Theorem cvmliftlem6 35262

Description: Lemma for cvmlift 35271. Induction step for cvmliftlem7 35263. 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 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ (1...𝑁))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13	cvmliftlem1 35257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
15	1	cvmsss 35239	. . . . . . . . . 10 ⊢ ((2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))) → (2nd ‘(𝑇‘𝑀)) ⊆ 𝐶)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (2nd ‘(𝑇‘𝑀)) ⊆ 𝐶)
17	4	adantr 480	. . . . . . . . . . 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 717	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
20		cvmcn 35234	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
21	2, 3	cnf 23149	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
22	17, 20, 21	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝐹:𝐵⟶𝑋)
23		ffn 6656	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
24		fniniseg 6998	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
28	26	simprd 495	. . . . . . . . . . . 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 13474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
31	13, 30	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ ℕ)
32	31	nnred 12161	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ ℝ)
33		peano2rem 11449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 − 1) ∈ ℝ)
35	8	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑁 ∈ ℕ)
36	34, 35	nndivred 12200	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
37	36	rexrd 11184	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
38	32, 35	nndivred 12200	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 / 𝑁) ∈ ℝ)
39	38	rexrd 11184	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 / 𝑁) ∈ ℝ*)
40	32	ltm1d 12075	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 − 1) < 𝑀)
41	35	nnred 12161	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑁 ∈ ℝ)
42	35	nngt0d 12195	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 0 < 𝑁)
43		ltdiv1 12007	. . . . . . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
45	40, 44	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
46	36, 38, 45	ltled 11282	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
47		lbicc2 13385	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
49	48, 29	eleqtrrdi 2839	. . . . . . . . . . . . 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 35259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇‘𝑀)))
51	28, 50	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))
52		eqid 2729	. . . . . . . . . . . 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 35249	. . . . . . . . . . 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 1374	. . . . . . . . . 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 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)))
56	16, 55	sseldd 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ 𝐶)
57		elssuni 4891	. . . . . . . 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 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ⊆ ∪ 𝐶)
59	58, 2	sseqtrrdi 3979	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ⊆ 𝐵)
60	1	cvmsf1o 35244	. . . . . . . . 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 1373	. . . . . . . 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 6780	. . . . . . . 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 6768	. . . . . . . 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 18	. . . . . . 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 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑧 ∈ 𝑊)
66	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 29, 65	cvmliftlem3 35259	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
67	64, 66	ffvelcdmd 7023	. . . . . 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 3938	. . . . 5 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) ∈ 𝐵)
69	68	anassrs 467	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑧 ∈ 𝑊) → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) ∈ 𝐵)
70	69	fmpttd 7053	. . 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 17	. . . . 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 35261	. . . . 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 591	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
75	74	feq1d 6638	. . 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 257	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑄‘𝑀):𝑊⟶𝐵)
77		fvres 6845	. . . . . . 7 ⊢ (𝑧 ∈ 𝑊 → ((𝐺 ↾ 𝑊)‘𝑧) = (𝐺‘𝑧))
78	65, 77	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝐺 ↾ 𝑊)‘𝑧) = (𝐺‘𝑧))
79		f1ocnvfv2 7218	. . . . . . 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 584	. . . . . 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 6845	. . . . . . 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 17	. . . . . 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 2771	. . . . 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 467	. . . 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 5188	. . 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 18	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶𝑋)
87	86	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐹:𝐵⟶𝑋)
88	87	feqmptd 6895	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦)))
89		fveq2 6826	. . . 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 7067	. . 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 24790	. . . . . . . 8 ⊢ (0[,]1) = ∪ II
92	91, 3	cnf 23149	. . . . . . 7 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
93	5, 92	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋)
94	93	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:(0[,]1)⟶𝑋)
95	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29	cvmliftlem2 35258	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1))
96	94, 95	fssresd 6695	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 ↾ 𝑊):𝑊⟶𝑋)
97	96	feqmptd 6895	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 ↾ 𝑊) = (𝑧 ∈ 𝑊 ↦ ((𝐺 ↾ 𝑊)‘𝑧)))
98	85, 90, 97	3eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊))
99	76, 98	jca 511	1 ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘𝑀):𝑊⟶𝐵 ∧ (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396  Vcvv 3438   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4553  {csn 4579  ⟨cop 4585  ∪ cuni 4861  ∪ ciun 4944   class class class wbr 5095   ↦ cmpt 5176   I cid 5517   × cxp 5621  ^◡ccnv 5622  ran crn 5624   ↾ cres 5625   “ cima 5626   ∘ ccom 5627   Fn wfn 6481  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  ℩crio 7309  (class class class)co 7353   ∈ cmpo 7355  1^st c1st 7929  2^nd c2nd 7930  ℝcr 11027  0cc0 11028  1c1 11029  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169   − cmin 11365   / cdiv 11795  ℕcn 12146  (,)cioo 13266  [,]cicc 13269  ...cfz 13428  seqcseq 13926   ↾_t crest 17342  topGenctg 17359   Cn ccn 23127  Homeochmeo 23656  IIcii 24784   CovMap ccvm 35227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fi 9320  df-sup 9351  df-inf 9352  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-q 12868  df-rp 12912  df-xneg 13032  df-xadd 13033  df-xmul 13034  df-icc 13273  df-fz 13429  df-seq 13927  df-exp 13987  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-rest 17344  df-topgen 17365  df-psmet 21271  df-xmet 21272  df-met 21273  df-bl 21274  df-mopn 21275  df-top 22797  df-topon 22814  df-bases 22849  df-cn 23130  df-hmeo 23658  df-ii 24786  df-cvm 35228

This theorem is referenced by:  cvmliftlem7  35263  cvmliftlem10  35266  cvmliftlem13  35268