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Theorem cvmliftlem10 35279
Description: Lemma for cvmlift 35284. 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 35275, cvmliftlem7 35276 (to show it is a function and a lift), cvmliftlem8 35277 (to show it is continuous), and cvmliftlem9 35278 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 23318 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 12919 . . . 4 ℕ = (ℤ‘1)
31, 2eleqtrdi 2849 . . 3 (𝜑𝑁 ∈ (ℤ‘1))
4 eluzfz2 13569 . . 3 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
53, 4syl 17 . 2 (𝜑𝑁 ∈ (1...𝑁))
6 eleq1 2827 . . . . . 6 (𝑦 = 1 → (𝑦 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
7 oveq2 7439 . . . . . . . . . . 11 (𝑦 = 1 → (1...𝑦) = (1...1))
8 1z 12645 . . . . . . . . . . . 12 1 ∈ ℤ
9 fzsn 13603 . . . . . . . . . . . 12 (1 ∈ ℤ → (1...1) = {1})
108, 9ax-mp 5 . . . . . . . . . . 11 (1...1) = {1}
117, 10eqtrdi 2791 . . . . . . . . . 10 (𝑦 = 1 → (1...𝑦) = {1})
1211iuneq1d 5024 . . . . . . . . 9 (𝑦 = 1 → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ {1} (𝑄𝑘))
13 1ex 11255 . . . . . . . . . 10 1 ∈ V
14 fveq2 6907 . . . . . . . . . 10 (𝑘 = 1 → (𝑄𝑘) = (𝑄‘1))
1513, 14iunxsn 5096 . . . . . . . . 9 𝑘 ∈ {1} (𝑄𝑘) = (𝑄‘1)
1612, 15eqtrdi 2791 . . . . . . . 8 (𝑦 = 1 → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = (𝑄‘1))
17 oveq1 7438 . . . . . . . . . . 11 (𝑦 = 1 → (𝑦 / 𝑁) = (1 / 𝑁))
1817oveq2d 7447 . . . . . . . . . 10 (𝑦 = 1 → (0[,](𝑦 / 𝑁)) = (0[,](1 / 𝑁)))
1918oveq2d 7447 . . . . . . . . 9 (𝑦 = 1 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](1 / 𝑁))))
2019oveq1d 7446 . . . . . . . 8 (𝑦 = 1 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶))
2116, 20eleq12d 2833 . . . . . . 7 (𝑦 = 1 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ (𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶)))
2216coeq2d 5876 . . . . . . . 8 (𝑦 = 1 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 ∘ (𝑄‘1)))
2318reseq2d 6000 . . . . . . . 8 (𝑦 = 1 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](1 / 𝑁))))
2422, 23eqeq12d 2751 . . . . . . 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 2827 . . . . . 6 (𝑦 = 𝑛 → (𝑦 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
29 oveq2 7439 . . . . . . . . 9 (𝑦 = 𝑛 → (1...𝑦) = (1...𝑛))
3029iuneq1d 5024 . . . . . . . 8 (𝑦 = 𝑛 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
31 oveq1 7438 . . . . . . . . . . 11 (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
3231oveq2d 7447 . . . . . . . . . 10 (𝑦 = 𝑛 → (0[,](𝑦 / 𝑁)) = (0[,](𝑛 / 𝑁)))
3332oveq2d 7447 . . . . . . . . 9 (𝑦 = 𝑛 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁))))
3433oveq1d 7446 . . . . . . . 8 (𝑦 = 𝑛 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶))
3530, 34eleq12d 2833 . . . . . . 7 (𝑦 = 𝑛 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶)))
3630coeq2d 5876 . . . . . . . 8 (𝑦 = 𝑛 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)))
3732reseq2d 6000 . . . . . . . 8 (𝑦 = 𝑛 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))
3836, 37eqeq12d 2751 . . . . . . 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 2827 . . . . . 6 (𝑦 = (𝑛 + 1) → (𝑦 ∈ (1...𝑁) ↔ (𝑛 + 1) ∈ (1...𝑁)))
43 oveq2 7439 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (1...𝑦) = (1...(𝑛 + 1)))
4443iuneq1d 5024 . . . . . . . 8 (𝑦 = (𝑛 + 1) → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘))
45 oveq1 7438 . . . . . . . . . . 11 (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
4645oveq2d 7447 . . . . . . . . . 10 (𝑦 = (𝑛 + 1) → (0[,](𝑦 / 𝑁)) = (0[,]((𝑛 + 1) / 𝑁)))
4746oveq2d 7447 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
4847oveq1d 7446 . . . . . . . 8 (𝑦 = (𝑛 + 1) → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
4944, 48eleq12d 2833 . . . . . . 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 5876 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)))
5146reseq2d 6000 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))
5250, 51eqeq12d 2751 . . . . . . 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 2827 . . . . . 6 (𝑦 = 𝑁 → (𝑦 ∈ (1...𝑁) ↔ 𝑁 ∈ (1...𝑁)))
57 oveq2 7439 . . . . . . . . . 10 (𝑦 = 𝑁 → (1...𝑦) = (1...𝑁))
