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Theorem cvmliftlem7 35275
Description: Lemma for cvmlift 35283. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 35274 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 13603 . . . 4 (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
2 cvmliftlem.n . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
32nncnd 12279 . . . . . . 7 (𝜑𝑁 ∈ ℂ)
43adantr 480 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
5 ax-1cn 11210 . . . . . 6 1 ∈ ℂ
6 npcan 11514 . . . . . 6 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
74, 5, 6sylancl 586 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑁 − 1) + 1) = 𝑁)
87oveq2d 7446 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (0...((𝑁 − 1) + 1)) = (0...𝑁))
91, 8sseqtrid 4047 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
10 simpr 484 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
11 elfzelz 13560 . . . . 5 (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℤ)
122nnzd 12637 . . . . 5 (𝜑𝑁 ∈ ℤ)
13 elfzm1b 13638 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
1411, 12, 13syl2anr 597 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
1510, 14mpbid 232 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
169, 15sseldd 3995 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...𝑁))
17 elfznn0 13656 . . . 4 ((𝑀 − 1) ∈ (0...𝑁) → (𝑀 − 1) ∈ ℕ0)
1817adantl 481 . . 3 ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → (𝑀 − 1) ∈ ℕ0)
19 eleq1 2826 . . . . . . 7 (𝑦 = 0 → (𝑦 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
20 fveq2 6906 . . . . . . . . 9 (𝑦 = 0 → (𝑄𝑦) = (𝑄‘0))
21 oveq1 7437 . . . . . . . . 9 (𝑦 = 0 → (𝑦 / 𝑁) = (0 / 𝑁))
2220, 21fveq12d 6913 . . . . . . . 8 (𝑦 = 0 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘0)‘(0 / 𝑁)))
23 fvoveq1 7453 . . . . . . . . . 10 (𝑦 = 0 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(0 / 𝑁)))
2423sneqd 4642 . . . . . . . . 9 (𝑦 = 0 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(0 / 𝑁))})
2524imaeq2d 6079 . . . . . . . 8 (𝑦 = 0 → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘(0 / 𝑁))}))
2622, 25eleq12d 2832 . . . . . . 7 (𝑦 = 0 → (((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘0)‘(0 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))})))
2719, 26imbi12d 344 . . . . . 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 340 . . . . 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 2826 . . . . . . 7 (𝑦 = 𝑛 → (𝑦 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
30 fveq2 6906 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑄𝑦) = (𝑄𝑛))
31 oveq1 7437 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
3230, 31fveq12d 6913 . . . . . . . 8 (𝑦 = 𝑛 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
33 fvoveq1 7453 . . . . . . . . . 10 (𝑦 = 𝑛 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
3433sneqd 4642 . . . . . . . . 9 (𝑦 = 𝑛 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
3534imaeq2d 6079 . . . . . . . 8 (𝑦 = 𝑛 → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
3632, 35eleq12d 2832 . . . . . . 7 (𝑦 = 𝑛 → (((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))})))
3729, 36imbi12d 344 . . . . . 6 (𝑦 = 𝑛 → ((𝑦 ∈ (0...𝑁) → ((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))})) ↔ (𝑛 ∈ (0...𝑁) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))))
3837imbi2d 340 . . . . 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 2826 . . . . . . 7 (𝑦 = (𝑛 + 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑛 + 1) ∈ (0...𝑁)))
40 fveq2 6906 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑄𝑦) = (𝑄‘(𝑛 + 1)))
41 oveq1 7437 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
4240, 41fveq12d 6913 . . . . . . . 8 (𝑦 = (𝑛 + 1) → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)))
43 fvoveq1 7453 . . . . . . . . . 10 (𝑦 = (𝑛 + 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
4443sneqd 4642 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑛 + 1) / 𝑁))})
4544imaeq2d 6079 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))
4642, 45eleq12d 2832 . . . . . . 7 (𝑦 = (𝑛 + 1) → (((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))})))
4739, 46imbi12d 344 . . . . . 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 340 . . . . 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 2826 . . . . . . 7 (𝑦 = (𝑀 − 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑀 − 1) ∈ (0...𝑁)))
50 fveq2 6906 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑄𝑦) = (𝑄‘(𝑀 − 1)))
51 oveq1 7437 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑦 / 𝑁) = ((𝑀 − 1) / 𝑁))
5250, 51fveq12d 6913 . . . . . . . 8 (𝑦 = (𝑀 − 1) → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
