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Theorem cvmliftlem10 35300
Description: Lemma for cvmlift 35305. The function 𝐾 is going to be our complete lifted path, formed by unioning together all the 𝑄 functions (each of which is defined on one segment [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁] of the interval). Here we prove by induction that 𝐾 is a continuous function and a lift of 𝐺 by applying cvmliftlem6 35296, cvmliftlem7 35297 (to show it is a function and a lift), cvmliftlem8 35298 (to show it is continuous), and cvmliftlem9 35299 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 23303 gives that 𝐾 is well-defined and continuous). (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⟩}⟩}))
cvmliftlem.k 𝐾 = 𝑘 ∈ (1...𝑁)(𝑄𝑘)
cvmliftlem10.1 (𝜒 ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))
Assertion
Ref Expression
cvmliftlem10 (𝜑 → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))
Distinct variable groups:   𝑣,𝑏,𝑧,𝐵   𝑗,𝑏,𝑘,𝑚,𝑛,𝑠,𝑢,𝑥,𝐹,𝑣,𝑧   𝑛,𝐿,𝑧   𝑃,𝑏,𝑘,𝑚,𝑛,𝑢,𝑣,𝑥,𝑧   𝐶,𝑏,𝑗,𝑘,𝑛,𝑠,𝑢,𝑣,𝑧   𝜑,𝑗,𝑛,𝑠,𝑥,𝑧   𝑁,𝑏,𝑘,𝑚,𝑛,𝑢,𝑣,𝑥,𝑧   𝑆,𝑏,𝑗,𝑘,𝑛,𝑠,𝑢,𝑣,𝑥,𝑧   𝑗,𝑋   𝐺,𝑏,𝑗,𝑘,𝑚,𝑛,𝑠,𝑢,𝑣,𝑥,𝑧   𝑇,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧   𝐽,𝑏,𝑗,𝑘,𝑛,𝑠,𝑢,𝑣,𝑥,𝑧   𝑄,𝑏,𝑘,𝑚,𝑛,𝑢,𝑣,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑣,𝑢,𝑘,𝑚,𝑏)   𝜒(𝑥,𝑧,𝑣,𝑢,𝑗,𝑘,𝑚,𝑛,𝑠,𝑏)   𝐵(𝑥,𝑢,𝑗,𝑘,𝑚,𝑛,𝑠)   𝐶(𝑥,𝑚)   𝑃(𝑗,𝑠)   𝑄(𝑗,𝑠)   𝑆(𝑚)   𝑇(𝑛)   𝐽(𝑚)   𝐾(𝑥,𝑧,𝑣,𝑢,𝑗,𝑘,𝑚,𝑛,𝑠,𝑏)   𝐿(𝑥,𝑣,𝑢,𝑗,𝑘,𝑚,𝑠,𝑏)   𝑁(𝑗,𝑠)   𝑋(𝑥,𝑧,𝑣,𝑢,𝑘,𝑚,𝑛,𝑠,𝑏)

Proof of Theorem cvmliftlem10
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cvmliftlem.n . . . 4 (𝜑𝑁 ∈ ℕ)
2 nnuz 12922 . . . 4 ℕ = (ℤ‘1)
31, 2eleqtrdi 2850 . . 3 (𝜑𝑁 ∈ (ℤ‘1))
4 eluzfz2 13573 . . 3 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
53, 4syl 17 . 2 (𝜑𝑁 ∈ (1...𝑁))
6 eleq1 2828 . . . . . 6 (𝑦 = 1 → (𝑦 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
7 oveq2 7440 . . . . . . . . . . 11 (𝑦 = 1 → (1...𝑦) = (1...1))
8 1z 12649 . . . . . . . . . . . 12 1 ∈ ℤ
9 fzsn 13607 . . . . . . . . . . . 12 (1 ∈ ℤ → (1...1) = {1})
108, 9ax-mp 5 . . . . . . . . . . 11 (1...1) = {1}
117, 10eqtrdi 2792 . . . . . . . . . 10 (𝑦 = 1 → (1...𝑦) = {1})
1211iuneq1d 5018 . . . . . . . . 9 (𝑦 = 1 → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ {1} (𝑄𝑘))
13 1ex 11258 . . . . . . . . . 10 1 ∈ V
14 fveq2 6905 . . . . . . . . . 10 (𝑘 = 1 → (𝑄𝑘) = (𝑄‘1))
1513, 14iunxsn 5090 . . . . . . . . 9 𝑘 ∈ {1} (𝑄𝑘) = (𝑄‘1)
1612, 15eqtrdi 2792 . . . . . . . 8 (𝑦 = 1 → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = (𝑄‘1))
17 oveq1 7439 . . . . . . . . . . 11 (𝑦 = 1 → (𝑦 / 𝑁) = (1 / 𝑁))
1817oveq2d 7448 . . . . . . . . . 10 (𝑦 = 1 → (0[,](𝑦 / 𝑁)) = (0[,](1 / 𝑁)))
1918oveq2d 7448 . . . . . . . . 9 (𝑦 = 1 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](1 / 𝑁))))
2019oveq1d 7447 . . . . . . . 8 (𝑦 = 1 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶))
2116, 20eleq12d 2834 . . . . . . 7 (𝑦 = 1 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ (𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶)))
2216coeq2d 5872 . . . . . . . 8 (𝑦 = 1 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 ∘ (𝑄‘1)))
2318reseq2d 5996 . . . . . . . 8 (𝑦 = 1 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](1 / 𝑁))))
2422, 23eqeq12d 2752 . . . . . . 7 (𝑦 = 1 → ((𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁)))))
2521, 24anbi12d 632 . . . . . 6 (𝑦 = 1 → (( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))) ↔ ((𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))))
266, 25imbi12d 344 . . . . 5 (𝑦 = 1 → ((𝑦 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))))) ↔ (1 ∈ (1...𝑁) → ((𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁)))))))
2726imbi2d 340 . . . 4 (𝑦 = 1 → ((𝜑 → (𝑦 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))))) ↔ (𝜑 → (1 ∈ (1...𝑁) → ((𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))))))
28 eleq1 2828 . . . . . 6 (𝑦 = 𝑛 → (𝑦 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
29 oveq2 7440 . . . . . . . . 9 (𝑦 = 𝑛 → (1...𝑦) = (1...𝑛))
3029iuneq1d 5018 . . . . . . . 8 (𝑦 = 𝑛 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
31 oveq1 7439 . . . . . . . . . . 11 (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
3231oveq2d 7448 . . . . . . . . . 10 (𝑦 = 𝑛 → (0[,](𝑦 / 𝑁)) = (0[,](𝑛 / 𝑁)))
3332oveq2d 7448 . . . . . . . . 9 (𝑦 = 𝑛 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁))))
3433oveq1d 7447 . . . . . . . 8 (𝑦 = 𝑛 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶))
3530, 34eleq12d 2834 . . . . . . 7 (𝑦 = 𝑛 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶)))
3630coeq2d 5872 . . . . . . . 8 (𝑦 = 𝑛 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)))
