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Theorem cvmliftlem7 35296
Description: Lemma for cvmlift 35304. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 35295 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 13607 . . . 4 (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
2 cvmliftlem.n . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
32nncnd 12282 . . . . . . 7 (𝜑𝑁 ∈ ℂ)
43adantr 480 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
5 ax-1cn 11213 . . . . . 6 1 ∈ ℂ
6 npcan 11517 . . . . . 6 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
74, 5, 6sylancl 586 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑁 − 1) + 1) = 𝑁)
87oveq2d 7447 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (0...((𝑁 − 1) + 1)) = (0...𝑁))
91, 8sseqtrid 4026 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
10 simpr 484 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
11 elfzelz 13564 . . . . 5 (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℤ)
122nnzd 12640 . . . . 5 (𝜑𝑁 ∈ ℤ)
13 elfzm1b 13642 . . . . 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 3984 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...𝑁))
17 elfznn0 13660 . . . 4 ((𝑀 − 1) ∈ (0...𝑁) → (𝑀 − 1) ∈ ℕ0)
1817adantl 481 . . 3 ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → (𝑀 − 1) ∈ ℕ0)
19 eleq1 2829 . . . . . . 7 (𝑦 = 0 → (𝑦 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
20 fveq2 6906 . . . . . . . . 9 (𝑦 = 0 → (𝑄𝑦) = (𝑄‘0))
21 oveq1 7438 . . . . . . . . 9 (𝑦 = 0 → (𝑦 / 𝑁) = (0 / 𝑁))
2220, 21fveq12d 6913 . . . . . . . 8 (𝑦 = 0 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘0)‘(0 / 𝑁)))
23 fvoveq1 7454 . . . . . . . . . 10 (𝑦 = 0 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(0 / 𝑁)))
2423sneqd 4638 . . . . . . . . 9 (𝑦 = 0 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(0 / 𝑁))})
2524imaeq2d 6078 . . . . . . . 8 (𝑦 = 0 → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘(0 / 𝑁))}))
2622, 25eleq12d 2835 . . . . . . 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 2829 . . . . . . 7 (𝑦 = 𝑛 → (𝑦 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
30 fveq2 6906 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑄𝑦) = (𝑄𝑛))
31 oveq1 7438 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
3230, 31fveq12d 6913 . . . . . . . 8 (𝑦 = 𝑛 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
33 fvoveq1 7454 . . . . . . . . . 10 (𝑦 = 𝑛 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
3433sneqd 4638 . . . . . . . . 9 (𝑦 = 𝑛 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
3534imaeq2d 6078 . . . . . . . 8 (𝑦 = 𝑛 → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
3632, 35eleq12d 2835 . . . . . . 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 2829 . . . . . . 7 (𝑦 = (𝑛 + 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑛 + 1) ∈ (0...𝑁)))
40 fveq2 6906 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑄𝑦) = (𝑄‘(𝑛 + 1)))
41 oveq1 7438 . . . . . . . . 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 7454 . . . . . . . . . 10 (𝑦 = (𝑛 + 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
4443sneqd 4638 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑛 + 1) / 𝑁))})
4544imaeq2d 6078 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘((𝑛 + 1) / 𝑁))}))
4642, 45eleq12d 2835 . . . . . . 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 2829 . . . . . . 7 (𝑦 = (𝑀 − 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑀 − 1) ∈ (0...𝑁)))
50 fveq2 6906 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑄𝑦) = (𝑄‘(𝑀 − 1)))
51 oveq1 7438 . . . . . . . . 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 7454 . . . . . . . . . 10 (𝑦 = (𝑀 − 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑀 − 1) / 𝑁)))
5453sneqd 4638 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑀 − 1) / 𝑁))})
5554imaeq2d 6078 . . . . . . . 8 (𝑦 = (𝑀 − 1) → (𝐹 “ {(𝐺‘(𝑦 / 𝑁))}) = (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
5652, 55eleq12d 2835 . . . . . . 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 35293 . . . . . . . . . 10 (𝑄‘0) = {⟨0, 𝑃⟩}
7170a1i 11 . . . . . . . . 9 (𝜑 → (𝑄‘0) = {⟨0, 𝑃⟩})
722nnne0d 12316 . . . . . . . . . 10 (𝜑𝑁 ≠ 0)
733, 72div0d 12042 . . . . . . . . 9 (𝜑 → (0 / 𝑁) = 0)
7471, 73fveq12d 6913 . . . . . . . 8 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = ({⟨0, 𝑃⟩}‘0))
75 0nn0 12541 . . . . . . . . 9 0 ∈ ℕ0
76 fvsng 7200 . . . . . . . . 9 ((0 ∈ ℕ0𝑃𝐵) → ({⟨0, 𝑃⟩}‘0) = 𝑃)
7775, 64, 76sylancr 587 . . . . . . . 8 (𝜑 → ({⟨0, 𝑃⟩}‘0) = 𝑃)
