Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmliftlem7 Structured version   Visualization version   GIF version

Theorem cvmliftlem7 31399
Description: Lemma for cvmlift 31407. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 31398 to show functionality and lifting of 𝑄). (Contributed by Mario Carneiro, 14-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
cvmliftlem7 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
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 cvmliftlem7
Dummy variables 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzssp1 12422 . . . 4 (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
2 cvmliftlem.n . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
32nncnd 11074 . . . . . . 7 (𝜑𝑁 ∈ ℂ)
43adantr 480 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
5 ax-1cn 10032 . . . . . 6 1 ∈ ℂ
6 npcan 10328 . . . . . 6 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
74, 5, 6sylancl 695 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑁 − 1) + 1) = 𝑁)
87oveq2d 6706 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (0...((𝑁 − 1) + 1)) = (0...𝑁))
91, 8syl5sseq 3686 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
10 simpr 476 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
11 elfzelz 12380 . . . . 5 (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℤ)
122nnzd 11519 . . . . 5 (𝜑𝑁 ∈ ℤ)
13 elfzm1b 12456 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
1411, 12, 13syl2anr 494 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
1510, 14mpbid 222 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
169, 15sseldd 3637 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...𝑁))
17 elfznn0 12471 . . . 4 ((𝑀 − 1) ∈ (0...𝑁) → (𝑀 − 1) ∈ ℕ0)
1817adantl 481 . . 3 ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → (𝑀 − 1) ∈ ℕ0)
19 eleq1 2718 . . . . . . 7 (𝑦 = 0 → (𝑦 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
20 fveq2 6229 . . . . . . . . 9 (𝑦 = 0 → (𝑄𝑦) = (𝑄‘0))
21 oveq1 6697 . . . . . . . . 9 (𝑦 = 0 → (𝑦 / 𝑁) = (0 / 𝑁))
2220, 21fveq12d 6235 . . . . . . . 8 (𝑦 = 0 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘0)‘(0 / 𝑁)))
2321fveq2d 6233 . . . . . . . . . 10 (𝑦 = 0 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(0 / 𝑁)))
2423sneqd 4222 . . . . . . . . 9 (𝑦 = 0 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(0 / 𝑁))})
2524imaeq2d 5501 . . . . . . . 8 (𝑦 = 0 → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘(0 / 𝑁))}))
2622, 25eleq12d 2724 . . . . . . 7 (𝑦 = 0 → (((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘0)‘(0 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))})))
2719, 26imbi12d 333 . . . . . 6 (𝑦 = 0 → ((𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ (0 ∈ (0...𝑁) → ((𝑄‘0)‘(0 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))}))))
2827imbi2d 329 . . . . 5 (𝑦 = 0 → ((𝜑 → (𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}))) ↔ (𝜑 → (0 ∈ (0...𝑁) → ((𝑄‘0)‘(0 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))})))))
29 eleq1 2718 . . . . . . 7 (𝑦 = 𝑛 → (𝑦 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
30 fveq2 6229 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑄𝑦) = (𝑄𝑛))
31 oveq1 6697 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
3230, 31fveq12d 6235 . . . . . . . 8 (𝑦 = 𝑛 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
3331fveq2d 6233 . . . . . . . . . 10 (𝑦 = 𝑛 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
3433sneqd 4222 . . . . . . . . 9 (𝑦 = 𝑛 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
3534imaeq2d 5501 . . . . . . . 8 (𝑦 = 𝑛 → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
3632, 35eleq12d 2724 . . . . . . 7 (𝑦 = 𝑛 → (((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))})))
3729, 36imbi12d 333 . . . . . 6 (𝑦 = 𝑛 → ((𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ (𝑛 ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))))
3837imbi2d 329 . . . . 5 (𝑦 = 𝑛 → ((𝜑 → (𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}))) ↔ (𝜑 → (𝑛 ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))})))))
39 eleq1 2718 . . . . . . 7 (𝑦 = (𝑛 + 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑛 + 1) ∈ (0...𝑁)))
40 fveq2 6229 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑄𝑦) = (𝑄‘(𝑛 + 1)))
41 oveq1 6697 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