3732reseq2d 5996 . . . . . . . 8 (𝑦 = 𝑛 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))
3836, 37eqeq12d 2752 . . . . . . 7 (𝑦 = 𝑛 → ((𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
3935, 38anbi12d 632 . . . . . 6 (𝑦 = 𝑛 → (( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))) ↔ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))
4028, 39imbi12d 344 . . . . 5 (𝑦 = 𝑛 → ((𝑦 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))))) ↔ (𝑛 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))))
4140imbi2d 340 . . . 4 (𝑦 = 𝑛 → ((𝜑 → (𝑦 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))))) ↔ (𝜑 → (𝑛 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))))
42 eleq1 2828 . . . . . 6 (𝑦 = (𝑛 + 1) → (𝑦 ∈ (1...𝑁) ↔ (𝑛 + 1) ∈ (1...𝑁)))
43 oveq2 7440 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (1...𝑦) = (1...(𝑛 + 1)))
4443iuneq1d 5018 . . . . . . . 8 (𝑦 = (𝑛 + 1) → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘))
45 oveq1 7439 . . . . . . . . . . 11 (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
4645oveq2d 7448 . . . . . . . . . 10 (𝑦 = (𝑛 + 1) → (0[,](𝑦 / 𝑁)) = (0[,]((𝑛 + 1) / 𝑁)))
4746oveq2d 7448 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
4847oveq1d 7447 . . . . . . . 8 (𝑦 = (𝑛 + 1) → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
4944, 48eleq12d 2834 . . . . . . 7 (𝑦 = (𝑛 + 1) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶)))
5044coeq2d 5872 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)))
5146reseq2d 5996 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))
5250, 51eqeq12d 2752 . . . . . . 7 (𝑦 = (𝑛 + 1) → ((𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))
5349, 52anbi12d 632 . . . . . 6 (𝑦 = (𝑛 + 1) → (( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))) ↔ ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))
5442, 53imbi12d 344 . . . . 5 (𝑦 = (𝑛 + 1) → ((𝑦 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))))) ↔ ((𝑛 + 1) ∈ (1...𝑁) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))))
5554imbi2d 340 . . . 4 (𝑦 = (𝑛 + 1) → ((𝜑 → (𝑦 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))))) ↔ (𝜑 → ((𝑛 + 1) ∈ (1...𝑁) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))))
56 eleq1 2828 . . . . . 6 (𝑦 = 𝑁 → (𝑦 ∈ (1...𝑁) ↔ 𝑁 ∈ (1...𝑁)))
57 oveq2 7440 . . . . . . . . . 10 (𝑦 = 𝑁 → (1...𝑦) = (1...𝑁))
5857iuneq1d 5018 . . . . . . . . 9 (𝑦 = 𝑁 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...𝑁)(𝑄𝑘))
59 cvmliftlem.k . . . . . . . . 9 𝐾 = 𝑘 ∈ (1...𝑁)(𝑄𝑘)
6058, 59eqtr4di 2794 . . . . . . . 8 (𝑦 = 𝑁 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝐾)
61 oveq1 7439 . . . . . . . . . . 11 (𝑦 = 𝑁 → (𝑦 / 𝑁) = (𝑁 / 𝑁))
6261oveq2d 7448 . . . . . . . . . 10 (𝑦 = 𝑁 → (0[,](𝑦 / 𝑁)) = (0[,](𝑁 / 𝑁)))
6362oveq2d 7448 . . . . . . . . 9 (𝑦 = 𝑁 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](𝑁 / 𝑁))))
6463oveq1d 7447 . . . . . . . 8 (𝑦 = 𝑁 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶))
6560, 64eleq12d 2834 . . . . . . 7 (𝑦 = 𝑁 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ 𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶)))
6660coeq2d 5872 . . . . . . . 8 (𝑦 = 𝑁 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹𝐾))
6762reseq2d 5996 . . . . . . . 8 (𝑦 = 𝑁 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))
6866, 67eqeq12d 2752 . . . . . . 7 (𝑦 = 𝑁 → ((𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))
6965, 68anbi12d 632 . . . . . 6 (𝑦 = 𝑁 → (( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))) ↔ (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))))
7056, 69imbi12d 344 . . . . 5 (𝑦 = 𝑁 → ((𝑦 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))))) ↔ (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))))
7170imbi2d 340 . . . 4 (𝑦 = 𝑁 → ((𝜑 → (𝑦 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁)))))) ↔ (𝜑 → (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))))))
72 eluzfz1 13572 . . . . . . . . 9 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
733, 72syl 17 . . . . . . . 8 (𝜑 → 1 ∈ (1...𝑁))
74 cvmliftlem.1 . . . . . . . . 9 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
75 cvmliftlem.b . . . . . . . . 9 𝐵 = 𝐶
76 cvmliftlem.x . . . . . . . . 9 𝑋 = 𝐽
77 cvmliftlem.f . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
78 cvmliftlem.g . . . . . . . . 9 (𝜑𝐺 ∈ (II Cn 𝐽))
79 cvmliftlem.p . . . . . . . . 9 (𝜑𝑃𝐵)
80 cvmliftlem.e . . . . . . . . 9 (𝜑 → (𝐹𝑃) = (𝐺‘0))
81 cvmliftlem.t . . . . . . . . 9 (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
82 cvmliftlem.a . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))
