Theorem cvmliftlem8 31274

Description: Lemma for cvmlift 31281. 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 12370	. . 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 31271	. . 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 491	. 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 31242	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
19		cnrest2r 21091	. . . 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 22567	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
22	12, 21	eqeltri 2697	. . . . 5 ⊢ 𝐿 ∈ (TopOn‘ℝ)
23		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
24	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14	cvmliftlem2 31268	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ (0[,]1))
25		unitssre 12319	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
26	24, 25	syl6ss 3615	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ ℝ)
27		resttopon 20965	. . . . 5 ⊢ ((𝐿 ∈ (TopOn‘ℝ) ∧ 𝑊 ⊆ ℝ) → (𝐿 ↾t 𝑊) ∈ (TopOn‘𝑊))
28	22, 26, 27	sylancr 695	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐿 ↾t 𝑊) ∈ (TopOn‘𝑊))
29		eqid 2622	. . . . . . 7 ⊢ (II ↾t 𝑊) = (II ↾t 𝑊)
30		iitopon 22682	. . . . . . . 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 22684	. . . . . . . . . . 11 ⊢ (0[,]1) = ∪ II
34	33, 4	cnf 21050	. . . . . . . . . 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 6249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐺 = (𝑧 ∈ (0[,]1) ↦ (𝐺‘𝑧)))
37	36, 32	eqeltrrd 2702	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ (0[,]1) ↦ (𝐺‘𝑧)) ∈ (II Cn 𝐽))
38	29, 31, 24, 37	cnmpt1res 21479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((II ↾t 𝑊) Cn 𝐽))
39		dfii2 22685	. . . . . . . . . 10 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
40	12	oveq1i 6660	. . . . . . . . . 10 ⊢ (𝐿 ↾t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1))
41	39, 40	eqtr4i 2647	. . . . . . . . 9 ⊢ II = (𝐿 ↾t (0[,]1))
42	41	oveq1i 6660	. . . . . . . 8 ⊢ (II ↾t 𝑊) = ((𝐿 ↾t (0[,]1)) ↾t 𝑊)
43		retop 22565	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
44	12, 43	eqeltri 2697	. . . . . . . . . 10 ⊢ 𝐿 ∈ Top
45	44	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐿 ∈ Top)
46		ovexd 6680	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (0[,]1) ∈ V)
47		restabs 20969	. . . . . . . . 9 ⊢ ((𝐿 ∈ Top ∧ 𝑊 ⊆ (0[,]1) ∧ (0[,]1) ∈ V) → ((𝐿 ↾t (0[,]1)) ↾t 𝑊) = (𝐿 ↾t 𝑊))
48	45, 24, 46, 47	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝐿 ↾t (0[,]1)) ↾t 𝑊) = (𝐿 ↾t 𝑊))
49	42, 48	syl5eq 2668	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (II ↾t 𝑊) = (𝐿 ↾t 𝑊))
50	49	oveq1d 6665	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((II ↾t 𝑊) Cn 𝐽) = ((𝐿 ↾t 𝑊) Cn 𝐽))
51	38, 50	eleqtrd 2703	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn 𝐽))
52		cvmtop2 31243	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
53	17, 52	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐽 ∈ Top)
54	4	toptopon 20722	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
55	53, 54	sylib 208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐽 ∈ (TopOn‘𝑋))
56		simprl 794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ (1...𝑁))
57		simprr 796	. . . . . . . . . 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 31269	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝑊)) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
59	58	anassrs 680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 ∈ 𝑊) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
60		eqid 2622	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) = (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧))
61	59, 60	fmptd 6385	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)):𝑊⟶(1st ‘(𝑇‘𝑀)))
62		frn 6053	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)):𝑊⟶(1st ‘(𝑇‘𝑀)) → ran (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ⊆ (1st ‘(𝑇‘𝑀)))
63	61, 62	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ran (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ⊆ (1st ‘(𝑇‘𝑀)))
64	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23	cvmliftlem1 31267	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
65	2	cvmsrcl 31246	. . . . . . . 8 ⊢ ((2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))) → (1st ‘(𝑇‘𝑀)) ∈ 𝐽)
66		elssuni 4467	. . . . . . . 8 ⊢ ((1st ‘(𝑇‘𝑀)) ∈ 𝐽 → (1st ‘(𝑇‘𝑀)) ⊆ ∪ 𝐽)
67	64, 65, 66	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇‘𝑀)) ⊆ ∪ 𝐽)
68	67, 4	syl6sseqr 3652	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇‘𝑀)) ⊆ 𝑋)
69		cnrest2 21090	. . . . . 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))))))
70	55, 63, 68, 69	syl3anc 1326	. . . . 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))))))
71	51, 70	mpbid 222	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑧 ∈ 𝑊 ↦ (𝐺‘𝑧)) ∈ ((𝐿 ↾t 𝑊) Cn (𝐽 ↾t (1st ‘(𝑇‘𝑀)))))
72	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cvmliftlem7 31273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
73		cvmcn 31244	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
74	3, 4	cnf 21050	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
75	17, 73, 74	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝐹:𝐵⟶𝑋)
76		ffn 6045	. . . . . . . . . . 11 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
77		fniniseg 6338	. . . . . . . . . . 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)))))
78	75, 76, 77	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)))))
79	72, 78	mpbid 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁))))
80	79	simpld 475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
81	79	simprd 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))
82	1	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℕ)
83	82	nnred 11035	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
84		peano2rem 10348	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ ℝ)
86	9	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
87	85, 86	nndivred 11069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
88	87	rexrd 10089	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
89	83, 86	nndivred 11069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ)
90	89	rexrd 10089	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ*)
91	83	ltm1d 10956	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑀 − 1) < 𝑀)
92	86	nnred 11035	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
93	86	nngt0d 11064	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → 0 < 𝑁)
94		ltdiv1 10887	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 − 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
95	85, 83, 92, 93, 94	syl112anc 1330	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
96	91, 95	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
97	87, 89, 96	ltled 10185	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
98		lbicc2 12288	. . . . . . . . . . . 12 ⊢ ((((𝑀 − 1) / 𝑁) ∈ ℝ* ∧ (𝑀 / 𝑁) ∈ ℝ* ∧ ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
99	88, 90, 97, 98	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
100	99, 14	syl6eleqr 2712	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ 𝑊)
101	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 100	cvmliftlem3 31269	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇‘𝑀)))
102	81, 101	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))
103		eqid 2622	. . . . . . . . 9 ⊢ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) = (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)
104	2, 3, 103	cvmsiota 31259	. . . . . . . 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)))
105	17, 64, 80, 102, 104	syl13anc 1328	. . . . . . 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)))
106	105	simpld 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)))
107	2	cvmshmeo 31253	. . . . . 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)))))
108	64, 106, 107	syl2anc 693	. . . . 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)))))
109		hmeocnvcn 21564	. . . . 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))))
110	108, 109	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))))
111	28, 71, 110	cnmpt11f 21467	. . 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))))
112	20, 111	sseldd 3604	. 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))
113	16, 112	eqeltrd 2701	1 ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑄‘𝑀) ∈ ((𝐿 ↾t 𝑊) Cn 𝐶))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  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  ran crn 5115   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  ℝcr 9935  0cc0 9936  1c1 9937  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  ℕcn 11020  (,)cioo 12175  [,]cicc 12178  ...cfz 12326  seqcseq 12801   ↾_t crest 16081  topGenctg 16098  Topctop 20698  TopOnctopon 20715   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-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-cn 21031  df-hmeo 21558  df-ii 22680  df-cvm 31238

This theorem is referenced by:  cvmliftlem10  31276