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Theorem cvmliftlem7 33885
Description: Lemma for cvmlift 33893. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 33884 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 13484 . . . 4 (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
2 cvmliftlem.n . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
32nncnd 12169 . . . . . . 7 (𝜑𝑁 ∈ ℂ)
43adantr 481 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℂ)
5 ax-1cn 11109 . . . . . 6 1 ∈ ℂ
6 npcan 11410 . . . . . 6 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
74, 5, 6sylancl 586 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑁 − 1) + 1) = 𝑁)
87oveq2d 7373 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (0...((𝑁 − 1) + 1)) = (0...𝑁))
91, 8sseqtrid 3996 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
10 simpr 485 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
11 elfzelz 13441 . . . . 5 (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℤ)
122nnzd 12526 . . . . 5 (𝜑𝑁 ∈ ℤ)
13 elfzm1b 13519 . . . . 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 3945 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ (0...𝑁))
17 elfznn0 13534 . . . 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 6842 . . . . . . . . 9 (𝑦 = 0 → (𝑄𝑦) = (𝑄‘0))
21 oveq1 7364 . . . . . . . . 9 (𝑦 = 0 → (𝑦 / 𝑁) = (0 / 𝑁))
2220, 21fveq12d 6849 . . . . . . . 8 (𝑦 = 0 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘0)‘(0 / 𝑁)))
23 fvoveq1 7380 . . . . . . . . . 10 (𝑦 = 0 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(0 / 𝑁)))
2423sneqd 4598 . . . . . . . . 9 (𝑦 = 0 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(0 / 𝑁))})
2524imaeq2d 6013 . . . . . . . 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 6842 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑄𝑦) = (𝑄𝑛))
31 oveq1 7364 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
3230, 31fveq12d 6849 . . . . . . . 8 (𝑦 = 𝑛 → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
33 fvoveq1 7380 . . . . . . . . . 10 (𝑦 = 𝑛 → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
3433sneqd 4598 . . . . . . . . 9 (𝑦 = 𝑛 → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
3534imaeq2d 6013 . . . . . . . 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 6842 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑄𝑦) = (𝑄‘(𝑛 + 1)))
41 oveq1 7364 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
4240, 41fveq12d 6849 . . . . . . . 8 (𝑦 = (𝑛 + 1) → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘((𝑛 + 1) / 𝑁)))
43 fvoveq1 7380 . . . . . . . . . 10 (𝑦 = (𝑛 + 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑛 + 1) / 𝑁)))
4443sneqd 4598 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑛 + 1) / 𝑁))})
4544imaeq2d 6013 . . . . . . . 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 6842 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑄𝑦) = (𝑄‘(𝑀 − 1)))
51 oveq1 7364 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → (𝑦 / 𝑁) = ((𝑀 − 1) / 𝑁))
5250, 51fveq12d 6849 . . . . . . . 8 (𝑦 = (𝑀 − 1) → ((𝑄𝑦)‘(𝑦 / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))
53 fvoveq1 7380 . . . . . . . . . 10 (𝑦 = (𝑀 − 1) → (𝐺‘(𝑦 / 𝑁)) = (𝐺‘((𝑀 − 1) / 𝑁)))
5453sneqd 4598 . . . . . . . . 9 (𝑦 = (𝑀 − 1) → {(𝐺‘(𝑦 / 𝑁))} = {(𝐺‘((𝑀 − 1) / 𝑁))})
5554imaeq2d 6013 . . . . . . . 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 33882 . . . . . . . . . 10 (𝑄‘0) = {⟨0, 𝑃⟩}
7170a1i 11 . . . . . . . . 9 (𝜑 → (𝑄‘0) = {⟨0, 𝑃⟩})
722nnne0d 12203 . . . . . . . . . 10 (𝜑𝑁 ≠ 0)
733, 72div0d 11930 . . . . . . . . 9 (𝜑 → (0 / 𝑁) = 0)
7471, 73fveq12d 6849 . . . . . . . 8 (𝜑 → ((𝑄‘0)‘(0 / 𝑁)) = ({⟨0, 𝑃⟩}‘0))
75 0nn0 12428 . . . . . . . . 9 0 ∈ ℕ0
76 fvsng 7126 . . . . . . . . 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 6846 . . . . . . . . 9 (𝜑 → (𝐺‘(0 / 𝑁)) = (𝐺‘0))
8065, 79eqtr4d 2779 . . . . . . . 8 (𝜑 → (𝐹𝑃) = (𝐺‘(0 / 𝑁)))
81 cvmcn 33856 . . . . . . . . . . 11 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
8262, 81syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Cn 𝐽))