83 cvmliftlem.l . . . . . . . . 9 𝐿 = (topGen‘ran (,))
84 cvmliftlem.q . . . . . . . . 9 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⟩}⟩}))
85 eqid 2736 . . . . . . . . 9 (((1 − 1) / 𝑁)[,](1 / 𝑁)) = (((1 − 1) / 𝑁)[,](1 / 𝑁))
8674, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85cvmliftlem8 35298 . . . . . . . 8 ((𝜑 ∧ 1 ∈ (1...𝑁)) → (𝑄‘1) ∈ ((𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶))
8773, 86mpdan 687 . . . . . . 7 (𝜑 → (𝑄‘1) ∈ ((𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶))
88 1m1e0 12339 . . . . . . . . . . . 12 (1 − 1) = 0
8988oveq1i 7442 . . . . . . . . . . 11 ((1 − 1) / 𝑁) = (0 / 𝑁)
901nncnd 12283 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
911nnne0d 12317 . . . . . . . . . . . 12 (𝜑𝑁 ≠ 0)
9290, 91div0d 12043 . . . . . . . . . . 11 (𝜑 → (0 / 𝑁) = 0)
9389, 92eqtrid 2788 . . . . . . . . . 10 (𝜑 → ((1 − 1) / 𝑁) = 0)
9493oveq1d 7447 . . . . . . . . 9 (𝜑 → (((1 − 1) / 𝑁)[,](1 / 𝑁)) = (0[,](1 / 𝑁)))
9594oveq2d 7448 . . . . . . . 8 (𝜑 → (𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) = (𝐿t (0[,](1 / 𝑁))))
9695oveq1d 7447 . . . . . . 7 (𝜑 → ((𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶))
9787, 96eleqtrd 2842 . . . . . 6 (𝜑 → (𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶))
98 simpr 484 . . . . . . . . . 10 ((𝜑 ∧ 1 ∈ (1...𝑁)) → 1 ∈ (1...𝑁))
9974, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85cvmliftlem7 35297 . . . . . . . . . 10 ((𝜑 ∧ 1 ∈ (1...𝑁)) → ((𝑄‘(1 − 1))‘((1 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((1 − 1) / 𝑁))}))
10074, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85, 98, 99cvmliftlem6 35296 . . . . . . . . 9 ((𝜑 ∧ 1 ∈ (1...𝑁)) → ((𝑄‘1):(((1 − 1) / 𝑁)[,](1 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁)))))
10173, 100mpdan 687 . . . . . . . 8 (𝜑 → ((𝑄‘1):(((1 − 1) / 𝑁)[,](1 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁)))))
102101simprd 495 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁))))
10394reseq2d 5996 . . . . . . 7 (𝜑 → (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁))) = (𝐺 ↾ (0[,](1 / 𝑁))))
104102, 103eqtrd 2776 . . . . . 6 (𝜑 → (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))
10597, 104jca 511 . . . . 5 (𝜑 → ((𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁)))))
106105a1d 25 . . . 4 (𝜑 → (1 ∈ (1...𝑁) → ((𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))))
107 elnnuz 12923 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
108107biimpi 216 . . . . . . . 8 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
109108adantl 481 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
110 peano2fzr 13578 . . . . . . . 8 ((𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
111110ex 412 . . . . . . 7 (𝑛 ∈ (ℤ‘1) → ((𝑛 + 1) ∈ (1...𝑁) → 𝑛 ∈ (1...𝑁)))
112109, 111syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝑛 + 1) ∈ (1...𝑁) → 𝑛 ∈ (1...𝑁)))
113112imim1d 82 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝑛 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))) → ((𝑛 + 1) ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))))
114 cvmliftlem10.1 . . . . . . 7 (𝜒 ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))
115 eqid 2736 . . . . . . . . 9 (𝐿t (0[,]((𝑛 + 1) / 𝑁))) = (𝐿t (0[,]((𝑛 + 1) / 𝑁)))
116 0re 11264 . . . . . . . . . . 11 0 ∈ ℝ
117114simplbi 497 . . . . . . . . . . . . . . . 16 (𝜒 → (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)))
118117adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)))
119118simprd 495 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (𝑛 + 1) ∈ (1...𝑁))
120 elfznn 13594 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (1...𝑁) → (𝑛 + 1) ∈ ℕ)
121119, 120syl 17 . . . . . . . . . . . . 13 ((𝜑𝜒) → (𝑛 + 1) ∈ ℕ)
122121nnred 12282 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝑛 + 1) ∈ ℝ)
1231adantr 480 . . . . . . . . . . . 12 ((𝜑𝜒) → 𝑁 ∈ ℕ)
124122, 123nndivred 12321 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
125 iccssre 13470 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ) → (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ)
126116, 124, 125sylancr 587 . . . . . . . . . 10 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ)
127117simpld 494 . . . . . . . . . . . . . . 15 (𝜒𝑛 ∈ ℕ)
128127adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 𝑛 ∈ ℕ)
129128nnred 12282 . . . . . . . . . . . . 13 ((𝜑𝜒) → 𝑛 ∈ ℝ)
130129, 123nndivred 12321 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ℝ)
131 icccld 24788 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ (𝑛 / 𝑁) ∈ ℝ) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
132116, 130, 131sylancr 587 . . . . . . . . . . 11 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
