 1327	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → 0 ≤ (𝑛 / 𝑁))
143	129	ltp1d 10954	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → 𝑛 < (𝑛 + 1))
144		ltdiv1 10887	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
145	129, 122, 139, 140, 144	syl112anc 1330	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
146	143, 145	mpbid 222	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁))
147	130, 124, 146	ltled 10185	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
148		elicc2 12238	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ) → ((𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)) ↔ ((𝑛 / 𝑁) ∈ ℝ ∧ 0 ≤ (𝑛 / 𝑁) ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))))
149	116, 124, 148	sylancr 695	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)) ↔ ((𝑛 / 𝑁) ∈ ℝ ∧ 0 ≤ (𝑛 / 𝑁) ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))))
150	130, 142, 147, 149	mpbir3and 1245	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)))
151		iccsplit 12305	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ ∧ (𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁))) → (0[,]((𝑛 + 1) / 𝑁)) = ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
152	136, 124, 150, 151	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → (0[,]((𝑛 + 1) / 𝑁)) = ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
153	135, 152	syl5sseqr 3654	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (0[,](𝑛 / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)))
154		uniretop 22566	. . . . . . . . . . . . . . . 16 ⊢ ℝ = ∪ (topGen‘ran (,))
155	83	unieqi 4445	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝐿 = ∪ (topGen‘ran (,))
156	154, 155	eqtr4i 2647	. . . . . . . . . . . . . . 15 ⊢ ℝ = ∪ 𝐿
157	156	restcldi 20977	. . . . . . . . . . . . . 14 ⊢ (((0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ ∧ (0[,](𝑛 / 𝑁)) ∈ (Clsd‘𝐿) ∧ (0[,](𝑛 / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁))) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁)))))
158	126, 134, 153, 157	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁)))))
159		icccld 22570	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 / 𝑁) ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
160	130, 124, 159	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
161	160, 133	syl6eleqr 2712	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘𝐿))
162		ssun2 3777	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
163	162, 152	syl5sseqr 3654	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)))
164	156	restcldi 20977	. . . . . . . . . . . . . 14 ⊢ (((0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ ∧ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘𝐿) ∧ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁))) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁)))))
165	126, 161, 163, 164	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁)))))
166		retop 22565	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) ∈ Top
167	83, 166	eqeltri 2697	. . . . . . . . . . . . . . 15 ⊢ 𝐿 ∈ Top
168	156	restuni 20966	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ Top ∧ (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ) → (0[,]((𝑛 + 1) / 𝑁)) = ∪ (𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))))
169	167, 126, 168	sylancr 695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (0[,]((𝑛 + 1) / 𝑁)) = ∪ (𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))))
170	152, 169	eqtr3d 2658	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = ∪ (𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))))
171	114	simprbi 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
172	171	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
173	172	simpld 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶))
174		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ (𝐿 ↾_t (0[,](𝑛 / 𝑁))) = ∪ (𝐿 ↾_t (0[,](𝑛 / 𝑁)))
175	174, 75	cnf 21050	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) → ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):∪ (𝐿 ↾_t (0[,](𝑛 / 𝑁)))⟶𝐵)
176	173, 175	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):∪ (𝐿 ↾_t (0[,](𝑛 / 𝑁)))⟶𝐵)
177		iccssre 12255	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ ∧ (𝑛 / 𝑁) ∈ ℝ) → (0[,](𝑛 / 𝑁)) ⊆ ℝ)
178	116, 130, 177	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (0[,](𝑛 / 𝑁)) ⊆ ℝ)
179	156	restuni 20966	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ Top ∧ (0[,](𝑛 / 𝑁)) ⊆ ℝ) → (0[,](𝑛 / 𝑁)) = ∪ (𝐿 ↾_t (0[,](𝑛 / 𝑁))))
180	167, 178, 179	sylancr 695	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → (0[,](𝑛 / 𝑁)) = ∪ (𝐿 ↾_t (0[,](𝑛 / 𝑁))))
181	180	feq2d 6031	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 ↔ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):∪ (𝐿 ↾_t (0[,](𝑛 / 𝑁)))⟶𝐵))
182	176, 181	mpbird 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):(0[,](𝑛 / 𝑁))⟶𝐵)
183		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))
184		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → (𝑛 + 1) ∈ (1...𝑁))
185	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 183	cvmliftlem7 31273	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))}))
186	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 183, 184, 185	cvmliftlem6 31272	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
187	119, 186	syldan 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
188	187	simpld 475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → (𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
189	128	nncnd 11036	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝜒) → 𝑛 ∈ ℂ)
190		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℂ
191		pncan 10287	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
192	189, 190, 191	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 + 1) − 1) = 𝑛)
193	192	oveq1d 6665	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
194	193	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
195	194	feq2d 6031	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ↔ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵))
196	188, 195	mpbid 222	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
