Theorem cvmliftlem6 35803

Description: Lemma for cvmlift 35812. Induction step for cvmliftlem7 35804. Assuming that 𝑄(𝑀 − 1) is defined at (𝑀 − 1) / 𝑁 and is a preimage of 𝐺((𝑀 − 1) / 𝑁), the next segment 𝑄(𝑀) is also defined and is a function on 𝑊 which is a lift 𝐺 for this segment. This follows explicitly from the definition 𝑄(𝑀) = ^◡(𝐹 ↾ 𝐼) ∘ 𝐺 since 𝐺 is in 1^st ‘(𝐹‘𝑀) for the entire interval so that ^◡(𝐹 ↾ 𝐼) maps this into 𝐼 and 𝐹 ∘ 𝑄 maps back to 𝐺. (Contributed by Mario Carneiro, 16-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))) ⊆ (1st ‘(𝑇‘𝑘)))
cvmliftlem.l	⊢ 𝐿 = (topGen‘ran (,))
cvmliftlem.q	⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
cvmliftlem5.3	⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))
cvmliftlem6.1	⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))
cvmliftlem6.2	⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))

Assertion

Ref	Expression
cvmliftlem6	⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘𝑀):𝑊⟶𝐵 ∧ (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊)))

Distinct variable groups:   𝑣,𝑏,𝑧,𝐵   𝑗,𝑏,𝑘,𝑚,𝑠,𝑢,𝑥,𝐹,𝑣,𝑧   𝑧,𝐿   𝑀,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧   𝑃,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧   𝐶,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑧   𝜑,𝑗,𝑠,𝑥,𝑧   𝜓,𝑧   𝑁,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧   𝑆,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑥,𝑧   𝑗,𝑋   𝐺,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧   𝑇,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧   𝐽,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑥,𝑧   𝑄,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧   𝑘,𝑊,𝑚,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑣, 𝑢, 𝑘, 𝑚, 𝑏)   𝜓(𝑥, 𝑣, 𝑢, 𝑗, 𝑘, 𝑚, 𝑠, 𝑏)   𝐵(𝑥, 𝑢, 𝑗, 𝑘, 𝑚, 𝑠)   𝐶(𝑥, 𝑚)   𝑃(𝑗, 𝑠)   𝑄(𝑗, 𝑠)   𝑆(𝑚)   𝐽(𝑚)   𝐿(𝑥, 𝑣, 𝑢, 𝑗, 𝑘, 𝑚, 𝑠, 𝑏)   𝑁(𝑗, 𝑠)   𝑊(𝑣, 𝑢, 𝑗, 𝑠, 𝑏)   𝑋(𝑥, 𝑧, 𝑣, 𝑢, 𝑘, 𝑚, 𝑠, 𝑏)