7874, 77eqtrd 2777 . . . . . . 7 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = 𝑃)
7973fveq2d 6910 . . . . . . . . 9 (𝜑 → (𝐺‘(0 / 𝑁)) = (𝐺‘0))
8065, 79eqtr4d 2780 . . . . . . . 8 (𝜑 → (𝐹𝑃) = (𝐺‘(0 / 𝑁)))
81 cvmcn 35267 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
8262, 81syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
8360, 61cnf 23254 . . . . . . . . . 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 7080 . . . . . . . . 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 2841 . . . . . 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 12920 . . . . . . . . . 10 0 = (ℤ‘0)
9391, 92eleqtrdi 2851 . . . . . . . . 9 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
9493adantl 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
95 peano2fzr 13577 . . . . . . . . 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 2737 . . . . . . . . . . 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 13568 . . . . . . . . . . . . 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 12565 . . . . . . . . . . . . . . 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 12921 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
107105, 106eleqtrdi 2851 . . . . . . . . . . . . 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 13556 . . . . . . . . . . . . 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 12589 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℂ)
114 pncan 11514 . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . 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 4638 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
121120imaeq2d 6078 . . . . . . . . . . . 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 2856 . . . . . . . . . . 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 35295 . . . . . . . . . 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 12588 . . . . . . . . . . . . 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 12320 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ)
128127rexrd 11311 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ*)
129 peano2re 11434 . . . . . . . . . . . . . 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 12320 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
132131rexrd 11311 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ*)
133125ltp1d 12198 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 < (𝑛 + 1))
134126nnred 12281 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℝ)
135126nngt0d 12315 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 0 < 𝑁)
136 ltdiv1 12132 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
137125, 130, 134, 135, 136syl112anc 1376 . . . . . . . . . . . . 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 11409 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
140 ubicc2 13505 . . . . . . . . . . 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 1373 . . . . . . . . . 10 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
142117oveq1d 7446 . . . . . . . . . 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 2844 . . . . . . . . 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 7105 . . . . . . . 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 5997 . . . . . . . . . . 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 2777 . . . . . . . . . 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 7008 . . . . . . . . . 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 2785 . . . . . . . 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 7080 . . . . . . . . 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 847 . . . . 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 12713 . . . 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 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  cop 4632   cuni 4907   ciun 4991   class class class wbr 5143  cmpt 5225   I cid 5577   × cxp 5683  ccnv 5684  ran crn 5686  cres 5687  cima 5688  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  crio 7387  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  0cn0 12526  cz 12613  cuz 12878  (,)cioo 13387  [,]cicc 13390  ...cfz 13547  seqcseq 14042  t crest 17465  topGenctg 17482   Cn ccn 23232  Homeochmeo 23761  IIcii 24901   CovMap ccvm 35260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-icc 13394  df-fz 13548  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-hmeo 23763  df-ii 24903  df-cvm 35261
This theorem is referenced by:  cvmliftlem8  35297  cvmliftlem9  35298  cvmliftlem10  35299  cvmliftlem13  35301
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