Theorem cvmliftlem8 33886

Description: Lemma for cvmlift 33893. 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 13470	. . 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 33883	. . 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 593	. 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
18		cvmtop1 33854	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
19		cnrest2r 22638	. . . 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 24127	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
22	12, 21	eqeltri 2834	. . . . 5 ⊢ 𝐿 ∈ (TopOn‘ℝ)
23		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
24	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14	cvmliftlem2 33880	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ (0[,]1))
25		unitssre 13416	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
26	24, 25	sstrdi 3956	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ ℝ)
27		resttopon 22512	. . . . 5 ⊢ ((𝐿 ∈ (TopOn‘ℝ) ∧ 𝑊 ⊆ ℝ) → (𝐿 ↾t 𝑊) ∈ (TopOn‘𝑊))
28	22, 26, 27	sylancr 587	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐿 ↾t 𝑊) ∈ (TopOn‘𝑊))
29		eqid 2736	. . . . . . 7 ⊢ (II ↾t 𝑊) = (II ↾t 𝑊)
30		iitopon 24242	. . . . . . . 8 ⊢ II ∈ (TopOn‘(0[,]1))
31	30	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → II ∈ (TopOn‘(0[,]1)))
32	6	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺 ∈ (II Cn 𝐽))
33		iiuni 24244	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
34	33, 4	cnf 22597	. . . . . . . . . 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 6910	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺 = (𝑧 ∈ (0[,]1) ↦ (𝐺‘𝑧)))
37	36, 32	eqeltrrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ (0[,]1) ↦ (𝐺‘𝑧)) ∈ (II Cn 𝐽))
38	29, 31, 24, 37	cnmpt1res 23027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((II ↾t 𝑊) Cn 𝐽))
39		dfii2 24245	. . . . . . . . . 10 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
40	12	oveq1i 7367	. . . . . . . . . 10 ⊢ (𝐿 ↾t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1))
41	39, 40	eqtr4i 2767	. . . . . . . . 9 ⊢ II = (𝐿 ↾t (0[,]1))
42	41	oveq1i 7367	. . . . . . . 8 ⊢ (II ↾t 𝑊) = ((𝐿 ↾t (0[,]1)) ↾t 𝑊)
43		retop 24125	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
44	12, 43	eqeltri 2834	. . . . . . . . . 10 ⊢ 𝐿 ∈ Top
45	44	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐿 ∈ Top)
46		ovexd 7392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (0[,]1) ∈ V)
47		restabs 22516	. . . . . . . . 9 ⊢ ((𝐿 ∈ Top ∧ 𝑊 ⊆ (0[,]1) ∧ (0[,]1) ∈ V) → ((𝐿 ↾t (0[,]1)) ↾t 𝑊) = (𝐿 ↾t 𝑊))
48	45, 24, 46, 47	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝐿 ↾t (0[,]1)) ↾t 𝑊) = (𝐿 ↾t 𝑊))
49	42, 48	eqtrid 2788	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (II ↾t 𝑊) = (𝐿 ↾t 𝑊))
50	49	oveq1d 7372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((II ↾t 𝑊) Cn 𝐽) = ((𝐿 ↾t 𝑊) Cn 𝐽))
51	38, 50	eleqtrd 2840	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn 𝐽))
52		cvmtop2 33855	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
53	17, 52	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐽 ∈ Top)
54	4	toptopon 22266	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
55	53, 54	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐽 ∈ (TopOn‘𝑋))
56		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ (1...𝑁))
57		simprr 771	. . . . . . . . . 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 33881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
59	58	anassrs 468	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝑊) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
60	59	fmpttd 7063	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)):𝑊⟶(1st ‘(𝑇‘𝑀)))
61	60	frnd 6676	. . . . . 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 33879	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
63	2	cvmsrcl 33858	. . . . . . . 8 ⊢ ((2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))) → (1st ‘(𝑇‘𝑀)) ∈ 𝐽)
64		elssuni 4898	. . . . . . . 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 3995	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇‘𝑀)) ⊆ 𝑋)
67		cnrest2 22637	. . . . . 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 1371	. . . . 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 231	. . . 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 33885	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
71		cvmcn 33856	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
72	3, 4	cnf 22597	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
73	17, 71, 72	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐹:𝐵⟶𝑋)
74		ffn 6668	. . . . . . . . . . 11 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
75		fniniseg 7010	. . . . . . . . . . 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 231	. . . . . . . . 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 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
79	77	simprd 496	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))
80	1	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℕ)
81	80	nnred 12168	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
82		peano2rem 11468	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
83	81, 82	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ ℝ)
84	9	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
85	83, 84	nndivred 12207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
86	85	rexrd 11205	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
87	81, 84	nndivred 12207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ)
88	87	rexrd 11205	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ*)
89	81	ltm1d 12087	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) < 𝑀)
90	84	nnred 12168	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
91	84	nngt0d 12202	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 0 < 𝑁)
92		ltdiv1 12019	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
94	89, 93	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
95	85, 87, 94	ltled 11303	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
96		lbicc2 13381	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
98	97, 14	eleqtrrdi 2849	. . . . . . . . . 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 33881	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇‘𝑀)))
100	79, 99	eqeltrd 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))
101		eqid 2736	. . . . . . . . 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 33871	. . . . . . . 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 1372	. . . . . . 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 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)))
105	2	cvmshmeo 33865	. . . . . 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 584	. . . . 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 23112	. . . . 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 23015	. . 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 3945	. 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 2838	1 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑄‘𝑀) ∈ ((𝐿 ↾t 𝑊) Cn 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  {csn 4586  ⟨cop 4592  ∪ cuni 4865  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188   I cid 5530   × cxp 5631  ^◡ccnv 5632  ran crn 5634   ↾ cres 5635   “ cima 5636   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  ℩crio 7312  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7919  2^nd c2nd 7920  ℝcr 11050  0cc0 11051  1c1 11052  ℝ^*cxr 11188   < clt 11189   ≤ cle 11190   − cmin 11385   / cdiv 11812  ℕcn 12153  (,)cioo 13264  [,]cicc 13267  ...cfz 13424  seqcseq 13906   ↾_t crest 17302  topGenctg 17319  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Homeochmeo 23104  IIcii 24238   CovMap ccvm 33849

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-icc 13271  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-hmeo 23106  df-ii 24240  df-cvm 33850

This theorem is referenced by:  cvmliftlem10  33888