Theorem cvmliftlem5 35321

Description: Lemma for cvmlift 35331. 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 12476	. . . 4 ⊢ 0 ∈ ℤ
2		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℕ)
3		nnuz 12772	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
4		1e0p1 12627	. . . . . . 7 ⊢ 1 = (0 + 1)
5	4	fveq2i 6825	. . . . . 6 ⊢ (ℤ≥‘1) = (ℤ≥‘(0 + 1))
6	3, 5	eqtri 2754	. . . . 5 ⊢ ℕ = (ℤ≥‘(0 + 1))
7	2, 6	eleqtrdi 2841	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ (ℤ≥‘(0 + 1)))
8		seqm1 13923	. . . 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 587	. . 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 6823	. . 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 6823	. . . 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 7356	. . 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 2791	. 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 12158	. . . . . 6 ⊢ ¬ 0 ∈ ℕ
16		disjsn 4664	. . . . . 6 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
17	15, 16	mpbir 231	. . . . 5 ⊢ (ℕ ∩ {0}) = ∅
18		fnresi 6610	. . . . . 6 ⊢ ( I ↾ ℕ) Fn ℕ
19		c0ex 11103	. . . . . . 7 ⊢ 0 ∈ V
20		snex 5374	. . . . . . 7 ⊢ {⟨0, 𝑃⟩} ∈ V
21	19, 20	fnsn 6539	. . . . . 6 ⊢ {⟨0, {⟨0, 𝑃⟩}⟩} Fn {0}
22		fvun1 6913	. . . . . 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 1453	. . . . 5 ⊢ (((ℕ ∩ {0}) = ∅ ∧ 𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
24	17, 2, 23	sylancr 587	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
25		fvresi 7107	. . . . 5 ⊢ (𝑀 ∈ ℕ → (( I ↾ ℕ)‘𝑀) = 𝑀)
26	25	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (( I ↾ ℕ)‘𝑀) = 𝑀)
27	24, 26	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = 𝑀)
28	27	oveq2d 7362	. 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 6837	. . 3 ⊢ (𝜑 → (𝑄‘(𝑀 − 1)) ∈ V)
30		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
31	30	oveq1d 7361	. . . . . . . 8 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑚 − 1) = (𝑀 − 1))
32	31	oveq1d 7361	. . . . . . 7 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝑚 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁))
33	30	oveq1d 7361	. . . . . . 7 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑚 / 𝑁) = (𝑀 / 𝑁))
34	32, 33	oveq12d 7364	. . . . . 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 2784	. . . . 5 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) = 𝑊)
37	30	fveq2d 6826	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑇‘𝑚) = (𝑇‘𝑀))
38	37	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (2nd ‘(𝑇‘𝑚)) = (2nd ‘(𝑇‘𝑀)))
39		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → 𝑥 = (𝑄‘(𝑀 − 1)))
40	39, 32	fveq12d 6829	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑥‘((𝑚 − 1) / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
41	40	eleq1d 2816	. . . . . . . . 9 ⊢ ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏 ↔ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
42	38, 41	riotaeqbidv 7306	. . . . . . . 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 5928	. . . . . . 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 5815	. . . . . 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 6824	. . . . 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 5178	. . . 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 2731	. . . 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 7379	. . . . . 6 ⊢ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ∈ V
49	35, 48	eqeltri 2827	. . . . 5 ⊢ 𝑊 ∈ V
50	49	mptex 7157	. . . 4 ⊢ (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))) ∈ V
51	46, 47, 50	ovmpoa 7501	. . 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 580	. 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 2770	1 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550  {csn 4576  ⟨cop 4582  ∪ cuni 4859  ∪ ciun 4941   ↦ cmpt 5172   I cid 5510   × cxp 5614  ^◡ccnv 5615  ran crn 5617   ↾ cres 5618   “ cima 5619   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  ℩crio 7302  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  0cc0 11003  1c1 11004   + caddc 11006   − cmin 11341   / cdiv 11771  ℕcn 12122  ℤcz 12465  ℤ_≥cuz 12729  (,)cioo 13242  [,]cicc 13245  ...cfz 13404  seqcseq 13905   ↾_t crest 17321  topGenctg 17338   Cn ccn 23137  Homeochmeo 23666  IIcii 24793   CovMap ccvm 35287

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-n0 12379  df-z 12466  df-uz 12730  df-seq 13906

This theorem is referenced by:  cvmliftlem6  35322  cvmliftlem8  35324  cvmliftlem9  35325