Proof of Theorem cvmliftlem6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftlem.1	. . . . . . . . . . 11 ⊢ 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))))})
2		cvmliftlem.b	. . . . . . . . . . 11 ⊢ 𝐵 = ∪ 𝐶
3		cvmliftlem.x	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ 𝐽
4		cvmliftlem.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
5		cvmliftlem.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
6		cvmliftlem.p	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
7		cvmliftlem.e	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
8		cvmliftlem.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9		cvmliftlem.t	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
10		cvmliftlem.a	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))
11		cvmliftlem.l	. . . . . . . . . . 11 ⊢ 𝐿 = (topGen‘ran (,))
12		cvmliftlem6.1	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))
13	12	adantrr 730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ (1...𝑁))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13	cvmliftlem1 35798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
15	1	cvmsss 35780	. . . . . . . . . 10 ⊢ ((2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))) → (2nd ‘(𝑇‘𝑀)) ⊆ 𝐶)
16	14, 15	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (2nd ‘(𝑇‘𝑀)) ⊆ 𝐶)
17	4	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
18		cvmliftlem6.2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
19	18	adantrr 730	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
20		cvmcn 35775	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
21	2, 3	cnf 23440	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
22	17, 20, 21	3syl 19	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝐹:𝐵⟶𝑋)
23		ffn 6712	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
24		fniniseg 7062	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐵 → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}) ↔ (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))))
25	22, 23, 24	3syl 19	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}) ↔ (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))))
26	19, 25	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁))))
27	26	simpld 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
28	26	simprd 501	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))
29		cvmliftlem5.3	. . . . . . . . . . . . 13 ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))
30		elfznn 13600	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
31	13, 30	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ ℕ)
32	31	nnred 12266	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ ℝ)
33		peano2rem 11543	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
34	32, 33	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 − 1) ∈ ℝ)
35	8	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑁 ∈ ℕ)
36	34, 35	nndivred 12308	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
37	36	rexrd 11277	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
38	32, 35	nndivred 12308	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 / 𝑁) ∈ ℝ)
39	38	rexrd 11277	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 / 𝑁) ∈ ℝ*)
40	32	ltm1d 12165	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 − 1) < 𝑀)
41	35	nnred 12266	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑁 ∈ ℝ)
42	35	nngt0d 12303	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 0 < 𝑁)
43		ltdiv1 12097	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 − 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
44	34, 32, 41, 42, 43	syl112anc 1401	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
45	40, 44	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
46	36, 38, 45	ltled 11376	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
47		lbicc2 13509	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 − 1) / 𝑁) ∈ ℝ* ∧ (𝑀 / 𝑁) ∈ ℝ* ∧ ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
48	37, 39, 46, 47	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
49	48, 29	eleqtrrdi 2877	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ 𝑊)
50	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 29, 49	cvmliftlem3 35800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇‘𝑀)))
51	28, 50	eqeltrd 2866	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))
52		eqid 2766	. . . . . . . . . . . 12 ⊢ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) = (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)
53	1, 2, 52	cvmsiota 35790	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ ((2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))) ∧ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))) → ((℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)) ∧ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)))
54	17, 14, 27, 51, 53	syl13anc 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)) ∧ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)))
55	54	simpld 500	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)))
56	16, 55	sseldd 3941	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ 𝐶)
57		elssuni 4909	. . . . . . . 8 ⊢ ((℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ 𝐶 → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ⊆ ∪ 𝐶)
58	56, 57	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ⊆ ∪ 𝐶)
59	58, 2	sseqtrrdi 3981	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ⊆ 𝐵)
60	1	cvmsf1o 35785	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))) ∧ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀))) → (𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)):(℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)–1-1-onto→(1st ‘(𝑇‘𝑀)))
61	17, 14, 55, 60	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)):(℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)–1-1-onto→(1st ‘(𝑇‘𝑀)))