4240, 41fveq12d 6235 . . . . . . . 8 (𝑦 = (𝑛 + 1) → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)))
4341fveq2d 6233 . . . . . . . . . 10 (𝑦 = (𝑛 + 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
4443sneqd 4222 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑛 + 1) / 𝑁))})
4544imaeq2d 5501 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))
4642, 45eleq12d 2724 . . . . . . 7 (𝑦 = (𝑛 + 1) → (((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))
4739, 46imbi12d 333 . . . . . 6 (𝑦 = (𝑛 + 1) → ((𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))))
4847imbi2d 329 . . . . 5 (𝑦 = (𝑛 + 1) → ((𝜑 → (𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}))) ↔ (𝜑 → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))))
49 eleq1 2718 . . . . . . 7 (𝑦 = (𝑀 − 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑀 − 1) ∈ (0...𝑁)))
50 fveq2 6229 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑄𝑦) = (𝑄‘(𝑀 − 1)))
51 oveq1 6697 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑦 / 𝑁) = ((𝑀 − 1) / 𝑁))
5250, 51fveq12d 6235 . . . . . . . 8 (𝑦 = (𝑀 − 1) → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
5351fveq2d 6233 . . . . . . . . . 10 (𝑦 = (𝑀 − 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑀 − 1) / 𝑁)))
5453sneqd 4222 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑀 − 1) / 𝑁))})
5554imaeq2d 5501 . . . . . . . 8 (𝑦 = (𝑀 − 1) → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
5652, 55eleq12d 2724 . . . . . . 7 (𝑦 = (𝑀 − 1) → (((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))
5749, 56imbi12d 333 . . . . . 6 (𝑦 = (𝑀 − 1) → ((𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))))
5857imbi2d 329 . . . . 5 (𝑦 = (𝑀 − 1) → ((𝜑 → (𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}))) ↔ (𝜑 → ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))))
59 cvmliftlem.1 . . . . . . . . . . 11 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
60 cvmliftlem.b . . . . . . . . . . 11 𝐵 = 𝐶
61 cvmliftlem.x . . . . . . . . . . 11 𝑋 = 𝐽
62 cvmliftlem.f . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
63 cvmliftlem.g . . . . . . . . . . 11 (𝜑𝐺 ∈ (II Cn 𝐽))
64 cvmliftlem.p . . . . . . . . . . 11 (𝜑𝑃𝐵)
65 cvmliftlem.e . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) = (𝐺‘0))
66 cvmliftlem.t . . . . . . . . . . 11 (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
67 cvmliftlem.a . . . . . . . . . . 11 (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))
68 cvmliftlem.l . . . . . . . . . . 11 𝐿 = (topGen‘ran (,))
69 cvmliftlem.q . . . . . . . . . . 11 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⟩}⟩}))
7059, 60, 61, 62, 63, 64, 65, 2, 66, 67, 68, 69cvmliftlem4 31396 . . . . . . . . . 10 (𝑄‘0) = {⟨0, 𝑃⟩}
7170a1i 11 . . . . . . . . 9 (𝜑 → (𝑄‘0) = {⟨0, 𝑃⟩})
722nnne0d 11103 . . . . . . . . . 10 (𝜑𝑁 ≠ 0)
733, 72div0d 10838 . . . . . . . . 9 (𝜑 → (0 / 𝑁) = 0)
7471, 73fveq12d 6235 . . . . . . . 8 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = ({⟨0, 𝑃⟩}‘0))
75 0nn0 11345 . . . . . . . . 9 0 ∈ ℕ0
76 fvsng 6488 . . . . . . . . 9 ((0 ∈ ℕ0𝑃𝐵) → ({⟨0, 𝑃⟩}‘0) = 𝑃)
7775, 64, 76sylancr 696 . . . . . . . 8 (𝜑 → ({⟨0, 𝑃⟩}‘0) = 𝑃)
7874, 77eqtrd 2685 . . . . . . 7 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = 𝑃)
7973fveq2d 6233 . . . . . . . . 9 (𝜑 → (𝐺‘(0 / 𝑁)) = (𝐺‘0))
8065, 79eqtr4d 2688 . . . . . . . 8 (𝜑 → (𝐹𝑃) = (𝐺‘(0 / 𝑁)))
81 cvmcn 31370 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
8262, 81syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
8360, 61cnf 21098 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
84 ffn 6083 . . . . . . . . . 10 (𝐹:𝐵𝑋𝐹 Fn 𝐵)
8582, 83, 843syl 18 . . . . . . . . 9 (𝜑𝐹 Fn 𝐵)
86 fniniseg 6378 . . . . . . . . 9 (𝐹 Fn 𝐵 → (𝑃 ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))}) ↔ (𝑃𝐵 ∧ (𝐹𝑃) = (𝐺‘(0 / 𝑁)))))
8785, 86syl 17 . . . . . . . 8 (𝜑 → (𝑃 ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))}) ↔ (𝑃𝐵 ∧ (𝐹𝑃) = (𝐺‘(0 / 𝑁)))))
8864, 80, 87mpbir2and 977 . . . . . . 7 (𝜑𝑃 ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))}))
8978, 88eqeltrd 2730 . . . . . 6 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))}))
9089a1d 25 . . . . 5 (𝜑 → (0 ∈ (0...𝑁) → ((𝑄‘0)‘(0 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))})))
91 id 22 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0𝑛 ∈ ℕ0)
