Theorem cvmliftlem8 35490

Description: Lemma for cvmlift 35497. The functions 𝑄 are continuous functions because they are defined as ^◡(𝐹 ↾ 𝐼) ∘ 𝐺 where 𝐺 is continuous and (𝐹 ↾ 𝐼) is a homeomorphism. (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))

Assertion

Ref	Expression
cvmliftlem8	⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑄‘𝑀) ∈ ((𝐿 ↾t 𝑊) Cn 𝐶))

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   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)   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 cvmliftlem8

Step	Hyp	Ref	Expression
1		elfznn 13498	. . 3 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
2		cvmliftlem.1	. . . 4 ⊢ 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))))})
3		cvmliftlem.b	. . . 4 ⊢ 𝐵 = ∪ 𝐶
4		cvmliftlem.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
5		cvmliftlem.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
6		cvmliftlem.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
7		cvmliftlem.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
8		cvmliftlem.e	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
9		cvmliftlem.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
10		cvmliftlem.t	. . . 4 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
11		cvmliftlem.a	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))
12		cvmliftlem.l	. . . 4 ⊢ 𝐿 = (topGen‘ran (,))
13		cvmliftlem.q	. . . 4 ⊢ 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⟩}⟩}))
14		cvmliftlem5.3	. . . 4 ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))
15	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cvmliftlem5 35487	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
16	1, 15	sylan2 594	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
17	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
18		cvmtop1 35458	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
19		cnrest2r 23262	. . . 4 ⊢ (𝐶 ∈ Top → ((𝐿 ↾t 𝑊) Cn (𝐶 ↾t (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))) ⊆ ((𝐿 ↾t 𝑊) Cn 𝐶))
20	17, 18, 19	3syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝐿 ↾t 𝑊) Cn (𝐶 ↾t (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))) ⊆ ((𝐿 ↾t 𝑊) Cn 𝐶))
21		retopon 24738	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
22	12, 21	eqeltri 2833	. . . . 5 ⊢ 𝐿 ∈ (TopOn‘ℝ)
23		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
24	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14	cvmliftlem2 35484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ (0[,]1))
25		unitssre 13443	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
26	24, 25	sstrdi 3935	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ ℝ)
27		resttopon 23136	. . . . 5 ⊢ ((𝐿 ∈ (TopOn‘ℝ) ∧ 𝑊 ⊆ ℝ) → (𝐿 ↾t 𝑊) ∈ (TopOn‘𝑊))
28	22, 26, 27	sylancr 588	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐿 ↾t 𝑊) ∈ (TopOn‘𝑊))
29		eqid 2737	. . . . . . 7 ⊢ (II ↾t 𝑊) = (II ↾t 𝑊)
30		iitopon 24856	. . . . . . . 8 ⊢ II ∈ (TopOn‘(0[,]1))
31	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → II ∈ (TopOn‘(0[,]1)))
32	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺 ∈ (II Cn 𝐽))
33		iiuni 24858	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
34	33, 4	cnf 23221	. . . . . . . . . 10 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
35	32, 34	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺:(0[,]1)⟶𝑋)
36	35	feqmptd 6902	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺 = (𝑧 ∈ (0[,]1) ↦ (𝐺‘𝑧)))
37	36, 32	eqeltrrd 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ (0[,]1) ↦ (𝐺‘𝑧)) ∈ (II Cn 𝐽))
38	29, 31, 24, 37	cnmpt1res 23651	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((II ↾t 𝑊) Cn 𝐽))
39		dfii2 24859	. . . . . . . . . 10 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
40	12	oveq1i 7370	. . . . . . . . . 10 ⊢ (𝐿 ↾t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1))
41	39, 40	eqtr4i 2763	. . . . . . . . 9 ⊢ II = (𝐿 ↾t (0[,]1))
42	41	oveq1i 7370	. . . . . . . 8 ⊢ (II ↾t 𝑊) = ((𝐿 ↾t (0[,]1)) ↾t 𝑊)
43		retop 24736	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
44	12, 43	eqeltri 2833	. . . . . . . . . 10 ⊢ 𝐿 ∈ Top
45	44	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐿 ∈ Top)
46		ovexd 7395	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (0[,]1) ∈ V)
47		restabs 23140	. . . . . . . . 9 ⊢ ((𝐿 ∈ Top ∧ 𝑊 ⊆ (0[,]1) ∧ (0[,]1) ∈ V) → ((𝐿 ↾t (0[,]1)) ↾t 𝑊) = (𝐿 ↾t 𝑊))
48	45, 24, 46, 47	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝐿 ↾t (0[,]1)) ↾t 𝑊) = (𝐿 ↾t 𝑊))
49	42, 48	eqtrid 2784	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (II ↾t 𝑊) = (𝐿 ↾t 𝑊))
50	49	oveq1d 7375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((II ↾t 𝑊) Cn 𝐽) = ((𝐿 ↾t 𝑊) Cn 𝐽))
51	38, 50	eleqtrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn 𝐽))
52		cvmtop2 35459	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
53	17, 52	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐽 ∈ Top)
54	4	toptopon 22892	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
55	53, 54	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐽 ∈ (TopOn‘𝑋))
56		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ (1...𝑁))
57		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → 𝑧 ∈ 𝑊)
58	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56, 14, 57	cvmliftlem3 35485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
59	58	anassrs 467	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝑊) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
60	59	fmpttd 7061	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)):𝑊⟶(1st ‘(𝑇‘𝑀)))
61	60	frnd 6670	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ran (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ⊆ (1st ‘(𝑇‘𝑀)))
62	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23	cvmliftlem1 35483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
63	2	cvmsrcl 35462	. . . . . . . 8 ⊢ ((2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))) → (1st ‘(𝑇‘𝑀)) ∈ 𝐽)
64		elssuni 4882	. . . . . . . 8 ⊢ ((1st ‘(𝑇‘𝑀)) ∈ 𝐽 → (1st ‘(𝑇‘𝑀)) ⊆ ∪ 𝐽)
65	62, 63, 64	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇‘𝑀)) ⊆ ∪ 𝐽)