62		f1ocnv 6840	. . . . . . . 8 ⊢ ((𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)):(℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)–1-1-onto→(1st ‘(𝑇‘𝑀)) → ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)):(1st ‘(𝑇‘𝑀))–1-1-onto→(℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
63		f1of 6827	. . . . . . . 8 ⊢ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)):(1st ‘(𝑇‘𝑀))–1-1-onto→(℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) → ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)):(1st ‘(𝑇‘𝑀))⟶(℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
64	61, 62, 63	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)):(1st ‘(𝑇‘𝑀))⟶(℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
65		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑧 ∈ 𝑊)
66	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 29, 65	cvmliftlem3 35800	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
67	64, 66	ffvelcdmd 7087	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) ∈ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
68	59, 67	sseldd 3941	. . . . 5 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) ∈ 𝐵)
69	68	anassrs 473	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑧 ∈ 𝑊) → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) ∈ 𝐵)
70	69	fmpttd 7117	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))):𝑊⟶𝐵)
71	12, 30	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
72		cvmliftlem.q	. . . . . 6 ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
73	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 72, 29	cvmliftlem5 35802	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
74	71, 73	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
75	74	feq1d 6694	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘𝑀):𝑊⟶𝐵 ↔ (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))):𝑊⟶𝐵))
76	70, 75	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑄‘𝑀):𝑊⟶𝐵)
77		fvres 6907	. . . . . . 7 ⊢ (𝑧 ∈ 𝑊 → ((𝐺 ↾ 𝑊)‘𝑧) = (𝐺‘𝑧))
78	65, 77	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝐺 ↾ 𝑊)‘𝑧) = (𝐺‘𝑧))
79		f1ocnvfv2 7286	. . . . . . 7 ⊢ (((𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)):(℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)–1-1-onto→(1st ‘(𝑇‘𝑀)) ∧ (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀))) → ((𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) = (𝐺‘𝑧))
80	61, 66, 79	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) = (𝐺‘𝑧))
81		fvres 6907	. . . . . . 7 ⊢ ((◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) ∈ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) → ((𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) = (𝐹‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
82	67, 81	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) = (𝐹‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
83	78, 80, 82	3eqtr2rd 2808	. . . . 5 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐹‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) = ((𝐺 ↾ 𝑊)‘𝑧))
84	83	anassrs 473	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑧 ∈ 𝑊) → (𝐹‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) = ((𝐺 ↾ 𝑊)‘𝑧))
85	84	mpteq2dva 5209	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝑊 ↦ (𝐹‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))) = (𝑧 ∈ 𝑊 ↦ ((𝐺 ↾ 𝑊)‘𝑧)))
86	4, 20, 21	3syl 19	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶𝑋)
87	86	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐹:𝐵⟶𝑋)
88	87	feqmptd 6956	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦)))
89		fveq2 6888	. . . 4 ⊢ (𝑦 = (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) → (𝐹‘𝑦) = (𝐹‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
90	69, 74, 88, 89	fmptco 7132	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐹 ∘ (𝑄‘𝑀)) = (𝑧 ∈ 𝑊 ↦ (𝐹‘(◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))))
91		iiuni 25077	. . . . . . . 8 ⊢ (0[,]1) = ∪ II
92	91, 3	cnf 23440	. . . . . . 7 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
93	5, 92	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋)
94	93	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:(0[,]1)⟶𝑋)
95	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29	cvmliftlem2 35799	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1))
96	94, 95	fssresd 6752	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 ↾ 𝑊):𝑊⟶𝑋)
97	96	feqmptd 6956	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 ↾ 𝑊) = (𝑧 ∈ 𝑊 ↦ ((𝐺 ↾ 𝑊)‘𝑧)))
98	85, 90, 97	3eqtr4d 2811	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊))
99	76, 98	jca 521	1 ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘𝑀):𝑊⟶𝐵 ∧ (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  {crab 3419  Vcvv 3458   ∖ cdif 3905   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  {csn 4594  ⟨cop 4600  ∪ cuni 4877  ∪ ciun 4961   class class class wbr 5114   ↦ cmpt 5197   I cid 5560   × cxp 5664  ^◡ccnv 5665  ran crn 5667   ↾ cres 5668   “ cima 5669   ∘ ccom 5670   Fn wfn 6538  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  ℩crio 7379  (class class class)co 7423   ∈ cmpo 7425  1^st c1st 7993  2^nd c2nd 7994  ℝcr 11117  0cc0 11118  1c1 11119  ℝ^*cxr 11260   < clt 11261   ≤ cle 11262   − cmin 11459   / cdiv 11889  ℕcn 12251  (,)cioo 13390  [,]cicc 13393  ...cfz 13553  seqcseq 14057   ↾_t crest 17498  topGenctg 17515   Cn ccn 23418  Homeochmeo 23947  IIcii 25071   CovMap ccvm 35768

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fi 9381  df-sup 9412  df-inf 9413  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-div 11890  df-nn 12252  df-2 12321  df-3 12322  df-n0 12523  df-z 12610  df-uz 12881  df-q 12991  df-rp 13035  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-icc 13397  df-fz 13554  df-seq 14058  df-exp 14118  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-rest 17500  df-topgen 17521  df-psmet 21551  df-xmet 21552  df-met 21553  df-bl 21554  df-mopn 21555  df-top 23088  df-topon 23105  df-bases 23140  df-cn 23421  df-hmeo 23949  df-ii 25073  df-cvm 35769

This theorem is used by:  cvmliftlem7  35804  cvmliftlem10  35807  cvmliftlem13  35809