Theorem cvmliftlem6 35355

Description: Lemma for cvmlift 35364. Induction step for cvmliftlem7 35356. 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 35350	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
15	1	cvmsss 35332	. . . . . . . . . 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 35327	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
21	2, 3	cnf 23162	. . . . . . . . . . . . . . 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 6999	. . . . . . . . . . . . . 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 13455	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
31	13, 30	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ ℕ)
32	31	nnred 12147	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑀 ∈ ℝ)
33		peano2rem 11435	. . . . . . . . . . . . . . . . . 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 12186	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
37	36	rexrd 11169	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
38	32, 35	nndivred 12186	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 / 𝑁) ∈ ℝ)
39	38	rexrd 11169	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 / 𝑁) ∈ ℝ*)
40	32	ltm1d 12061	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝑀 − 1) < 𝑀)
41	35	nnred 12147	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 𝑁 ∈ ℝ)
42	35	nngt0d 12181	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → 0 < 𝑁)
43		ltdiv1 11993	. . . . . . . . . . . . . . . . . 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 11268	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
47		lbicc2 13366	. . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . 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 35352	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇‘𝑀)))
51	28, 50	eqeltrd 2833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇‘𝑀)))
52		eqid 2733	. . . . . . . . . . . 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 35342	. . . . . . . . . . 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 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ 𝐶)
57		elssuni 4889	. . . . . . . 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 3972	. . . . . 6 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ⊆ 𝐵)
60	1	cvmsf1o 35337	. . . . . . . . 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 35352	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝑧 ∈ 𝑊)) → (𝐺‘𝑧) ∈ (1st ‘(𝑇‘𝑀)))
67	64, 66	ffvelcdmd 7024	. . . . . 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 3931	. . . . 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 7054	. . 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 35354	. . . . 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 6847	. . . . . . 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 7217	. . . . . . 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 6847	. . . . . . 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 2775	. . . . 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 5186	. . 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 6896	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦)))
89		fveq2 6828	. . . 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 7068	. . 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 24802	. . . . . . . 8 ⊢ (0[,]1) = ∪ II
92	91, 3	cnf 23162	. . . . . . 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 35351	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1))
96	94, 95	fssresd 6695	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 ↾ 𝑊):𝑊⟶𝑋)
97	96	feqmptd 6896	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐺 ↾ 𝑊) = (𝑧 ∈ 𝑊 ↦ ((𝐺 ↾ 𝑊)‘𝑧)))
98	85, 90, 97	3eqtr4d 2778	. 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 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396  Vcvv 3437   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  {csn 4575  ⟨cop 4581  ∪ cuni 4858  ∪ ciun 4941   class class class wbr 5093   ↦ cmpt 5174   I cid 5513   × cxp 5617  ^◡ccnv 5618  ran crn 5620   ↾ cres 5621   “ cima 5622   ∘ ccom 5623   Fn wfn 6481  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  ℩crio 7308  (class class class)co 7352   ∈ cmpo 7354  1^st c1st 7925  2^nd c2nd 7926  ℝcr 11012  0cc0 11013  1c1 11014  ℝ^*cxr 11152   < clt 11153   ≤ cle 11154   − cmin 11351   / cdiv 11781  ℕcn 12132  (,)cioo 13247  [,]cicc 13250  ...cfz 13409  seqcseq 13910   ↾_t crest 17326  topGenctg 17343   Cn ccn 23140  Homeochmeo 23669  IIcii 24796   CovMap ccvm 35320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  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 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fi 9302  df-sup 9333  df-inf 9334  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-z 12476  df-uz 12739  df-q 12849  df-rp 12893  df-xneg 13013  df-xadd 13014  df-xmul 13015  df-icc 13254  df-fz 13410  df-seq 13911  df-exp 13971  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-rest 17328  df-topgen 17349  df-psmet 21285  df-xmet 21286  df-met 21287  df-bl 21288  df-mopn 21289  df-top 22810  df-topon 22827  df-bases 22862  df-cn 23143  df-hmeo 23671  df-ii 24798  df-cvm 35321

This theorem is referenced by:  cvmliftlem7  35356  cvmliftlem10  35359  cvmliftlem13  35361