Theorem cvmliftlem5 36023

Description: Lemma for cvmlift 36033. Definition of 𝑄 at a successor. This is a function defined on 𝑊 as ^◡(𝑇 ↾ 𝐼) ∘ 𝐺 where 𝐼 is the unique covering set of 2^nd ‘(𝑇‘𝑀) that contains 𝑄(𝑀 − 1) evaluated at the last defined point, namely (𝑀 − 1) / 𝑁 (note that for 𝑀 = 1 this is using the seed value 𝑄(0)(0) = 𝑃). (Contributed by Mario Carneiro, 15-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
cvmliftlem5	⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))

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 cvmliftlem5

Step	Hyp	Ref	Expression
1		0z 12685	. . . 4 ⊢ 0 ∈ ℤ
2		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℕ)
3		nnuz 12985	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
4		1e0p1 12842	. . . . . . 7 ⊢ 1 = (0 + 1)
5	4	fveq2i 6880	. . . . . 6 ⊢ (ℤ≥‘1) = (ℤ≥‘(0 + 1))
6	3, 5	eqtri 2784	. . . . 5 ⊢ ℕ = (ℤ≥‘(0 + 1))
7	2, 6	eleqtrdi 2871	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ (ℤ≥‘(0 + 1)))
8		seqm1 14142	. . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ (ℤ≥‘(0 + 1))) → (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⟩}⟩}))‘𝑀) = ((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⟩}⟩}))‘(𝑀 − 1))(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⟩}⟩})‘𝑀)))
9	1, 7, 8	sylancr 599	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (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⟩}⟩}))‘𝑀) = ((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⟩}⟩}))‘(𝑀 − 1))(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⟩}⟩})‘𝑀)))
10		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⟩}⟩}))
11	10	fveq1i 6878	. . 3 ⊢ (𝑄‘𝑀) = (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⟩}⟩}))‘𝑀)
12	10	fveq1i 6878	. . . 4 ⊢ (𝑄‘(𝑀 − 1)) = (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⟩}⟩}))‘(𝑀 − 1))
13	12	oveq1i 7422	. . 3 ⊢ ((𝑄‘(𝑀 − 1))(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⟩}⟩})‘𝑀)) = ((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⟩}⟩}))‘(𝑀 − 1))(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⟩}⟩})‘𝑀))
14	9, 11, 13	3eqtr4g 2821	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑄‘𝑀) = ((𝑄‘(𝑀 − 1))(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⟩}⟩})‘𝑀)))
15		0nnn 12355	. . . . . 6 ⊢ ¬ 0 ∈ ℕ
16		disjsn 4672	. . . . . 6 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
17	15, 16	mpbir 234	. . . . 5 ⊢ (ℕ ∩ {0}) = ∅
18		fnresi 6660	. . . . . 6 ⊢ ( I ↾ ℕ) Fn ℕ
19		c0ex 11281	. . . . . . 7 ⊢ 0 ∈ V
20		snex 5397	. . . . . . 7 ⊢ {⟨0, 𝑃⟩} ∈ V
21	19, 20	fnsn 6590	. . . . . 6 ⊢ {⟨0, {⟨0, 𝑃⟩}⟩} Fn {0}
22		fvun1 6968	. . . . . 6 ⊢ ((( I ↾ ℕ) Fn ℕ ∧ {⟨0, {⟨0, 𝑃⟩}⟩} Fn {0} ∧ ((ℕ ∩ {0}) = ∅ ∧ 𝑀 ∈ ℕ)) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
23	18, 21, 22	mp3an12 1480	. . . . 5 ⊢ (((ℕ ∩ {0}) = ∅ ∧ 𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
24	17, 2, 23	sylancr 599	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
25		fvresi 7170	. . . . 5 ⊢ (𝑀 ∈ ℕ → (( I ↾ ℕ)‘𝑀) = 𝑀)
26	25	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (( I ↾ ℕ)‘𝑀) = 𝑀)
27	24, 26	eqtrd 2796	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = 𝑀)
28	27	oveq2d 7428	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ((𝑄‘(𝑀 − 1))(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⟩}⟩})‘𝑀)) = ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))𝑀))
29		fvexd 6892	. . 3 ⊢ (𝜑 → (𝑄‘(𝑀 − 1)) ∈ V)
30		simpr 490	. . . . . . . . 9 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
31	30	oveq1d 7427	. . . . . . . 8 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑚 − 1) = (𝑀 − 1))
32	31	oveq1d 7427	. . . . . . 7 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝑚 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁))
33	30	oveq1d 7427	. . . . . . 7 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑚 / 𝑁) = (𝑀 / 𝑁))
34	32, 33	oveq12d 7430	. . . . . 6 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
35		cvmliftlem5.3	. . . . . 6 ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))
36	34, 35	eqtr4di 2814	. . . . 5 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) = 𝑊)
37	30	fveq2d 6881	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑇‘𝑚) = (𝑇‘𝑀))
38	37	fveq2d 6881	. . . . . . . . 9 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (2nd ‘(𝑇‘𝑚)) = (2nd ‘(𝑇‘𝑀)))
39		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → 𝑥 = (𝑄‘(𝑀 − 1)))
40	39, 32	fveq12d 6884	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑥‘((𝑚 − 1) / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
41	40	eleq1d 2846	. . . . . . . . 9 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏 ↔ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
42	38, 41	riotaeqbidv 7372	. . . . . . . 8 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏) = (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
43	42	reseq2d 5970	. . . . . . 7 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏)) = (𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)))
44	43	cnveqd 5853	. . . . . 6 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏)) = ◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)))
45	44	fveq1d 6879	. . . . 5 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)) = (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))
46	36, 45	mpteq12dv 5192	. . . 4 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
47		eqid 2761	. . . 4 ⊢ (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))) = (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
48		ovex 7445	. . . . . 6 ⊢ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ∈ V
49	35, 48	eqeltri 2857	. . . . 5 ⊢ 𝑊 ∈ V
50	49	mptex 7221	. . . 4 ⊢ (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) ∈ V
51	46, 47, 50	ovmpoa 7567	. . 3 ⊢ (((𝑄‘(𝑀 − 1)) ∈ V ∧ 𝑀 ∈ ℕ) → ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
52	29, 51	sylan 592	. 2 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))
53	14, 28, 52	3eqtrd 2800	1 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951   ↦ cmpt 5186   I cid 5545   × cxp 5649  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  ℩crio 7368  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7988  2^nd c2nd 7989  0cc0 11181  1c1 11182   + caddc 11184   − cmin 11522   / cdiv 11954  ℕcn 12316  ℤcz 12674  ℤ_≥cuz 12946  (,)cioo 13457  [,]cicc 13460  ...cfz 13620  seqcseq 14124   ↾_t crest 17571  topGenctg 17588   Cn ccn 23522  Homeochmeo 24052  IIcii 25176   CovMap ccvm 35989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-n0 12588  df-z 12675  df-uz 12947  df-seq 14125

This theorem is used by:  cvmliftlem6  36024  cvmliftlem8  36026  cvmliftlem9  36027