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Theorem cvmliftlem5 31609
Description: Lemma for cvmlift 31619. Definition of 𝑄 at a successor. This is a function defined on 𝑊 as (𝑇𝐼) ∘ 𝐺 where 𝐼 is the unique covering set of 2nd ‘(𝑇𝑀) 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
cvmliftlem.b 𝐵 = 𝐶
cvmliftlem.x 𝑋 = 𝐽
cvmliftlem.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftlem.g (𝜑𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p (𝜑𝑃𝐵)
cvmliftlem.e (𝜑 → (𝐹𝑃) = (𝐺‘0))
cvmliftlem.n (𝜑𝑁 ∈ ℕ)
cvmliftlem.t (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
cvmliftlem.a (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))
cvmliftlem.l 𝐿 = (topGen‘ran (,))
cvmliftlem.q 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
cvmliftlem5.3 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))
Assertion
Ref Expression
cvmliftlem5 ((𝜑𝑀 ∈ ℕ) → (𝑄𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
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
StepHypRef Expression
1 0z 11595 . . . 4 0 ∈ ℤ
2 simpr 471 . . . . 5 ((𝜑𝑀 ∈ ℕ) → 𝑀 ∈ ℕ)
3 nnuz 11930 . . . . . 6 ℕ = (ℤ‘1)
4 1e0p1 11759 . . . . . . 7 1 = (0 + 1)
54fveq2i 6336 . . . . . 6 (ℤ‘1) = (ℤ‘(0 + 1))
63, 5eqtri 2793 . . . . 5 ℕ = (ℤ‘(0 + 1))
72, 6syl6eleq 2860 . . . 4 ((𝜑𝑀 ∈ ℕ) → 𝑀 ∈ (ℤ‘(0 + 1)))
8 seqm1 13025 . . . 4 ((0 ∈ ℤ ∧ 𝑀 ∈ (ℤ‘(0 + 1))) → (seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘𝑀) = ((seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀)))
91, 7, 8sylancr 575 . . 3 ((𝜑𝑀 ∈ ℕ) → (seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘𝑀) = ((seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))((( 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 ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
1110fveq1i 6334 . . 3 (𝑄𝑀) = (seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘𝑀)
1210fveq1i 6334 . . . 4 (𝑄‘(𝑀 − 1)) = (seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘(𝑀 − 1))
1312oveq1i 6806 . . 3 ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀)) = ((seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀))
149, 11, 133eqtr4g 2830 . 2 ((𝜑𝑀 ∈ ℕ) → (𝑄𝑀) = ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀)))
15 0nnn 11258 . . . . . 6 ¬ 0 ∈ ℕ
16 disjsn 4384 . . . . . 6 ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
1715, 16mpbir 221 . . . . 5 (ℕ ∩ {0}) = ∅
18 fnresi 6147 . . . . . 6 ( I ↾ ℕ) Fn ℕ
19 c0ex 10240 . . . . . . 7 0 ∈ V
20 snex 5037 . . . . . . 7 {⟨0, 𝑃⟩} ∈ V
2119, 20fnsn 6086 . . . . . 6 {⟨0, {⟨0, 𝑃⟩}⟩} Fn {0}
22 fvun1 6413 . . . . . 6 ((( I ↾ ℕ) Fn ℕ ∧ {⟨0, {⟨0, 𝑃⟩}⟩} Fn {0} ∧ ((ℕ ∩ {0}) = ∅ ∧ 𝑀 ∈ ℕ)) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
2318, 21, 22mp3an12 1562 . . . . 5 (((ℕ ∩ {0}) = ∅ ∧ 𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
2417, 2, 23sylancr 575 . . . 4 ((𝜑𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
25 fvresi 6586 . . . . 5 (𝑀 ∈ ℕ → (( I ↾ ℕ)‘𝑀) = 𝑀)
2625adantl 467 . . . 4 ((𝜑𝑀 ∈ ℕ) → (( I ↾ ℕ)‘𝑀) = 𝑀)
2724, 26eqtrd 2805 . . 3 ((𝜑𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = 𝑀)
2827oveq2d 6812 . 2 ((𝜑𝑀 ∈ ℕ) → ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀)) = ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))𝑀))
29 fvexd 6346 . . 3 (𝜑 → (𝑄‘(𝑀 − 1)) ∈ V)
30 simpr 471 . . . . . . . . 9 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
3130oveq1d 6811 . . . . . . . 8 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑚 − 1) = (𝑀 − 1))
3231oveq1d 6811 . . . . . . 7 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝑚 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁))
3330oveq1d 6811 . . . . . . 7 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑚 / 𝑁) = (𝑀 / 𝑁))
3432, 33oveq12d 6814 . . . . . 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))
3634, 35syl6eqr 2823 . . . . 5 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) = 𝑊)
3730fveq2d 6337 . . . . . . . . . 10 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑇𝑚) = (𝑇𝑀))
3837fveq2d 6337 . . . . . . . . 9 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (2nd ‘(𝑇𝑚)) = (2nd ‘(𝑇𝑀)))
39 simpl 468 . . . . . . . . . . 11 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → 𝑥 = (𝑄‘(𝑀 − 1)))
4039, 32fveq12d 6340 . . . . . . . . . 10 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑥‘((𝑚 − 1) / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
4140eleq1d 2835 . . . . . . . . 9 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏 ↔ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
4238, 41riotaeqbidv 6760 . . . . . . . 8 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
4342reseq2d 5533 . . . . . . 7 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)))
4443cnveqd 5435 . . . . . 6 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)))
4544fveq1d 6335 . . . . 5 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)) = ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))
4636, 45mpteq12dv 4868 . . . 4 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
47 eqid 2771 . . . 4 (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))) = (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
48 ovex 6827 . . . . . 6 (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ∈ V
4935, 48eqeltri 2846 . . . . 5 𝑊 ∈ V
5049mptex 6633 . . . 4 (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))) ∈ V
5146, 47, 50ovmpt2a 6942 . . 3 (((𝑄‘(𝑀 − 1)) ∈ V ∧ 𝑀 ∈ ℕ) → ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
5229, 51sylan 569 . 2 ((𝜑𝑀 ∈ ℕ) → ((𝑄‘(𝑀 − 1))(𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
5314, 28, 523eqtrd 2809 1 ((𝜑𝑀 ∈ ℕ) → (𝑄𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  𝒫 cpw 4298  {csn 4317  cop 4323   cuni 4575   ciun 4655  cmpt 4864   I cid 5157   × cxp 5248  ccnv 5249  ran crn 5251  cres 5252  cima 5253   Fn wfn 6025  wf 6026  cfv 6030  crio 6756  (class class class)co 6796  cmpt2 6798  1st c1st 7317  2nd c2nd 7318  0cc0 10142  1c1 10143   + caddc 10145  cmin 10472   / cdiv 10890  cn 11226  cz 11584  cuz 11893  (,)cioo 12380  [,]cicc 12383  ...cfz 12533  seqcseq 13008  t crest 16289  topGenctg 16306   Cn ccn 21249  Homeochmeo 21777  IIcii 22898   CovMap ccvm 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-n0 11500  df-z 11585  df-uz 11894  df-seq 13009
This theorem is referenced by:  cvmliftlem6  31610  cvmliftlem8  31612  cvmliftlem9  31613
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