Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmliftlem5 Structured version   Visualization version   GIF version

Theorem cvmliftlem5 32433
Description: Lemma for cvmlift 32443. 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 11980 . . . 4 0 ∈ ℤ
2 simpr 485 . . . . 5 ((𝜑𝑀 ∈ ℕ) → 𝑀 ∈ ℕ)
3 nnuz 12269 . . . . . 6 ℕ = (ℤ‘1)
4 1e0p1 12128 . . . . . . 7 1 = (0 + 1)
54fveq2i 6666 . . . . . 6 (ℤ‘1) = (ℤ‘(0 + 1))
63, 5eqtri 2841 . . . . 5 ℕ = (ℤ‘(0 + 1))
72, 6eleqtrdi 2920 . . . 4 ((𝜑𝑀 ∈ ℕ) → 𝑀 ∈ (ℤ‘(0 + 1)))
8 seqm1 13375 . . . 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 587 . . 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 6664 . . 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 6664 . . . 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 7155 . . 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 2878 . 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 11661 . . . . . 6 ¬ 0 ∈ ℕ
16 disjsn 4639 . . . . . 6 ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
1715, 16mpbir 232 . . . . 5 (ℕ ∩ {0}) = ∅
18 fnresi 6469 . . . . . 6 ( I ↾ ℕ) Fn ℕ
19 c0ex 10623 . . . . . . 7 0 ∈ V
20 snex 5322 . . . . . . 7 {⟨0, 𝑃⟩} ∈ V
2119, 20fnsn 6405 . . . . . 6 {⟨0, {⟨0, 𝑃⟩}⟩} Fn {0}
22 fvun1 6747 . . . . . 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 1442 . . . . 5 (((ℕ ∩ {0}) = ∅ ∧ 𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
2417, 2, 23sylancr 587 . . . 4 ((𝜑𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = (( I ↾ ℕ)‘𝑀))
25 fvresi 6927 . . . . 5 (𝑀 ∈ ℕ → (( I ↾ ℕ)‘𝑀) = 𝑀)
2625adantl 482 . . . 4 ((𝜑𝑀 ∈ ℕ) → (( I ↾ ℕ)‘𝑀) = 𝑀)
2724, 26eqtrd 2853 . . 3 ((𝜑𝑀 ∈ ℕ) → ((( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})‘𝑀) = 𝑀)
2827oveq2d 7161 . 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 6678 . . 3 (𝜑 → (𝑄‘(𝑀 − 1)) ∈ V)
30 simpr 485 . . . . . . . . 9 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
3130oveq1d 7160 . . . . . . . 8 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑚 − 1) = (𝑀 − 1))
3231oveq1d 7160 . . . . . . 7 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝑚 − 1) / 𝑁) = ((𝑀 − 1) / 𝑁))
3330oveq1d 7160 . . . . . . 7 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑚 / 𝑁) = (𝑀 / 𝑁))
3432, 33oveq12d 7163 . . . . . 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 2871 . . . . 5 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) = 𝑊)
3730fveq2d 6667 . . . . . . . . . 10 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑇𝑚) = (𝑇𝑀))
3837fveq2d 6667 . . . . . . . . 9 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (2nd ‘(𝑇𝑚)) = (2nd ‘(𝑇𝑀)))
39 simpl 483 . . . . . . . . . . 11 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → 𝑥 = (𝑄‘(𝑀 − 1)))
4039, 32fveq12d 6670 . . . . . . . . . 10 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → (𝑥‘((𝑚 − 1) / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
4140eleq1d 2894 . . . . . . . . 9 ((𝑥 = (𝑄‘(𝑀 − 1)) ∧ 𝑚 = 𝑀) → ((𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏 ↔ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))
4238, 41riotaeqbidv 7106 . . . . . . . 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 5846 . . . . . . 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 5739 . . . . . 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 6665 . . . . 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 5142 . . . 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 2818 . . . 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 7178 . . . . . 6 (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ∈ V
4935, 48eqeltri 2906 . . . . 5 𝑊 ∈ V
5049mptex 6977 . . . 4 (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))) ∈ V
5146, 47, 50ovmpoa 7294 . . 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 580 . 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 2857 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 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557  cop 4563   cuni 4830   ciun 4910  cmpt 5137   I cid 5452   × cxp 5546  ccnv 5547  ran crn 5549  cres 5550  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  crio 7102  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  0cc0 10525  1c1 10526   + caddc 10528  cmin 10858   / cdiv 11285  cn 11626  cz 11969  cuz 12231  (,)cioo 12726  [,]cicc 12729  ...cfz 12880  seqcseq 13357  t crest 16682  topGenctg 16699   Cn ccn 21760  Homeochmeo 22289  IIcii 23410   CovMap ccvm 32399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13358
This theorem is referenced by:  cvmliftlem6  32434  cvmliftlem8  32436  cvmliftlem9  32437
  Copyright terms: Public domain W3C validator