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Theorem cvmliftlem7 33754
Description: Lemma for cvmlift 33762. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 33753 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 13476 . . . 4 (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
2 cvmliftlem.n . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
32nncnd 12165 . . . . . . 7 (𝜑𝑁 ∈ ℂ)
43adantr 481 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
5 ax-1cn 11105 . . . . . 6 1 ∈ ℂ
6 npcan 11406 . . . . . 6 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
74, 5, 6sylancl 586 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑁 − 1) + 1) = 𝑁)
87oveq2d 7369 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (0...((𝑁 − 1) + 1)) = (0...𝑁))
91, 8sseqtrid 3994 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
10 simpr 485 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
11 elfzelz 13433 . . . . 5 (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℤ)
122nnzd 12522 . . . . 5 (𝜑𝑁 ∈ ℤ)
13 elfzm1b 13511 . . . . 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 231 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
169, 15sseldd 3943 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...𝑁))
17 elfznn0 13526 . . . 4 ((𝑀 − 1) ∈ (0...𝑁) → (𝑀 − 1) ∈ ℕ0)
1817adantl 482 . . 3 ((𝜑 ∧ (𝑀 − 1) ∈ (0...𝑁)) → (𝑀 − 1) ∈ ℕ0)
19 eleq1 2825 . . . . . . 7 (𝑦 = 0 → (𝑦 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
20 fveq2 6839 . . . . . . . . 9 (𝑦 = 0 → (𝑄𝑦) = (𝑄‘0))
21 oveq1 7360 . . . . . . . . 9 (𝑦 = 0 → (𝑦 / 𝑁) = (0 / 𝑁))
2220, 21fveq12d 6846 . . . . . . . 8 (𝑦 = 0 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘0)‘(0 / 𝑁)))
23 fvoveq1 7376 . . . . . . . . . 10 (𝑦 = 0 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(0 / 𝑁)))
2423sneqd 4596 . . . . . . . . 9 (𝑦 = 0 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(0 / 𝑁))})
2524imaeq2d 6011 . . . . . . . 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 2825 . . . . . . 7 (𝑦 = 𝑛 → (𝑦 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...𝑁)))
30 fveq2 6839 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑄𝑦) = (𝑄𝑛))
31 oveq1 7360 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
3230, 31fveq12d 6846 . . . . . . . 8 (𝑦 = 𝑛 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
33 fvoveq1 7376 . . . . . . . . . 10 (𝑦 = 𝑛 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
3433sneqd 4596 . . . . . . . . 9 (𝑦 = 𝑛 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
3534imaeq2d 6011 . . . . . . . 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 2825 . . . . . . 7 (𝑦 = (𝑛 + 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑛 + 1) ∈ (0...𝑁)))
40 fveq2 6839 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑄𝑦) = (𝑄‘(𝑛 + 1)))
41 oveq1 7360 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
4240, 41fveq12d 6846 . . . . . . . 8 (𝑦 = (𝑛 + 1) → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)))
43 fvoveq1 7376 . . . . . . . . . 10 (𝑦 = (𝑛 + 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
4443sneqd 4596 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑛 + 1) / 𝑁))})
4544imaeq2d 6011 . . . . . . . 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 2825 . . . . . . 7 (𝑦 = (𝑀 − 1) → (𝑦 ∈ (0...𝑁) ↔ (𝑀 − 1) ∈ (0...𝑁)))
50 fveq2 6839 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑄𝑦) = (𝑄‘(𝑀 − 1)))
51 oveq1 7360 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑦 / 𝑁) = ((𝑀 − 1) / 𝑁))
5250, 51fveq12d 6846 . . . . . . . 8 (𝑦 = (𝑀 − 1) → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
53 fvoveq1 7376 . . . . . . . . . 10 (𝑦 = (𝑀 − 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑀 − 1) / 𝑁)))
5453sneqd 4596 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑀 − 1) / 𝑁))})
5554imaeq2d 6011 . . . . . . . 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 33751 . . . . . . . . . 10 (𝑄‘0) = {⟨0, 𝑃⟩}
7170a1i 11 . . . . . . . . 9 (𝜑 → (𝑄‘0) = {⟨0, 𝑃⟩})
722nnne0d 12199 . . . . . . . . . 10 (𝜑𝑁 ≠ 0)
733, 72div0d 11926 . . . . . . . . 9 (𝜑 → (0 / 𝑁) = 0)
7471, 73fveq12d 6846 . . . . . . . 8 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = ({⟨0, 𝑃⟩}‘0))
75 0nn0 12424 . . . . . . . . 9 0 ∈ ℕ0
76 fvsng 7122 . . . . . . . . 9 ((0 ∈ ℕ0𝑃𝐵) → ({⟨0, 𝑃⟩}‘0) = 𝑃)
7775, 64, 76sylancr 587 . . . . . . . 8 (𝜑 → ({⟨0, 𝑃⟩}‘0) = 𝑃)
7874, 77eqtrd 2776 . . . . . . 7 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = 𝑃)
7973fveq2d 6843 . . . . . . . . 9 (𝜑 → (𝐺‘(0 / 𝑁)) = (𝐺‘0))
8065, 79eqtr4d 2779 . . . . . . . 8 (𝜑 → (𝐹𝑃) = (𝐺‘(0 / 𝑁)))
81 cvmcn 33725 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
8262, 81syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
8360, 61cnf 22581 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
84 ffn 6665 . . . . . . . . . 10 (𝐹:𝐵𝑋𝐹 Fn 𝐵)
8582, 83, 843syl 18 . . . . . . . . 9 (𝜑𝐹 Fn 𝐵)
86 fniniseg 7007 . . . . . . . . 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 711 . . . . . . 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 12797 . . . . . . . . . 10 0 = (ℤ‘0)
9391, 92eleqtrdi 2848 . . . . . . . . 9 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
9493adantl 482 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
95 peano2fzr 13446 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘0) ∧ (𝑛 + 1) ∈ (0...𝑁)) → 𝑛 ∈ (0...𝑁))
9695ex 413 . . . . . . . 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 2736 . . . . . . . . . . 11 ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))
100 simprlr 778 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (0...𝑁))
101 elfzle2 13437 . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℕ0)