92 nn0uz 11760 . . . . . . . . . . . 12 0 = (ℤ‘0)
9391, 92syl6eleq 2740 . . . . . . . . . . 11 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
9493adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
95 peano2fzr 12392 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘0) ∧ (𝑛 + 1) ∈ (0...𝑁)) → 𝑛 ∈ (0...𝑁))
9695ex 449 . . . . . . . . . 10 (𝑛 ∈ (ℤ‘0) → ((𝑛 + 1) ∈ (0...𝑁) → 𝑛 ∈ (0...𝑁)))
9794, 96syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) ∈ (0...𝑁) → 𝑛 ∈ (0...𝑁)))
9897imim1d 82 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))})) → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))))
99 eqid 2651 . . . . . . . . . . . . . . 15 ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))
100 simprlr 820 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (0...𝑁))
101 elfzle2 12383 . . . . . . . . . . . . . . . . 17 ((𝑛 + 1) ∈ (0...𝑁) → (𝑛 + 1) ≤ 𝑁)
102100, 101syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ≤ 𝑁)
103 simprll 819 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℕ0)
104 nn0p1nn 11370 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
105103, 104syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ ℕ)
106 nnuz 11761 . . . . . . . . . . . . . . . . . 18 ℕ = (ℤ‘1)
107105, 106syl6eleq 2740 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (ℤ‘1))
10812adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℤ)
109 elfz5 12372 . . . . . . . . . . . . . . . . 17 (((𝑛 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → ((𝑛 + 1) ∈ (1...𝑁) ↔ (𝑛 + 1) ≤ 𝑁))
110107, 108, 109syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) ∈ (1...𝑁) ↔ (𝑛 + 1) ≤ 𝑁))
111102, 110mpbird 247 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (1...𝑁))
112 simprr 811 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
113103nn0cnd 11391 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℂ)
114 pncan 10325 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
115113, 5, 114sylancl 695 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) − 1) = 𝑛)
116115fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄𝑛))
117115oveq1d 6705 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
118116, 117fveq12d 6235 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
119117fveq2d 6233 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐺‘(((𝑛 + 1) − 1) / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
120119sneqd 4222 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
121120imaeq2d 5501 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐹 “ {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))}) = (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
122112, 118, 1213eltr4d 2745 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))}))
12359, 60, 61, 62, 63, 64, 65, 2, 66, 67, 68, 69, 99, 111, 122cvmliftlem6 31398 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
124123simpld 474 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
125103nn0red 11390 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℝ)
1262adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℕ)
127125, 126nndivred 11107 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ)
128127rexrd 10127 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ*)
129 peano2re 10247 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
130125, 129syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ ℝ)
131130, 126nndivred 11107 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
132131rexrd 10127 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ*)
133125ltp1d 10992 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 < (𝑛 + 1))
134126nnred 11073 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℝ)
135126nngt0d 11102 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 0 < 𝑁)
136 ltdiv1 10925 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
137125, 130, 134, 135, 136syl112anc 1370 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
138133, 137mpbid 222 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁))
139127, 131, 138ltled 10223 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
140 ubicc2 12327 . . . . . . . . . . . . . . 15 (((𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ* ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁)) → ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