197		ffun 6048	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 → Fun ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘))
198	182, 197	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝜒) → Fun ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘))
199	128, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝜒) → 𝑛 ∈ (ℤ_≥‘1))
200		eluzfz2 12349	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (ℤ_≥‘1) → 𝑛 ∈ (1...𝑛))
201	199, 200	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝜒) → 𝑛 ∈ (1...𝑛))
202		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 𝑛 → (𝑄‘𝑘) = (𝑄‘𝑛))
203	202	ssiun2s 4564	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (1...𝑛) → (𝑄‘𝑛) ⊆ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘))
204	201, 203	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝜒) → (𝑄‘𝑛) ⊆ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘))
205		peano2rem 10348	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
206	129, 205	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 − 1) ∈ ℝ)
207	206, 123	nndivred 11069	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 − 1) / 𝑁) ∈ ℝ)
208	207	rexrd 10089	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 − 1) / 𝑁) ∈ ℝ^*)
209	130	rexrd 10089	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) ∈ ℝ^*)
210	129	ltm1d 10956	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 − 1) < 𝑛)
211		ltdiv1 10887	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑛 − 1) < 𝑛 ↔ ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁)))
212	206, 129, 139, 140, 211	syl112anc 1330	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 − 1) < 𝑛 ↔ ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁)))
213	210, 212	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁))
214	207, 130, 213	ltled 10185	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 − 1) / 𝑁) ≤ (𝑛 / 𝑁))
215		ubicc2 12289	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑛 − 1) / 𝑁) ∈ ℝ^* ∧ (𝑛 / 𝑁) ∈ ℝ^* ∧ ((𝑛 − 1) / 𝑁) ≤ (𝑛 / 𝑁)) → (𝑛 / 𝑁) ∈ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
216	208, 209, 214, 215	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) ∈ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
217	199, 119, 110	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝜒) → 𝑛 ∈ (1...𝑁))
218		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))
219		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
220	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 218	cvmliftlem7 31273	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝑄‘(𝑛 − 1))‘((𝑛 − 1) / 𝑁)) ∈ (^◡𝐹 “ {(𝐺‘((𝑛 − 1) / 𝑁))}))
221	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 218, 219, 220	cvmliftlem6 31272	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝑄‘𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘𝑛)) = (𝐺 ↾ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))))
222	217, 221	syldan 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘𝑛)) = (𝐺 ↾ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))))
223	222	simpld 475	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝜒) → (𝑄‘𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵)
224		fdm 6051	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑄‘𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵 → dom (𝑄‘𝑛) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
225	223, 224	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝜒) → dom (𝑄‘𝑛) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
226	216, 225	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) ∈ dom (𝑄‘𝑛))
227		funssfv 6209	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∧ (𝑄‘𝑛) ⊆ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∧ (𝑛 / 𝑁) ∈ dom (𝑄‘𝑛)) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)‘(𝑛 / 𝑁)) = ((𝑄‘𝑛)‘(𝑛 / 𝑁)))
228	198, 204, 226, 227	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)‘(𝑛 / 𝑁)) = ((𝑄‘𝑛)‘(𝑛 / 𝑁)))
229	192	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝜒) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄‘𝑛))
230	229, 193	fveq12d 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘𝑛)‘(𝑛 / 𝑁)))
231	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84	cvmliftlem9 31275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)))
232	119, 231	syldan 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)))
233	193	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
234	232, 233	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
235	228, 230, 234	3eqtr2d 2662	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)‘(𝑛 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
236	235	opeq2d 4409	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → ⟨(𝑛 / 𝑁), (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)‘(𝑛 / 𝑁))⟩ = ⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩)
237	236	sneqd 4189	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → {⟨(𝑛 / 𝑁), (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)‘(𝑛 / 𝑁))⟩} = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
238		ffn 6045	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 → ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) Fn (0[,](𝑛 / 𝑁)))
239	182, 238	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) Fn (0[,](𝑛 / 𝑁)))
240		0xr 10086	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ^*
241	240	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜒) → 0 ∈ ℝ^*)
242		ubicc2 12289	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ^* ∧ (𝑛 / 𝑁) ∈ ℝ^* ∧ 0 ≤ (𝑛 / 𝑁)) → (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁)))
243	241, 209, 142, 242	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁)))
244		fnressn 6425	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) Fn (0[,](𝑛 / 𝑁)) ∧ (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁))) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)‘(𝑛 / 𝑁))⟩})
245	239, 243, 244	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)‘(𝑛 / 𝑁))⟩})
246		ffn 6045	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 → (𝑄‘(𝑛 + 1)) Fn ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
247	196, 246	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (𝑄‘(𝑛 + 1)) Fn ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