13383fveq2i 6908 . . . . . . . . . . 11 (Clsd‘𝐿) = (Clsd‘(topGen‘ran (,)))
134132, 133eleqtrrdi 2851 . . . . . . . . . 10 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘𝐿))
135 ssun1 4177 . . . . . . . . . . 11 (0[,](𝑛 / 𝑁)) ⊆ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
136116a1i 11 . . . . . . . . . . . 12 ((𝜑𝜒) → 0 ∈ ℝ)
137128nnnn0d 12589 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → 𝑛 ∈ ℕ0)
138137nn0ge0d 12592 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 0 ≤ 𝑛)
139123nnred 12282 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 𝑁 ∈ ℝ)
140123nngt0d 12316 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 0 < 𝑁)
141 divge0 12138 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ (𝑛 / 𝑁))
142129, 138, 139, 140, 141syl22anc 838 . . . . . . . . . . . . 13 ((𝜑𝜒) → 0 ≤ (𝑛 / 𝑁))
143129ltp1d 12199 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → 𝑛 < (𝑛 + 1))
144 ltdiv1 12133 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
145129, 122, 139, 140, 144syl112anc 1375 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
146143, 145mpbid 232 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁))
147130, 124, 146ltled 11410 . . . . . . . . . . . . 13 ((𝜑𝜒) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
148 elicc2 13453 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ) → ((𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)) ↔ ((𝑛 / 𝑁) ∈ ℝ ∧ 0 ≤ (𝑛 / 𝑁) ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))))
149116, 124, 148sylancr 587 . . . . . . . . . . . . 13 ((𝜑𝜒) → ((𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)) ↔ ((𝑛 / 𝑁) ∈ ℝ ∧ 0 ≤ (𝑛 / 𝑁) ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))))
150130, 142, 147, 149mpbir3and 1342 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)))
151 iccsplit 13526 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ ∧ (𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁))) → (0[,]((𝑛 + 1) / 𝑁)) = ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
152136, 124, 150, 151syl3anc 1372 . . . . . . . . . . 11 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) = ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
153135, 152sseqtrrid 4026 . . . . . . . . . 10 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)))
154 uniretop 24784 . . . . . . . . . . . 12 ℝ = (topGen‘ran (,))
15583unieqi 4918 . . . . . . . . . . . 12 𝐿 = (topGen‘ran (,))
156154, 155eqtr4i 2767 . . . . . . . . . . 11 ℝ = 𝐿
157156restcldi 23182 . . . . . . . . . 10 (((0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ ∧ (0[,](𝑛 / 𝑁)) ∈ (Clsd‘𝐿) ∧ (0[,](𝑛 / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁))) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(𝐿t (0[,]((𝑛 + 1) / 𝑁)))))
158126, 134, 153, 157syl3anc 1372 . . . . . . . . 9 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(𝐿t (0[,]((𝑛 + 1) / 𝑁)))))
159 icccld 24788 . . . . . . . . . . . 12 (((𝑛 / 𝑁) ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
160130, 124, 159syl2anc 584 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
161160, 133eleqtrrdi 2851 . . . . . . . . . 10 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘𝐿))
162 ssun2 4178 . . . . . . . . . . 11 ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
163162, 152sseqtrrid 4026 . . . . . . . . . 10 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)))
164156restcldi 23182 . . . . . . . . . 10 (((0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ ∧ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘𝐿) ∧ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁))) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(𝐿t (0[,]((𝑛 + 1) / 𝑁)))))
165126, 161, 163, 164syl3anc 1372 . . . . . . . . 9 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(𝐿t (0[,]((𝑛 + 1) / 𝑁)))))
166 retop 24783 . . . . . . . . . . . 12 (topGen‘ran (,)) ∈ Top
16783, 166eqeltri 2836 . . . . . . . . . . 11 𝐿 ∈ Top
168156restuni 23171 . . . . . . . . . . 11 ((𝐿 ∈ Top ∧ (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ) → (0[,]((𝑛 + 1) / 𝑁)) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
169167, 126, 168sylancr 587 . . . . . . . . . 10 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
170152, 169eqtr3d 2778 . . . . . . . . 9 ((𝜑𝜒) → ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
171114simprbi 496 . . . . . . . . . . . . . . . 16 (𝜒 → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
172171adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
173172simpld 494 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶))