141128, 132, 139, 140syl3anc 1366 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
142117oveq1d 6705 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
143141, 142eleqtrrd 2733 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)))
144124, 143ffvelrnd 6400 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ 𝐵)
145123simprd 478 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))))
146142reseq2d 5428 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
147145, 146eqtrd 2685 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
148147fveq1d 6231 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝐹 ∘ (𝑄‘(𝑛 + 1)))‘((𝑛 + 1) / 𝑁)) = ((𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))‘((𝑛 + 1) / 𝑁)))
149142feq2d 6069 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ↔ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵))
150124, 149mpbid 222 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
151 fvco3 6314 . . . . . . . . . . . . . 14 (((𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) → ((𝐹 ∘ (𝑄‘(𝑛 + 1)))‘((𝑛 + 1) / 𝑁)) = (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))))
152150, 141, 151syl2anc 694 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝐹 ∘ (𝑄‘(𝑛 + 1)))‘((𝑛 + 1) / 𝑁)) = (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))))
153 fvres 6245 . . . . . . . . . . . . . 14 (((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) → ((𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))‘((𝑛 + 1) / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
154141, 153syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))‘((𝑛 + 1) / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
155148, 152, 1543eqtr3d 2693 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))) = (𝐺‘((𝑛 + 1) / 𝑁)))
15685adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝐹 Fn 𝐵)
157 fniniseg 6378 . . . . . . . . . . . . 13 (𝐹 Fn 𝐵 → (((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}) ↔ (((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))) = (𝐺‘((𝑛 + 1) / 𝑁)))))
158156, 157syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}) ↔ (((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁))) = (𝐺‘((𝑛 + 1) / 𝑁)))))
159144, 155, 158mpbir2and 977 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))
160159expr 642 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁))) → (((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))
161160expr 642 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) ∈ (0...𝑁) → (((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))))
162161a2d 29 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → (((𝑛 + 1) ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))})) → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))))
16398, 162syld 47 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))})) → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))))
164163expcom 450 . . . . . 6 (𝑛 ∈ ℕ0 → (𝜑 → ((𝑛 ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))})) → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))))
165164a2d 29 . . . . 5 (𝑛 ∈ ℕ0 → ((𝜑 → (𝑛 ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝜑 → ((𝑛 + 1) ∈ (0...𝑁) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))))
16628, 38, 48, 58, 90, 165nn0ind 11510 . . . 4 ((𝑀 − 1) ∈ ℕ0 → (𝜑 → ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))))
167166impd 446 . . 3 ((𝑀 − 1) ∈ ℕ0 → ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))
16818, 167mpcom 38 . 2 ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
16916, 168syldan 486 1 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   I cid 5052   × cxp 5141  ccnv 5142  ran crn 5144  cres 5145  cima 5146  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  crio 6650  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  0cn0 11330  cz 11415  cuz 11725  (,)cioo 12213  [,]cicc 12216  ...cfz 12364  seqcseq 12841  t crest 16128  topGenctg 16145   Cn ccn 21076  Homeochmeo 21604  IIcii 22725   CovMap ccvm 31363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-icc 12220  df-fz 12365  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-hmeo 21606  df-ii 22727  df-cvm 31364
This theorem is referenced by:  cvmliftlem8  31400  cvmliftlem9  31401  cvmliftlem10  31402  cvmliftlem13  31404
  Copyright terms: Public domain W3C validator