8360, 61cnf 22597 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
84 ffn 6668 . . . . . . . . . 10 (𝐹:𝐵𝑋𝐹 Fn 𝐵)
8582, 83, 843syl 18 . . . . . . . . 9 (𝜑𝐹 Fn 𝐵)
86 fniniseg 7010 . . . . . . . . 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 12805 . . . . . . . . . 10 0 = (ℤ‘0)
9391, 92eleqtrdi 2848 . . . . . . . . 9 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
9493adantl 482 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
95 peano2fzr 13454 . . . . . . . . 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 13445 . . . . . . . . . . . . 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 12452 . . . . . . . . . . . . . . 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 12806 . . . . . . . . . . . . . 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 13433 . . . . . . . . . . . . 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 12475 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 ∈ ℂ)
114 pncan 11407 . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄𝑛))
117115oveq1d 7372 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
118116, 117fveq12d 6849 . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝐺‘(((𝑛 + 1) − 1) / 𝑁)) = (𝐺‘(𝑛 / 𝑁)))
120119sneqd 4598 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))} = {(𝐺‘(𝑛 / 𝑁))})
121120imaeq2d 6013 . . . . . . . . . . . 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 33884 . . . . . . . . . 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 12474 . . . . . . . . . . . . 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 12207 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ)
128127rexrd 11205 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ∈ ℝ*)
129 peano2re 11328 . . . . . . . . . . . . . 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 12207 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
132131rexrd 11205 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → ((𝑛 + 1) / 𝑁) ∈ ℝ*)
133125ltp1d 12085 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑛 < (𝑛 + 1))
134126nnred 12168 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 𝑁 ∈ ℝ)
135126nngt0d 12202 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → 0 < 𝑁)
136 ltdiv1 12019 . . . . . . . . . . . . . 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 11303 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑛 ∈ ℕ0 ∧ (𝑛 + 1) ∈ (0...𝑁)) ∧ ((𝑄𝑛)‘(𝑛 / 𝑁)) ∈ (𝐹 “ {(𝐺‘(𝑛 / 𝑁))}))) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
140 ubicc2 13382 . . . . . . . . . . 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 7372 . . . . . . . . . 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 7036 . . . . . . . 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 5937 . . . . . . . . . . 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 6844 . . . . . . . . 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 6654 . . . . . . . . . . 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 6940 . . . . . . . . . 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 6861 . . . . . . . . . 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 7010 . . . . . . . . 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 12598 . . . 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 3064  {crab 3407  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   ciun 4954   class class class wbr 5105  cmpt 5188   I cid 5530   × cxp 5631  ccnv 5632  ran crn 5634  cres 5635  cima 5636  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  crio 7312  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  0cn0 12413  cz 12499  cuz 12763  (,)cioo 13264  [,]cicc 13267  ...cfz 13424  seqcseq 13906  t crest 17302  topGenctg 17319   Cn ccn 22575  Homeochmeo 23104  IIcii 24238   CovMap ccvm 33849
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 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-hmeo 23106  df-ii 24240  df-cvm 33850
This theorem is referenced by:  cvmliftlem8  33886  cvmliftlem9  33887  cvmliftlem10  33888  cvmliftlem13  33890
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