174 eqid 2736 . . . . . . . . . . . . . . 15 (𝐿t (0[,](𝑛 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁)))
175174, 75cnf 23255 . . . . . . . . . . . . . 14 ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) → 𝑘 ∈ (1...𝑛)(𝑄𝑘): (𝐿t (0[,](𝑛 / 𝑁)))⟶𝐵)
176173, 175syl 17 . . . . . . . . . . . . 13 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘): (𝐿t (0[,](𝑛 / 𝑁)))⟶𝐵)
177 iccssre 13470 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ (𝑛 / 𝑁) ∈ ℝ) → (0[,](𝑛 / 𝑁)) ⊆ ℝ)
178116, 130, 177sylancr 587 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ⊆ ℝ)
179156restuni 23171 . . . . . . . . . . . . . . 15 ((𝐿 ∈ Top ∧ (0[,](𝑛 / 𝑁)) ⊆ ℝ) → (0[,](𝑛 / 𝑁)) = (𝐿t (0[,](𝑛 / 𝑁))))
180167, 178, 179sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) = (𝐿t (0[,](𝑛 / 𝑁))))
181180feq2d 6721 . . . . . . . . . . . . 13 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 𝑘 ∈ (1...𝑛)(𝑄𝑘): (𝐿t (0[,](𝑛 / 𝑁)))⟶𝐵))
182176, 181mpbird 257 . . . . . . . . . . . 12 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵)
183 eqid 2736 . . . . . . . . . . . . . . . 16 ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))
184 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → (𝑛 + 1) ∈ (1...𝑁))
18574, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 183cvmliftlem7 35297 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘(((𝑛 + 1) − 1) / 𝑁))}))
18674, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 183, 184, 185cvmliftlem6 35296 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
187119, 186syldan 591 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
188187simpld 494 . . . . . . . . . . . . 13 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
189128nncnd 12283 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → 𝑛 ∈ ℂ)
190 ax-1cn 11214 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
191 pncan 11515 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
192189, 190, 191sylancl 586 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → ((𝑛 + 1) − 1) = 𝑛)
193192oveq1d 7447 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
194193oveq1d 7447 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
195194feq2d 6721 . . . . . . . . . . . . 13 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ↔ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵))
196188, 195mpbid 232 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
197176ffund 6739 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → Fun 𝑘 ∈ (1...𝑛)(𝑄𝑘))
198128, 108syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → 𝑛 ∈ (ℤ‘1))
199 eluzfz2 13573 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ (1...𝑛))
200198, 199syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → 𝑛 ∈ (1...𝑛))
201 fveq2 6905 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑛 → (𝑄𝑘) = (𝑄𝑛))
202201ssiun2s 5047 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑛) → (𝑄𝑛) ⊆ 𝑘 ∈ (1...𝑛)(𝑄𝑘))
203200, 202syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (𝑄𝑛) ⊆ 𝑘 ∈ (1...𝑛)(𝑄𝑘))
204 peano2rem 11577 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
205129, 204syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → (𝑛 − 1) ∈ ℝ)
206205, 123nndivred 12321 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ∈ ℝ)
207206rexrd 11312 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ∈ ℝ*)
208130rexrd 11312 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ℝ*)
209129ltm1d 12201 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → (𝑛 − 1) < 𝑛)
210 ltdiv1 12133 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑛 − 1) < 𝑛 ↔ ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁)))
211205, 129, 139, 140, 210syl112anc 1375 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → ((𝑛 − 1) < 𝑛 ↔ ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁)))
212209, 211mpbid 232 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁))
213206, 130, 212ltled 11410 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ≤ (𝑛 / 𝑁))
214 ubicc2 13506 . . . . . . . . . . . . . . . . . . . 20 ((((𝑛 − 1) / 𝑁) ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 − 1) / 𝑁) ≤ (𝑛 / 𝑁)) → (𝑛 / 𝑁) ∈ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
215207, 208, 213, 214syl3anc 1372 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
216198, 119, 110syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → 𝑛 ∈ (1...𝑁))
217 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))
218 simpr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
21974, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 217cvmliftlem7 35297 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝑄‘(𝑛 − 1))‘((𝑛 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 − 1) / 𝑁))}))