5857iuneq1d 5024 . . . . . . . . 9 (𝑦 = 𝑁 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...𝑁)(𝑄𝑘))
59 cvmliftlem.k . . . . . . . . 9 𝐾 = 𝑘 ∈ (1...𝑁)(𝑄𝑘)
6058, 59eqtr4di 2793 . . . . . . . 8 (𝑦 = 𝑁 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝐾)
61 oveq1 7438 . . . . . . . . . . 11 (𝑦 = 𝑁 → (𝑦 / 𝑁) = (𝑁 / 𝑁))
6261oveq2d 7447 . . . . . . . . . 10 (𝑦 = 𝑁 → (0[,](𝑦 / 𝑁)) = (0[,](𝑁 / 𝑁)))
6362oveq2d 7447 . . . . . . . . 9 (𝑦 = 𝑁 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](𝑁 / 𝑁))))
6463oveq1d 7446 . . . . . . . 8 (𝑦 = 𝑁 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶))
6560, 64eleq12d 2833 . . . . . . 7 (𝑦 = 𝑁 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ 𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶)))
6660coeq2d 5876 . . . . . . . 8 (𝑦 = 𝑁 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹𝐾))
6762reseq2d 6000 . . . . . . . 8 (𝑦 = 𝑁 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))
6866, 67eqeq12d 2751 . . . . . . 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 13568 . . . . . . . . 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 2735 . . . . . . . . 9 (((1 − 1) / 𝑁)[,](1 / 𝑁)) = (((1 − 1) / 𝑁)[,](1 / 𝑁))
8674, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85cvmliftlem8 35277 . . . . . . . 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 12336 . . . . . . . . . . . 12 (1 − 1) = 0
8988oveq1i 7441 . . . . . . . . . . 11 ((1 − 1) / 𝑁) = (0 / 𝑁)
901nncnd 12280 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
911nnne0d 12314 . . . . . . . . . . . 12 (𝜑𝑁 ≠ 0)
9290, 91div0d 12040 . . . . . . . . . . 11 (𝜑 → (0 / 𝑁) = 0)
9389, 92eqtrid 2787 . . . . . . . . . 10 (𝜑 → ((1 − 1) / 𝑁) = 0)
9493oveq1d 7446 . . . . . . . . 9 (𝜑 → (((1 − 1) / 𝑁)[,](1 / 𝑁)) = (0[,](1 / 𝑁)))
9594oveq2d 7447 . . . . . . . 8 (𝜑 → (𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) = (𝐿t (0[,](1 / 𝑁))))
9695oveq1d 7446 . . . . . . 7 (𝜑 → ((𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶))
9787, 96eleqtrd 2841 . . . . . 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 35276 . . . . . . . . . 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 35275 . . . . . . . . 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 6000 . . . . . . 7 (𝜑 → (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁))) = (𝐺 ↾ (0[,](1 / 𝑁))))
104102, 103eqtrd 2775 . . . . . 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 12920 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
108107biimpi 216 . . . . . . . 8 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
109108adantl 481 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
110 peano2fzr 13574 . . . . . . . 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 2735 . . . . . . . . 9 (𝐿t (0[,]((𝑛 + 1) / 𝑁))) = (𝐿t (0[,]((𝑛 + 1) / 𝑁)))
116 0re 11261 . . . . . . . . . . 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 13590 . . . . . . . . . . . . . 14 ((𝑛 + 1) ∈ (1...𝑁) → (𝑛 + 1) ∈ ℕ)
121119, 120syl 17 . . . . . . . . . . . . 13 ((𝜑𝜒) → (𝑛 + 1) ∈ ℕ)
122121nnred 12279 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝑛 + 1) ∈ ℝ)
1231adantr 480 . . . . . . . . . . . 12 ((𝜑𝜒) → 𝑁 ∈ ℕ)
124122, 123nndivred 12318 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
125 iccssre 13466 . . . . . . . . . . 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 12279 . . . . . . . . . . . . 13 ((𝜑𝜒) → 𝑛 ∈ ℝ)
130129, 123nndivred 12318 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ℝ)
131 icccld 24803 . . . . . . . . . . . 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 6910 . . . . . . . . . . 11 (Clsd‘𝐿) = (Clsd‘(topGen‘ran (,)))
134132, 133eleqtrrdi 2850 . . . . . . . . . 10 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘𝐿))
135 ssun1 4188 . . . . . . . . . . 11 (0[,](𝑛 / 𝑁)) ⊆ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
136116a1i 11 . . . . . . . . . . . 12 ((𝜑𝜒) → 0 ∈ ℝ)
137128nnnn0d 12585 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → 𝑛 ∈ ℕ0)
138137nn0ge0d 12588 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 0 ≤ 𝑛)
139123nnred 12279 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 𝑁 ∈ ℝ)