248	124	rexrd 10089	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 + 1) / 𝑁) ∈ ℝ^*)
249		lbicc2 12288	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 / 𝑁) ∈ ℝ^* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ^* ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁)) → (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
250	209, 248, 147, 249	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
251		fnressn 6425	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑄‘(𝑛 + 1)) Fn ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∧ (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) → ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
252	247, 250, 251	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
253	237, 245, 252	3eqtr4d 2666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ {(𝑛 / 𝑁)}) = ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}))
254		df-icc 12182	. . . . . . . . . . . . . . . . . . . . 21 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
255		xrmaxle 12014	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ^* ∧ (𝑛 / 𝑁) ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0) ≤ 𝑧 ↔ (0 ≤ 𝑧 ∧ (𝑛 / 𝑁) ≤ 𝑧)))
256		xrlemin 12015	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝑛 / 𝑁) ∈ ℝ^* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ^*) → (𝑧 ≤ if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁)) ↔ (𝑧 ≤ (𝑛 / 𝑁) ∧ 𝑧 ≤ ((𝑛 + 1) / 𝑁))))
257	254, 255, 256	ixxin 12192	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0 ∈ ℝ^* ∧ (𝑛 / 𝑁) ∈ ℝ^) ∧ ((𝑛 / 𝑁) ∈ ℝ^ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ^*)) → ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0)[,]if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁))))
258	241, 209, 209, 248, 257	syl22anc 1327	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0)[,]if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁))))
259	142	iftrued 4094	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜒) → if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0) = (𝑛 / 𝑁))
260	147	iftrued 4094	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜒) → if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁)) = (𝑛 / 𝑁))
261	259, 260	oveq12d 6668	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0)[,]if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁))) = ((𝑛 / 𝑁)[,](𝑛 / 𝑁)))
262		iccid 12220	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 / 𝑁) ∈ ℝ^* → ((𝑛 / 𝑁)[,](𝑛 / 𝑁)) = {(𝑛 / 𝑁)})
263	209, 262	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜒) → ((𝑛 / 𝑁)[,](𝑛 / 𝑁)) = {(𝑛 / 𝑁)})
264	258, 261, 263	3eqtrd 2660	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = {(𝑛 / 𝑁)})
265	264	reseq2d 5396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ {(𝑛 / 𝑁)}))
266	264	reseq2d 5396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}))
267	253, 265, 266	3eqtr4d 2666	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
268		fresaun 6075	. . . . . . . . . . . . . . . 16 ⊢ ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
269	182, 196, 267, 268	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
270		fzsuc 12388	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (ℤ_≥‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
271	199, 270	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜒) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
272	271	iuneq1d 4545	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜒) → ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) = ∪ 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄‘𝑘))
273		iunxun 4605	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄‘𝑘) = (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄‘𝑘))
274		ovex 6678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 + 1) ∈ V
275		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (𝑛 + 1) → (𝑄‘𝑘) = (𝑄‘(𝑛 + 1)))
276	274, 275	iunxsn 4603	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄‘𝑘) = (𝑄‘(𝑛 + 1))
277	276	uneq2i 3764	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄‘𝑘)) = (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1)))
278	273, 277	eqtri 2644	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄‘𝑘) = (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1)))
279	272, 278	syl6req 2673	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))) = ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘))
280	279	feq1d 6030	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵 ↔ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵))
281	269, 280	mpbid 222	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
282	170	feq2d 6031	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵 ↔ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘):∪ (𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁)))⟶𝐵))
283	281, 282	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘):∪ (𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
284	279	reseq1d 5395	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ↾ (0[,](𝑛 / 𝑁))))
285		fresaunres1 6077	. . . . . . . . . . . . . . . 16 ⊢ ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘))
286	182, 196, 267, 285	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘))
287	284, 286	eqtr3d 2658	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ↾ (0[,](𝑛 / 𝑁))) = ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘))
288	167	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → 𝐿 ∈ Top)
289		ovex 6678	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]((𝑛 + 1) / 𝑁)) ∈ V
290	289	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (0[,]((𝑛 + 1) / 𝑁)) ∈ V)
291		restabs 20969	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ Top ∧ (0[,](𝑛 / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)) ∧ (0[,]((𝑛 + 1) / 𝑁)) ∈ V) → ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) ↾_t (0[,](𝑛 / 𝑁))) = (𝐿 ↾_t (0[,](𝑛 / 𝑁))))
292	288, 153, 290, 291	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) ↾_t (0[,](𝑛 / 𝑁))) = (𝐿 ↾_t (0[,](𝑛 / 𝑁))))
293	292	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) = ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶))