22074, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 217, 218, 219cvmliftlem6 35296 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝑄𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄𝑛)) = (𝐺 ↾ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))))
221216, 220syldan 591 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → ((𝑄𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄𝑛)) = (𝐺 ↾ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))))
222221simpld 494 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → (𝑄𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵)
223222fdmd 6745 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → dom (𝑄𝑛) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
224215, 223eleqtrrd 2843 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ dom (𝑄𝑛))
225 funssfv 6926 . . . . . . . . . . . . . . . . . 18 ((Fun 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∧ (𝑄𝑛) ⊆ 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∧ (𝑛 / 𝑁) ∈ dom (𝑄𝑛)) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
226197, 203, 224, 225syl3anc 1372 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
227192fveq2d 6909 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄𝑛))
228227, 193fveq12d 6912 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
22974, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84cvmliftlem9 35299 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)))
230119, 229syldan 591 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)))
231193fveq2d 6909 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
232230, 231eqtr3d 2778 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
233226, 228, 2323eqtr2d 2782 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
234233opeq2d 4879 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩ = ⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩)
235234sneqd 4637 . . . . . . . . . . . . . 14 ((𝜑𝜒) → {⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩} = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
236182ffnd 6736 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘) Fn (0[,](𝑛 / 𝑁)))
237 0xr 11309 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
238237a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → 0 ∈ ℝ*)
239 ubicc2 13506 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ* ∧ 0 ≤ (𝑛 / 𝑁)) → (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁)))
240238, 208, 142, 239syl3anc 1372 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁)))
241 fnressn 7177 . . . . . . . . . . . . . . 15 (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) Fn (0[,](𝑛 / 𝑁)) ∧ (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁))) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩})
242236, 240, 241syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩})
243196ffnd 6736 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)) Fn ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
244124rexrd 11312 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → ((𝑛 + 1) / 𝑁) ∈ ℝ*)
245 lbicc2 13505 . . . . . . . . . . . . . . . 16 (((𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ* ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁)) → (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
246208, 244, 147, 245syl3anc 1372 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
247 fnressn 7177 . . . . . . . . . . . . . . 15 (((𝑄‘(𝑛 + 1)) Fn ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∧ (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) → ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
248243, 246, 247syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
249235, 242, 2483eqtr4d 2786 . . . . . . . . . . . . 13 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}) = ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}))
250 df-icc 13395 . . . . . . . . . . . . . . . . 17 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
251 xrmaxle 13226 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ*𝑧 ∈ ℝ*) → (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0) ≤ 𝑧 ↔ (0 ≤ 𝑧 ∧ (𝑛 / 𝑁) ≤ 𝑧)))
252 xrlemin 13227 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ*) → (𝑧 ≤ if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁)) ↔ (𝑧 ≤ (𝑛 / 𝑁) ∧ 𝑧 ≤ ((𝑛 + 1) / 𝑁))))
253250, 251, 252ixxin 13405 . . . . . . . . . . . . . . . 16 (((0 ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ*) ∧ ((𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ*)) → ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0)[,]if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁))))