140123nngt0d 12313 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 0 < 𝑁)
141 divge0 12135 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ (𝑛 / 𝑁))
142129, 138, 139, 140, 141syl22anc 839 . . . . . . . . . . . . 13 ((𝜑𝜒) → 0 ≤ (𝑛 / 𝑁))
143129ltp1d 12196 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → 𝑛 < (𝑛 + 1))
144 ltdiv1 12130 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
145129, 122, 139, 140, 144syl112anc 1373 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
146143, 145mpbid 232 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁))
147130, 124, 146ltled 11407 . . . . . . . . . . . . 13 ((𝜑𝜒) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
148 elicc2 13449 . . . . . . . . . . . . . 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 1341 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)))
151 iccsplit 13522 . . . . . . . . . . . 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 1370 . . . . . . . . . . 11 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) = ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
153135, 152sseqtrrid 4049 . . . . . . . . . 10 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)))
154 uniretop 24799 . . . . . . . . . . . 12 ℝ = (topGen‘ran (,))
15583unieqi 4924 . . . . . . . . . . . 12 𝐿 = (topGen‘ran (,))
156154, 155eqtr4i 2766 . . . . . . . . . . 11 ℝ = 𝐿
157156restcldi 23197 . . . . . . . . . 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 1370 . . . . . . . . 9 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(𝐿t (0[,]((𝑛 + 1) / 𝑁)))))
159 icccld 24803 . . . . . . . . . . . 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 2850 . . . . . . . . . 10 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘𝐿))
162 ssun2 4189 . . . . . . . . . . 11 ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
163162, 152sseqtrrid 4049 . . . . . . . . . 10 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)))
164156restcldi 23197 . . . . . . . . . 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 1370 . . . . . . . . 9 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(𝐿t (0[,]((𝑛 + 1) / 𝑁)))))
166 retop 24798 . . . . . . . . . . . 12 (topGen‘ran (,)) ∈ Top
16783, 166eqeltri 2835 . . . . . . . . . . 11 𝐿 ∈ Top
168156restuni 23186 . . . . . . . . . . 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 2777 . . . . . . . . 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 2735 . . . . . . . . . . . . . . 15 (𝐿t (0[,](𝑛 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁)))
175174, 75cnf 23270 . . . . . . . . . . . . . 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 13466 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ (𝑛 / 𝑁) ∈ ℝ) → (0[,](𝑛 / 𝑁)) ⊆ ℝ)
178116, 130, 177sylancr 587 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ⊆ ℝ)
179156restuni 23186 . . . . . . . . . . . . . . 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 6723 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . 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 35276 . . . . . . . . . . . . . . . 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 35275 . . . . . . . . . . . . . . 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 12280 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → 𝑛 ∈ ℂ)
190 ax-1cn 11211 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
191 pncan 11512 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
192189, 190, 191sylancl 586 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → ((𝑛 + 1) − 1) = 𝑛)
193192oveq1d 7446 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
194193oveq1d 7446 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
195194feq2d 6723 . . . . . . . . . . . . 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 6741 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → Fun 𝑘 ∈ (1...𝑛)(𝑄𝑘))
198128, 108syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → 𝑛 ∈ (ℤ‘1))
199 eluzfz2 13569 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ (1...𝑛))
200198, 199syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → 𝑛 ∈ (1...𝑛))
201 fveq2 6907 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑛 → (𝑄𝑘) = (𝑄𝑛))