53 fvoveq1 7453 . . . . . . . . . 10 (𝑦 = (𝑀 − 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑀 − 1) / 𝑁)))
5453sneqd 4642 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑀 − 1) / 𝑁))})
5554imaeq2d 6079 . . . . . . . 8 (𝑦 = (𝑀 − 1) → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
5652, 55eleq12d 2832 . . . . . . 7 (𝑦 = (𝑀 − 1) → (((𝑄𝑦)‘(𝑦 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) ↔ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))
5749, 56imbi12d 344 . . . . . 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 340 . . . . 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 35272 . . . . . . . . . 10 (𝑄‘0) = {⟨0, 𝑃⟩}
7170a1i 11 . . . . . . . . 9 (𝜑 → (𝑄‘0) = {⟨0, 𝑃⟩})
722nnne0d 12313 . . . . . . . . . 10 (𝜑𝑁 ≠ 0)
733, 72div0d 12039 . . . . . . . . 9 (𝜑 → (0 / 𝑁) = 0)
7471, 73fveq12d 6913 . . . . . . . 8 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = ({⟨0, 𝑃⟩}‘0))
75 0nn0 12538 . . . . . . . . 9 0 ∈ ℕ0
76 fvsng 7199 . . . . . . . . 9 ((0 ∈ ℕ0𝑃𝐵) → ({⟨0, 𝑃⟩}‘0) = 𝑃)
7775, 64, 76sylancr 587 . . . . . . . 8 (𝜑 → ({⟨0, 𝑃⟩}‘0) = 𝑃)
7874, 77eqtrd 2774 . . . . . . 7 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = 𝑃)
7973fveq2d 6910 . . . . . . . . 9 (𝜑 → (𝐺‘(0 / 𝑁)) = (𝐺‘0))
8065, 79eqtr4d 2777 . . . . . . . 8 (𝜑 → (𝐹𝑃) = (𝐺‘(0 / 𝑁)))
81 cvmcn 35246 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
8262, 81syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
8360, 61cnf 23269 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
84 ffn 6736 . . . . . . . . . 10 (𝐹:𝐵𝑋𝐹 Fn 𝐵)
8582, 83, 843syl 18 . . . . . . . . 9 (𝜑𝐹 Fn 𝐵)
86 fniniseg 7079 . . . . . . . . 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 713 . . . . . . 7 (𝜑𝑃 ∈ (𝐹 “ {(𝐺‘(0 / 𝑁))}))
8978, 88eqeltrd 2838 . . . . . 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 . . . . . . . . . 10 (𝑛 ∈ ℕ0𝑛 ∈ ℕ0)
92 nn0uz 12917 . . . . . . . . . 10 0 = (ℤ‘0)
9391, 92eleqtrdi 2848 . . . . . . . . 9 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
9493adantl 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
95 peano2fzr 13573 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘0) ∧ (𝑛 + 1) ∈ (0...𝑁)) → 𝑛 ∈ (0...𝑁))
9695ex 412 . . . . . . . 8 (𝑛 ∈ (ℤ‘0) → ((𝑛 + 1) ∈ (0...𝑁) → 𝑛 ∈ (0...𝑁)))
9794, 96syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) ∈ (0...𝑁) → 𝑛 ∈ (0...𝑁)))
9897imim1d 82 . . . . . 6 ((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 2734 . . . . . . . . . . 11 ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))
100 simprlr 780 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (0...𝑁))
101 elfzle2 13564 . . . . . . . . . . . . 13 ((𝑛 + 1) ∈ (0...𝑁) → (𝑛 + 1) ≤ 𝑁)
102100, 101syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ≤ 𝑁)
103 simprll 779 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℕ0)
104 nn0p1nn 12562 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
105103, 104syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ ℕ)
106 nnuz 12918 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
107105, 106eleqtrdi 2848 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (ℤ‘1))
10812adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℤ)
109 elfz5 13552 . . . . . . . . . . . . 13 (((𝑛 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → ((𝑛 + 1) ∈ (1...𝑁) ↔ (𝑛 + 1) ≤ 𝑁))
110107, 108, 109syl2anc 584 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) ∈ (1...𝑁) ↔ (𝑛 + 1) ≤ 𝑁))
111102, 110mpbird 257 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (1...𝑁))
112 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
113103nn0cnd 12586 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℂ)
114 pncan 11511 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
115113, 5, 114sylancl 586 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) − 1) = 𝑛)
116115fveq2d 6910 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄𝑛))
117115oveq1d 7445 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
118116, 117fveq12d 6913 . . . . . . . . . . . 12 ((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 6910 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐺‘(((𝑛 + 1) − 1) / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
120119sneqd 4642 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
121120imaeq2d 6079 . . . . . . . . . . . 12 ((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 2853 . . . . . . . . . . 11 ((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 35274 . . . . . . . . . 10 ((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 494 . . . . . . . . 9 ((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 12585 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℝ)
1262adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℕ)