254238, 208, 208, 244, 253syl22anc 838 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0)[,]if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁))))
255142iftrued 4532 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0) = (𝑛 / 𝑁))
256147iftrued 4532 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁)) = (𝑛 / 𝑁))
257255, 256oveq12d 7450 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0)[,]if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁))) = ((𝑛 / 𝑁)[,](𝑛 / 𝑁)))
258 iccid 13433 . . . . . . . . . . . . . . . 16 ((𝑛 / 𝑁) ∈ ℝ* → ((𝑛 / 𝑁)[,](𝑛 / 𝑁)) = {(𝑛 / 𝑁)})
259208, 258syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ((𝑛 / 𝑁)[,](𝑛 / 𝑁)) = {(𝑛 / 𝑁)})
260254, 257, 2593eqtrd 2780 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = {(𝑛 / 𝑁)})
261260reseq2d 5996 . . . . . . . . . . . . 13 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}))
262260reseq2d 5996 . . . . . . . . . . . . 13 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}))
263249, 261, 2623eqtr4d 2786 . . . . . . . . . . . 12 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
264 fresaun 6778 . . . . . . . . . . . 12 (( 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
265182, 196, 263, 264syl3anc 1372 . . . . . . . . . . 11 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
266 fzsuc 13612 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
267198, 266syl 17 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
268267iuneq1d 5018 . . . . . . . . . . . . 13 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) = 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘))
269 iunxun 5093 . . . . . . . . . . . . . 14 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘))
270 ovex 7465 . . . . . . . . . . . . . . . 16 (𝑛 + 1) ∈ V
271 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → (𝑄𝑘) = (𝑄‘(𝑛 + 1)))
272270, 271iunxsn 5090 . . . . . . . . . . . . . . 15 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘) = (𝑄‘(𝑛 + 1))
273272uneq2i 4164 . . . . . . . . . . . . . 14 ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘)) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))
274269, 273eqtri 2764 . . . . . . . . . . . . 13 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))
275268, 274eqtr2di 2793 . . . . . . . . . . . 12 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) = 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘))
276275feq1d 6719 . . . . . . . . . . 11 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵))
277265, 276mpbid 232 . . . . . . . . . 10 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
278170feq2d 6721 . . . . . . . . . 10 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘): (𝐿t (0[,]((𝑛 + 1) / 𝑁)))⟶𝐵))
279277, 278mpbid 232 . . . . . . . . 9 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘): (𝐿t (0[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
280275reseq1d 5995 . . . . . . . . . . 11 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ (0[,](𝑛 / 𝑁))))
281 fresaunres1 6780 . . . . . . . . . . . 12 (( 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
282182, 196, 263, 281syl3anc 1372 . . . . . . . . . . 11 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
283280, 282eqtr3d 2778 . . . . . . . . . 10 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ (0[,](𝑛 / 𝑁))) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
284167a1i 11 . . . . . . . . . . . 12 ((𝜑𝜒) → 𝐿 ∈ Top)
285 ovex 7465 . . . . . . . . . . . . 13 (0[,]((𝑛 + 1) / 𝑁)) ∈ V
286285a1i 11 . . . . . . . . . . . 12 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) ∈ V)
287 restabs 23174 . . . . . . . . . . . 12 ((𝐿 ∈ Top ∧ (0[,](𝑛 / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)) ∧ (0[,]((𝑛 + 1) / 𝑁)) ∈ V) → ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁))))
288284, 153, 286, 287syl3anc 1372 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁))))
289288oveq1d 7447 . . . . . . . . . 10 ((𝜑𝜒) → (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶))
290173, 283, 2893eltr4d 2855 . . . . . . . . 9 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ (0[,](𝑛 / 𝑁))) ∈ (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶))
29174, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 183cvmliftlem8 35298 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
292119, 291syldan 591 . . . . . . . . . . 11 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
293194oveq2d 7448 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
294293oveq1d 7447 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) = ((𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