202201ssiun2s 5053 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑛) → (𝑄𝑛) ⊆ 𝑘 ∈ (1...𝑛)(𝑄𝑘))
203200, 202syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (𝑄𝑛) ⊆ 𝑘 ∈ (1...𝑛)(𝑄𝑘))
204 peano2rem 11574 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
205129, 204syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → (𝑛 − 1) ∈ ℝ)
206205, 123nndivred 12318 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ∈ ℝ)
207206rexrd 11309 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ∈ ℝ*)
208130rexrd 11309 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ℝ*)
209129ltm1d 12198 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → (𝑛 − 1) < 𝑛)
210 ltdiv1 12130 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑛 − 1) < 𝑛 ↔ ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁)))
211205, 129, 139, 140, 210syl112anc 1373 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → ((𝑛 − 1) < 𝑛 ↔ ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁)))
212209, 211mpbid 232 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁))
213206, 130, 212ltled 11407 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ≤ (𝑛 / 𝑁))
214 ubicc2 13502 . . . . . . . . . . . . . . . . . . . 20 ((((𝑛 − 1) / 𝑁) ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 − 1) / 𝑁) ≤ (𝑛 / 𝑁)) → (𝑛 / 𝑁) ∈ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
215207, 208, 213, 214syl3anc 1370 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
216198, 119, 110syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → 𝑛 ∈ (1...𝑁))
217 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . 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 35276 . . . . . . . . . . . . . . . . . . . . . . 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 35275 . . . . . . . . . . . . . . . . . . . . . 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 6747 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → dom (𝑄𝑛) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
224215, 223eleqtrrd 2842 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ dom (𝑄𝑛))
225 funssfv 6928 . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
227192fveq2d 6911 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄𝑛))
228227, 193fveq12d 6914 . . . . . . . . . . . . . . . . 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 35278 . . . . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
232230, 231eqtr3d 2777 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
233226, 228, 2323eqtr2d 2781 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
234233opeq2d 4885 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩ = ⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩)
235234sneqd 4643 . . . . . . . . . . . . . 14 ((𝜑𝜒) → {⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩} = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
236182ffnd 6738 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘) Fn (0[,](𝑛 / 𝑁)))
237 0xr 11306 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
238237a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → 0 ∈ ℝ*)
239 ubicc2 13502 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ* ∧ 0 ≤ (𝑛 / 𝑁)) → (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁)))
240238, 208, 142, 239syl3anc 1370 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁)))
241 fnressn 7178 . . . . . . . . . . . . . . 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 6738 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)) Fn ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
244124rexrd 11309 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → ((𝑛 + 1) / 𝑁) ∈ ℝ*)
245 lbicc2 13501 . . . . . . . . . . . . . . . 16 (((𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ* ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁)) → (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
246208, 244, 147, 245syl3anc 1370 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
247 fnressn 7178 . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . 13 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}) = ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}))