294	173, 287, 293	3eltr4d 2716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ↾ (0[,](𝑛 / 𝑁))) ∈ (((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶))
295	74, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 183	cvmliftlem8 31274	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿 ↾_t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
296	119, 295	syldan 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿 ↾_t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
297	194	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (𝐿 ↾_t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿 ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
298	297	oveq1d 6665	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((𝐿 ↾_t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) = ((𝐿 ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
299	296, 298	eleqtrd 2703	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿 ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
300	279	reseq1d 5395	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
301		fresaunres2 6076	. . . . . . . . . . . . . . . 16 ⊢ ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
302	182, 196, 267, 301	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
303	300, 302	eqtr3d 2658	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
304		restabs 20969	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ Top ∧ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)) ∧ (0[,]((𝑛 + 1) / 𝑁)) ∈ V) → ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿 ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
305	288, 163, 290, 304	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿 ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
306	305	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → (((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) = ((𝐿 ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
307	299, 303, 306	3eltr4d 2716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) ∈ (((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) ↾_t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
308	115, 75, 158, 165, 170, 283, 294, 307	paste 21098	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜒) → ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
309	152	reseq2d 5396	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
310	172	simprd 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))
311	187	simprd 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))))
312	194	reseq2d 5396	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜒) → (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
313	311, 312	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜒) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
314	310, 313	uneq12d 3768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜒) → ((𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) ∪ (𝐹 ∘ (𝑄‘(𝑛 + 1)))) = ((𝐺 ↾ (0[,](𝑛 / 𝑁))) ∪ (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
315		coundi 5636	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∘ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1)))) = ((𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) ∪ (𝐹 ∘ (𝑄‘(𝑛 + 1))))
316		resundi 5410	. . . . . . . . . . . . . 14 ⊢ (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝐺 ↾ (0[,](𝑛 / 𝑁))) ∪ (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
317	314, 315, 316	3eqtr4g 2681	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → (𝐹 ∘ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1)))) = (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
318	279	coeq2d 5284	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜒) → (𝐹 ∘ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∪ (𝑄‘(𝑛 + 1)))) = (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)))
319	309, 317, 318	3eqtr2rd 2663	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜒) → (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))
320	308, 319	jca 554	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜒) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))
321	114, 320	sylan2br 493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))
322	321	expr 643	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁))) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))
323	322	expr 643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) ∈ (1...𝑁) → ((∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))))
324	323	a2d 29	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑛 + 1) ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))) → ((𝑛 + 1) ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))))
325	113, 324	syld 47	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))) → ((𝑛 + 1) ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))))
326	325	expcom 451	. . . . 5 ⊢ (𝑛 ∈ ℕ → (𝜑 → ((𝑛 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))) → ((𝑛 + 1) ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))))
327	326	a2d 29	. . . 4 ⊢ (𝑛 ∈ ℕ → ((𝜑 → (𝑛 ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))) → (𝜑 → ((𝑛 + 1) ∈ (1...𝑁) → (∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...(𝑛 + 1))(𝑄‘𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))))
328	27, 41, 55, 71, 106, 327	nnind 11038	. . 3 ⊢ (𝑁 ∈ ℕ → (𝜑 → (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))))
329	1, 328	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))))
330	5, 329	mpd 15	1 ⊢ (𝜑 → (𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ifcif 4086 𝒫 cpw 4158 {csn 4177 ⟨cop 4183 ∪ cuni 4436 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ∘ ccom 5118 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 ↦ cmpt2 6652 1^st c1st 7166 2^nd c2nd 7167 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℤcz 11377 ℤ_≥cuz 11687 (,)cioo 12175 [,]cicc 12178 ...cfz 12326 seqcseq 12801 ↾_t crest 16081 topGenctg 16098 Topctop 20698 Clsdccld 20820 Cn ccn 21028 Homeochmeo 21556 IIcii 22678 CovMap ccvm 31237

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-rest 16083 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-cn 21031 df-hmeo 21558 df-ii 22680 df-cvm 31238

This theorem is referenced by: cvmliftlem11 31277

W3C validator