295292, 294eleqtrd 2842 . . . . . . . . . 10 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
296275reseq1d 5995 . . . . . . . . . . 11 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
297 fresaunres2 6779 . . . . . . . . . . . 12 (( 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ∧ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
298182, 196, 263, 297syl3anc 1372 . . . . . . . . . . 11 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
299296, 298eqtr3d 2778 . . . . . . . . . 10 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
300 restabs 23174 . . . . . . . . . . . 12 ((𝐿 ∈ Top ∧ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)) ∧ (0[,]((𝑛 + 1) / 𝑁)) ∈ V) → ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
301284, 163, 286, 300syl3anc 1372 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
302301oveq1d 7447 . . . . . . . . . 10 ((𝜑𝜒) → (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) = ((𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
303295, 299, 3023eltr4d 2855 . . . . . . . . 9 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) ∈ (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
304115, 75, 158, 165, 170, 279, 290, 303paste 23303 . . . . . . . 8 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
305152reseq2d 5996 . . . . . . . . 9 ((𝜑𝜒) → (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
306172simprd 495 . . . . . . . . . . 11 ((𝜑𝜒) → (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))
307187simprd 495 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))))
308194reseq2d 5996 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
309307, 308eqtrd 2776 . . . . . . . . . . 11 ((𝜑𝜒) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
310306, 309uneq12d 4168 . . . . . . . . . 10 ((𝜑𝜒) → ((𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) ∪ (𝐹 ∘ (𝑄‘(𝑛 + 1)))) = ((𝐺 ↾ (0[,](𝑛 / 𝑁))) ∪ (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
311 coundi 6266 . . . . . . . . . 10 (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = ((𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) ∪ (𝐹 ∘ (𝑄‘(𝑛 + 1))))
312 resundi 6010 . . . . . . . . . 10 (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝐺 ↾ (0[,](𝑛 / 𝑁))) ∪ (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
313310, 311, 3123eqtr4g 2801 . . . . . . . . 9 ((𝜑𝜒) → (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
314275coeq2d 5872 . . . . . . . . 9 ((𝜑𝜒) → (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)))
315305, 313, 3143eqtr2rd 2783 . . . . . . . 8 ((𝜑𝜒) → (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))
316304, 315jca 511 . . . . . . 7 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))
317114, 316sylan2br 595 . . . . . 6 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))
318317expr 456 . . . . 5 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁))) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))
319113, 318animpimp2impd 846 . . . 4 (𝑛 ∈ ℕ → ((𝜑 → (𝑛 ∈ (1...𝑁) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))) → (𝜑 → ((𝑛 + 1) ∈ (1...𝑁) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))))))
32027, 41, 55, 71, 106, 319nnind 12285 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))))
3211, 320mpcom 38 . 2 (𝜑 → (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))))
3225, 321mpd 15 1 (𝜑 → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  ifcif 4524  𝒫 cpw 4599  {csn 4625  cop 4631   cuni 4906   ciun 4990   class class class wbr 5142  cmpt 5224   I cid 5576   × cxp 5682  ccnv 5683  dom cdm 5684  ran crn 5685  cres 5686  cima 5687  ccom 5688  Fun wfun 6554   Fn wfn 6555  wf 6556  cfv 6560  crio 7388  (class class class)co 7432  cmpo 7434  1st c1st 8013  2nd c2nd 8014  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159  *cxr 11295   < clt 11296  cle 11297  cmin 11493   / cdiv 11921  cn 12267  cz 12615  cuz 12879  (,)cioo 13388  [,]cicc 13391  ...cfz 13548  seqcseq 14043  t crest 17466  topGenctg 17483  Topctop 22900  Clsdccld 23025   Cn ccn 23233  Homeochmeo 23762  IIcii 24902   CovMap ccvm 35261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fi 9452  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ioo 13392  df-icc 13395  df-fz 13549  df-seq 14044  df-exp 14104  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-rest 17468  df-topgen 17489  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-top 22901  df-topon 22918  df-bases 22954  df-cld 23028  df-cn 23236  df-hmeo 23764  df-ii 24904  df-cvm 35262
This theorem is referenced by:  cvmliftlem11  35301
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