66	65, 4	sseqtrrdi 3964	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇‘𝑀)) ⊆ 𝑋)
67		cnrest2 23261	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ran (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ⊆ (1st ‘(𝑇‘𝑀)) ∧ (1st ‘(𝑇‘𝑀)) ⊆ 𝑋) → ((𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn 𝐽) ↔ (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn (𝐽 ↾t (1st ‘(𝑇‘𝑀))))))
68	55, 61, 66, 67	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn 𝐽) ↔ (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn (𝐽 ↾t (1st ‘(𝑇‘𝑀))))))
69	51, 68	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn (𝐽 ↾t (1st ‘(𝑇‘𝑀)))))
70	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cvmliftlem7 35489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
71		cvmcn 35460	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
72	3, 4	cnf 23221	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
73	17, 71, 72	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐹:𝐵⟶𝑋)
74		ffn 6662	. . . . . . . . . . 11 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
75		fniniseg 7006	. . . . . . . . . . 11 ⊢ (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)))))
76	73, 74, 75	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (((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)))))
77	70, 76	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁))))
78	77	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
79	77	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))
80	1	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℕ)
81	80	nnred 12180	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
82		peano2rem 11452	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
83	81, 82	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ ℝ)
84	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
85	83, 84	nndivred 12222	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
86	85	rexrd 11186	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
87	81, 84	nndivred 12222	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ)
88	87	rexrd 11186	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ*)
89	81	ltm1d 12079	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) < 𝑀)
90	84	nnred 12180	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
91	84	nngt0d 12217	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 0 < 𝑁)
92		ltdiv1 12011	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 − 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
93	83, 81, 90, 91, 92	syl112anc 1377	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
94	89, 93	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
95	85, 87, 94	ltled 11285	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
96		lbicc2 13408	. . . . . . . . . . . 12 ⊢ ((((𝑀 − 1) / 𝑁) ∈ ℝ* ∧ (𝑀 / 𝑁) ∈ ℝ* ∧ ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
97	86, 88, 95, 96	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
98	97, 14	eleqtrrdi 2848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ 𝑊)
99	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 98	cvmliftlem3 35485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇‘𝑀)))
100	79, 99	eqeltrd 2837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))
101		eqid 2737	. . . . . . . . 9 ⊢ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) = (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)
102	2, 3, 101	cvmsiota 35475	. . . . . . . 8 ⊢ ((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)))
103	17, 62, 78, 100, 102	syl13anc 1375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((℩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)))
104	103	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)))
105	2	cvmshmeo 35469	. . . . . 6 ⊢ (((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)) ∈ ((𝐶 ↾t (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽 ↾t (1st ‘(𝑇‘𝑀)))))
106	62, 104, 105	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐶 ↾t (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽 ↾t (1st ‘(𝑇‘𝑀)))))
107		hmeocnvcn 23736	. . . . 5 ⊢ ((𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐶 ↾t (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽 ↾t (1st ‘(𝑇‘𝑀)))) → ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐽 ↾t (1st ‘(𝑇‘𝑀))) Cn (𝐶 ↾t (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))))
108	106, 107	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐽 ↾t (1st ‘(𝑇‘𝑀))) Cn (𝐶 ↾t (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))))
109	28, 69, 108	cnmpt11f 23639	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) ∈ ((𝐿 ↾t 𝑊) Cn (𝐶 ↾t (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))))
110	20, 109	sseldd 3923	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) ∈ ((𝐿 ↾t 𝑊) Cn 𝐶))
111	16, 110	eqeltrd 2837	1 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑄‘𝑀) ∈ ((𝐿 ↾t 𝑊) Cn 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  {csn 4568  ⟨cop 4574  ∪ cuni 4851  ∪ ciun 4934   class class class wbr 5086   ↦ cmpt 5167   I cid 5518   × cxp 5622  ^◡ccnv 5623  ran crn 5625   ↾ cres 5626   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  ℩crio 7316  (class class class)co 7360   ∈ cmpo 7362  1^st c1st 7933  2^nd c2nd 7934  ℝcr 11028  0cc0 11029  1c1 11030  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171   − cmin 11368   / cdiv 11798  ℕcn 12165  (,)cioo 13289  [,]cicc 13292  ...cfz 13452  seqcseq 13954   ↾_t crest 17374  topGenctg 17391  Topctop 22868  TopOnctopon 22885   Cn ccn 23199  Homeochmeo 23728  IIcii 24852   CovMap ccvm 35453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fi 9317  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-ioo 13293  df-icc 13296  df-fz 13453  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-rest 17376  df-topgen 17397  df-psmet 21336  df-xmet 21337  df-met 21338  df-bl 21339  df-mopn 21340  df-top 22869  df-topon 22886  df-bases 22921  df-cn 23202  df-hmeo 23730  df-ii 24854  df-cvm 35454

This theorem is referenced by:  cvmliftlem10  35492