Theorem cvmliftlem8 36026

Description: Lemma for cvmlift 36033. 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 13667	. . 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 36023	. . 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 605	. 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
18		cvmtop1 35994	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
19		cnrest2r 23585	. . . 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 19	. . 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 25062	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
22	12, 21	eqeltri 2857	. . . . 5 ⊢ 𝐿 ∈ (TopOn‘ℝ)
23		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
24	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14	cvmliftlem2 36020	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ (0[,]1))
25		unitssre 13611	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
26	24, 25	sstrdi 3943	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ ℝ)
27		resttopon 23459	. . . . 5 ⊢ ((𝐿 ∈ (TopOn‘ℝ) ∧ 𝑊 ⊆ ℝ) → (𝐿 ↾t 𝑊) ∈ (TopOn‘𝑊))
28	22, 26, 27	sylancr 599	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐿 ↾t 𝑊) ∈ (TopOn‘𝑊))
29		eqid 2761	. . . . . . 7 ⊢ (II ↾t 𝑊) = (II ↾t 𝑊)
30		iitopon 25180	. . . . . . . 8 ⊢ II ∈ (TopOn‘(0[,]1))
31	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → II ∈ (TopOn‘(0[,]1)))
32	6	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺 ∈ (II Cn 𝐽))
33		iiuni 25182	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
34	33, 4	cnf 23544	. . . . . . . . . 10 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
35	32, 34	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺:(0[,]1)⟶𝑋)
36	35	feqmptd 6945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺 = (𝑧 ∈ (0[,]1) ↦ (𝐺‘𝑧)))
37	36, 32	eqeltrrd 2862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ (0[,]1) ↦ (𝐺‘𝑧)) ∈ (II Cn 𝐽))
38	29, 31, 24, 37	cnmpt1res 23975	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((II ↾t 𝑊) Cn 𝐽))
39		dfii2 25183	. . . . . . . . . 10 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
40	12	oveq1i 7422	. . . . . . . . . 10 ⊢ (𝐿 ↾t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1))
41	39, 40	eqtr4i 2787	. . . . . . . . 9 ⊢ II = (𝐿 ↾t (0[,]1))
42	41	oveq1i 7422	. . . . . . . 8 ⊢ (II ↾t 𝑊) = ((𝐿 ↾t (0[,]1)) ↾t 𝑊)
43		retop 25060	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
44	12, 43	eqeltri 2857	. . . . . . . . . 10 ⊢ 𝐿 ∈ Top
45	44	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐿 ∈ Top)
46		ovexd 7447	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (0[,]1) ∈ V)
47		restabs 23463	. . . . . . . . 9 ⊢ ((𝐿 ∈ Top ∧ 𝑊 ⊆ (0[,]1) ∧ (0[,]1) ∈ V) → ((𝐿 ↾t (0[,]1)) ↾t 𝑊) = (𝐿 ↾t 𝑊))
48	45, 24, 46, 47	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝐿 ↾t (0[,]1)) ↾t 𝑊) = (𝐿 ↾t 𝑊))
49	42, 48	eqtrid 2808	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (II ↾t 𝑊) = (𝐿 ↾t 𝑊))
50	49	oveq1d 7427	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((II ↾t 𝑊) Cn 𝐽) = ((𝐿 ↾t 𝑊) Cn 𝐽))
51	38, 50	eleqtrd 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn 𝐽))
52		cvmtop2 35995	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
53	17, 52	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐽 ∈ Top)
54	4	toptopon 23215	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
55	53, 54	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐽 ∈ (TopOn‘𝑋))
56		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ (1...𝑁))
57		simprr 785	. . . . . . . . . 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 36021	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
59	58	anassrs 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝑊) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
60	59	fmpttd 7107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)):𝑊⟶(1st ‘(𝑇‘𝑀)))
61	60	frnd 6710	. . . . . 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 36019	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
63	2	cvmsrcl 35998	. . . . . . . 8 ⊢ ((2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))) → (1st ‘(𝑇‘𝑀)) ∈ 𝐽)
64		elssuni 4899	. . . . . . . 8 ⊢ ((1st ‘(𝑇‘𝑀)) ∈ 𝐽 → (1st ‘(𝑇‘𝑀)) ⊆ ∪ 𝐽)
65	62, 63, 64	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇‘𝑀)) ⊆ ∪ 𝐽)
66	65, 4	sseqtrrdi 3972	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇‘𝑀)) ⊆ 𝑋)
67		cnrest2 23584	. . . . . 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 1398	. . . . 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 235	. . . 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 36025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
71		cvmcn 35996	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
72	3, 4	cnf 23544	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
73	17, 71, 72	3syl 19	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐹:𝐵⟶𝑋)
74		ffn 6701	. . . . . . . . . . 11 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
75		fniniseg 7051	. . . . . . . . . . 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 19	. . . . . . . . . 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 235	. . . . . . . . 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 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
79	77	simprd 501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))
80	1	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℕ)
81	80	nnred 12331	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
82		peano2rem 11606	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
83	81, 82	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ ℝ)
84	9	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
85	83, 84	nndivred 12373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
86	85	rexrd 11340	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
87	81, 84	nndivred 12373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ)
88	87	rexrd 11340	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ*)
89	81	ltm1d 12230	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) < 𝑀)
90	84	nnred 12331	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
91	84	nngt0d 12368	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 0 < 𝑁)
92		ltdiv1 12162	. . . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
94	89, 93	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
95	85, 87, 94	ltled 11439	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
96		lbicc2 13576	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
98	97, 14	eleqtrrdi 2872	. . . . . . . . . 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 36021	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇‘𝑀)))
100	79, 99	eqeltrd 2861	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))
101		eqid 2761	. . . . . . . . 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 36011	. . . . . . . 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 1399	. . . . . . 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 500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)))
105	2	cvmshmeo 36005	. . . . . 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 596	. . . . 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 24060	. . . . 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 18	. . . 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 23963	. . 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 3932	. 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 2861	1 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑄‘𝑀) ∈ ((𝐿 ↾t 𝑊) Cn 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186   I cid 5545   × cxp 5649  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  ℩crio 7368  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7988  2^nd c2nd 7989  ℝcr 11180  0cc0 11181  1c1 11182  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325   − cmin 11522   / cdiv 11954  ℕcn 12316  (,)cioo 13457  [,]cicc 13460  ...cfz 13620  seqcseq 14124   ↾_t crest 17571  topGenctg 17588  Topctop 23191  TopOnctopon 23208   Cn ccn 23522  Homeochmeo 24052  IIcii 25176   CovMap ccvm 35989

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fi 9387  df-sup 9418  df-inf 9419  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-q 13057  df-rp 13102  df-xneg 13222  df-xadd 13223  df-xmul 13224  df-ioo 13461  df-icc 13464  df-fz 13621  df-seq 14125  df-exp 14185  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-rest 17573  df-topgen 17594  df-psmet 21650  df-xmet 21651  df-met 21652  df-bl 21653  df-mopn 21654  df-top 23192  df-topon 23209  df-bases 23244  df-cn 23525  df-hmeo 24054  df-ii 25178  df-cvm 35990

This theorem is used by:  cvmliftlem10  36028