Theorem cvmliftlem6 32532

Description: Lemma for cvmlift 32541. Induction step for cvmliftlem7 32533. 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 715	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ (1...𝑁))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13	cvmliftlem1 32527	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
15	1	cvmsss 32509	. . . . . . . . . 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 483	. . . . . . . . . . 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 715	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
20		cvmcn 32504	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
21	2, 3	cnf 21848	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
22	17, 20, 21	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝐹:𝐵⟶𝑋)
23		ffn 6509	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐵⟶𝑋 → 𝐹 Fn 𝐵)
24		fniniseg 6825	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 497	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
28	26	simprd 498	. . . . . . . . . . . 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 12930	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
31	13, 30	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ ℕ)
32	31	nnred 11647	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ ℝ)
33		peano2rem 10947	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 − 1) ∈ ℝ)
35	8	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑁 ∈ ℕ)
36	34, 35	nndivred 11685	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
37	36	rexrd 10685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
38	32, 35	nndivred 11685	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 / 𝑁) ∈ ℝ)
39	38	rexrd 10685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 / 𝑁) ∈ ℝ*)
40	32	ltm1d 11566	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 − 1) < 𝑀)
41	35	nnred 11647	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑁 ∈ ℝ)
42	35	nngt0d 11680	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 0 < 𝑁)
43		ltdiv1 11498	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
45	40, 44	mpbid 234	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
46	36, 38, 45	ltled 10782	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
47		lbicc2 12846	. . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
49	48, 29	eleqtrrdi 2924	. . . . . . . . . . . . 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 32529	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇‘𝑀)))
51	28, 50	eqeltrd 2913	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))
52		eqid 2821	. . . . . . . . . . . 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 32519	. . . . . . . . . . 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 1368	. . . . . . . . . 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 497	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇‘𝑀)))
56	16, 55	sseldd 3968	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ 𝐶)
57		elssuni 4861	. . . . . . . 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 4018	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ⊆ 𝐵)
60	1	cvmsf1o 32514	. . . . . . . . 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 1367	. . . . . . . 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 6622	. . . . . . . 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 6610	. . . . . . . 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 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑧 ∈ 𝑊)
66	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 29, 65	cvmliftlem3 32529	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
67	64, 66	ffvelrnd 6847	. . . . . 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 3968	. . . . 5 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) ∈ 𝐵)
69	68	anassrs 470	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑧 ∈ 𝑊) → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) ∈ 𝐵)
70	69	fmpttd 6874	. . 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 32531	. . . . 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 593	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
75	74	feq1d 6494	. . 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 259	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑄‘𝑀):𝑊⟶𝐵)
77		fvres 6684	. . . . . . 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 7028	. . . . . . 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 586	. . . . . 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 6684	. . . . . . 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 2863	. . . . 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 470	. . . 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 5154	. . 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 483	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐹:𝐵⟶𝑋)
88	87	feqmptd 6728	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦)))
89		fveq2 6665	. . . 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 6886	. . 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 23483	. . . . . . . 8 ⊢ (0[,]1) = ∪ II
92	91, 3	cnf 21848	. . . . . . 7 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
93	5, 92	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋)
94	93	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:(0[,]1)⟶𝑋)
95	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29	cvmliftlem2 32528	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1))
96	94, 95	fssresd 6540	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 ↾ 𝑊):𝑊⟶𝑋)
97	96	feqmptd 6728	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 ↾ 𝑊) = (𝑧 ∈ 𝑊 ↦ ((𝐺 ↾ 𝑊)‘𝑧)))
98	85, 90, 97	3eqtr4d 2866	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊))
99	76, 98	jca 514	1 ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘𝑀):𝑊⟶𝐵 ∧ (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142  Vcvv 3495   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4561  ⟨cop 4567  ∪ cuni 4832  ∪ ciun 4912   class class class wbr 5059   ↦ cmpt 5139   I cid 5454   × cxp 5548  ^◡ccnv 5549  ran crn 5551   ↾ cres 5552   “ cima 5553   ∘ ccom 5554   Fn wfn 6345  ⟶wf 6346  –1-1-onto→wf1o 6349  ‘cfv 6350  ℩crio 7107  (class class class)co 7150   ∈ cmpo 7152  1^st c1st 7681  2^nd c2nd 7682  ℝcr 10530  0cc0 10531  1c1 10532  ℝ^*cxr 10668   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  ℕcn 11632  (,)cioo 12732  [,]cicc 12735  ...cfz 12886  seqcseq 13363   ↾_t crest 16688  topGenctg 16705   Cn ccn 21826  Homeochmeo 22355  IIcii 23477   CovMap ccvm 32497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-icc 12739  df-fz 12887  df-seq 13364  df-exp 13424  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-rest 16690  df-topgen 16711  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-mopn 20535  df-top 21496  df-topon 21513  df-bases 21548  df-cn 21829  df-hmeo 22357  df-ii 23479  df-cvm 32498

This theorem is referenced by:  cvmliftlem7  32533  cvmliftlem10  32536  cvmliftlem13  32538