127125, 126nndivred 12317 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ)
128127rexrd 11308 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ*)
129 peano2re 11431 . . . . . . . . . . . . . 14 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
130125, 129syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ ℝ)
131130, 126nndivred 12317 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
132131rexrd 11308 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ*)
133125ltp1d 12195 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 < (𝑛 + 1))
134126nnred 12278 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℝ)
135126nngt0d 12312 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 0 < 𝑁)
136 ltdiv1 12129 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
137125, 130, 134, 135, 136syl112anc 1373 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
138133, 137mpbid 232 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁))
139127, 131, 138ltled 11406 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
140 ubicc2 13501 . . . . . . . . . . 11 (((𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ* ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁)) → ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
141128, 132, 139, 140syl3anc 1370 . . . . . . . . . 10 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
142117oveq1d 7445 . . . . . . . . . 10 ((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 2841 . . . . . . . . 9 ((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, 143ffvelcdmd 7104 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ 𝐵)
145123simprd 495 . . . . . . . . . . 11 ((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 5999 . . . . . . . . . . 11 ((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 2774 . . . . . . . . . 10 ((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 6908 . . . . . . . . 9 ((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 6722 . . . . . . . . . . 11 ((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 232 . . . . . . . . . 10 ((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 7007 . . . . . . . . . 10 (((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 584 . . . . . . . . 9 ((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 6925 . . . . . . . . . 10 (((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) → ((𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))‘((𝑛 + 1) / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
154141, 153syl 17 . . . . . . . . 9 ((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 2782 . . . . . . . 8 ((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 . . . . . . . . 9 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝐹 Fn 𝐵)
157 fniniseg 7079 . . . . . . . . 9 (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 . . . . . . . 8 ((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 713 . . . . . . 7 ((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 456 . . . . . 6 ((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))})))
16198, 160animpimp2impd 846 . . . . 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))})))))
16228, 38, 48, 58, 90, 161nn0ind 12710 . . . 4 ((𝑀 − 1) ∈ ℕ0 → (𝜑 → ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))))
163162impd 410 . . 3 ((𝑀 − 1) ∈ ℕ0 → ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})))
16418, 163mpcom 38 . 2 ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
16516, 164syldan 591 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 206  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630  cop 4636   cuni 4911   ciun 4995   class class class wbr 5147  cmpt 5230   I cid 5581   × cxp 5686  ccnv 5687  ran crn 5689  cres 5690  cima 5691  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  crio 7386  (class class class)co 7430  cmpo 7432  1st c1st 8010  2nd c2nd 8011  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155  *cxr 11291   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  cn 12263  0cn0 12523  cz 12610  cuz 12875  (,)cioo 13383  [,]cicc 13386  ...cfz 13543  seqcseq 14038  t crest 17466  topGenctg 17483   Cn ccn 23247  Homeochmeo 23776  IIcii 24914   CovMap ccvm 35239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-icc 13390  df-fz 13544  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-rest 17468  df-topgen 17489  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-top 22915  df-topon 22932  df-bases 22968  df-cn 23250  df-hmeo 23778  df-ii 24916  df-cvm 35240
This theorem is referenced by:  cvmliftlem8  35276  cvmliftlem9  35277  cvmliftlem10  35278  cvmliftlem13  35280
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