104 nn0p1nn 12448 . . . . . . . . . . . . . . 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 12798 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℤ)
109 elfz5 13425 . . . . . . . . . . . . 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 256 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 + 1) ∈ (1...𝑁))
112 simprr 771 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))
113103nn0cnd 12471 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℂ)
114 pncan 11403 . . . . . . . . . . . . . . 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 6843 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄𝑛))
117115oveq1d 7368 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
118116, 117fveq12d 6846 . . . . . . . . . . . 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 6843 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐺‘(((𝑛 + 1) − 1) / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
120119sneqd 4596 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
121120imaeq2d 6011 . . . . . . . . . . . 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 33753 . . . . . . . . . 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 495 . . . . . . . . 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 12470 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℝ)
1262adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℕ)
127125, 126nndivred 12203 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ)
128127rexrd 11201 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ*)
129 peano2re 11324 . . . . . . . . . . . . . 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 12203 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
132131rexrd 11201 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ*)
133125ltp1d 12081 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 < (𝑛 + 1))
134126nnred 12164 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℝ)
135126nngt0d 12198 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 0 < 𝑁)
136 ltdiv1 12015 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
137125, 130, 134, 135, 136syl112anc 1374 . . . . . . . . . . . . 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 231 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁))
139127, 131, 138ltled 11299 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
140 ubicc2 13374 . . . . . . . . . . 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 1371 . . . . . . . . . 10 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
142117oveq1d 7368 . . . . . . . . . 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 7032 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)) ∈ 𝐵)
145123simprd 496 . . . . . . . . . . 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 5935 . . . . . . . . . . 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 2776 . . . . . . . . . 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 6841 . . . . . . . . 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 6651 . . . . . . . . . . 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 231 . . . . . . . . . 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 6937 . . . . . . . . . 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 6858 . . . . . . . . . 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 2784 . . . . . . . 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 481 . . . . . . . . 9 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝐹 Fn 𝐵)
157 fniniseg 7007 . . . . . . . . 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 711 . . . . . . 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 457 . . . . . 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 844 . . . . 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 12594 . . . 4 ((𝑀 − 1) ∈ ℕ0 → (𝜑 → ((𝑀 − 1) ∈ (0...𝑁) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))))
163162impd 411 . . 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 205  wa 396   = wceq 1541  wcel 2106  wral 3062  {crab 3405  Vcvv 3443  cdif 3905  cun 3906  cin 3907  wss 3908  c0 4280  𝒫 cpw 4558  {csn 4584  cop 4590   cuni 4863   ciun 4952   class class class wbr 5103  cmpt 5186   I cid 5528   × cxp 5629  ccnv 5630  ran crn 5632  cres 5633  cima 5634  ccom 5635   Fn wfn 6488  wf 6489  cfv 6493  crio 7308  (class class class)co 7353  cmpo 7355  1st c1st 7915  2nd c2nd 7916  cc 11045  cr 11046  0cc0 11047  1c1 11048   + caddc 11050  *cxr 11184   < clt 11185  cle 11186  cmin 11381   / cdiv 11808  cn 12149  0cn0 12409  cz 12495  cuz 12759  (,)cioo 13256  [,]cicc 13259  ...cfz 13416  seqcseq 13898  t crest 17294  topGenctg 17311   Cn ccn 22559  Homeochmeo 23088  IIcii 24222   CovMap ccvm 33718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124  ax-pre-sup 11125
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-er 8644  df-map 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fi 9343  df-sup 9374  df-inf 9375  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-div 11809  df-nn 12150  df-2 12212  df-3 12213  df-n0 12410  df-z 12496  df-uz 12760  df-q 12866  df-rp 12908  df-xneg 13025  df-xadd 13026  df-xmul 13027  df-icc 13263  df-fz 13417  df-seq 13899  df-exp 13960  df-cj 14976  df-re 14977  df-im 14978  df-sqrt 15112  df-abs 15113  df-rest 17296  df-topgen 17317  df-psmet 20773  df-xmet 20774  df-met 20775  df-bl 20776  df-mopn 20777  df-top 22227  df-topon 22244  df-bases 22280  df-cn 22562  df-hmeo 23090  df-ii 24224  df-cvm 33719
This theorem is referenced by:  cvmliftlem8  33755  cvmliftlem9  33756  cvmliftlem10  33757  cvmliftlem13  33759
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