250 df-icc 13391 . . . . . . . . . . . . . . . . 17 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
251 xrmaxle 13222 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ*𝑧 ∈ ℝ*) → (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0) ≤ 𝑧 ↔ (0 ≤ 𝑧 ∧ (𝑛 / 𝑁) ≤ 𝑧)))
252 xrlemin 13223 . . . . . . . . . . . . . . . . 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 13401 . . . . . . . . . . . . . . . 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 839 . . . . . . . . . . . . . . 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 4539 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0) = (𝑛 / 𝑁))
256147iftrued 4539 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁)) = (𝑛 / 𝑁))
257255, 256oveq12d 7449 . . . . . . . . . . . . . . 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 13429 . . . . . . . . . . . . . . . 16 ((𝑛 / 𝑁) ∈ ℝ* → ((𝑛 / 𝑁)[,](𝑛 / 𝑁)) = {(𝑛 / 𝑁)})
259208, 258syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ((𝑛 / 𝑁)[,](𝑛 / 𝑁)) = {(𝑛 / 𝑁)})
260254, 257, 2593eqtrd 2779 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = {(𝑛 / 𝑁)})
261260reseq2d 6000 . . . . . . . . . . . . 13 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}))
262260reseq2d 6000 . . . . . . . . . . . . 13 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}))
263249, 261, 2623eqtr4d 2785 . . . . . . . . . . . 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 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)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
265182, 196, 263, 264syl3anc 1370 . . . . . . . . . . 11 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
266 fzsuc 13608 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
267198, 266syl 17 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
268267iuneq1d 5024 . . . . . . . . . . . . 13 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) = 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘))
269 iunxun 5099 . . . . . . . . . . . . . 14 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘))
270 ovex 7464 . . . . . . . . . . . . . . . 16 (𝑛 + 1) ∈ V
271 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑛 + 1) → (𝑄𝑘) = (𝑄‘(𝑛 + 1)))
272270, 271iunxsn 5096 . . . . . . . . . . . . . . 15 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘) = (𝑄‘(𝑛 + 1))
273272uneq2i 4175 . . . . . . . . . . . . . 14 ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘)) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))
274269, 273eqtri 2763 . . . . . . . . . . . . 13 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))
275268, 274eqtr2di 2792 . . . . . . . . . . . 12 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) = 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘))
276275feq1d 6721 . . . . . . . . . . 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 6723 . . . . . . . . . 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 5999 . . . . . . . . . . 11 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ (0[,](𝑛 / 𝑁))))
281 fresaunres1 6782 . . . . . . . . . . . 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 1370 . . . . . . . . . . 11 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
283280, 282eqtr3d 2777 . . . . . . . . . 10 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ (0[,](𝑛 / 𝑁))) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
284167a1i 11 . . . . . . . . . . . 12 ((𝜑𝜒) → 𝐿 ∈ Top)
285 ovex 7464 . . . . . . . . . . . . 13 (0[,]((𝑛 + 1) / 𝑁)) ∈ V
286285a1i 11 . . . . . . . . . . . 12 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) ∈ V)
287 restabs 23189 . . . . . . . . . . . 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 1370 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁))))
289288oveq1d 7446 . . . . . . . . . 10 ((𝜑𝜒) → (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶))
290173, 283, 2893eltr4d 2854 . . . . . . . . 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 35277 . . . . . . . . . . . 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 7447 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
294293oveq1d 7446 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) = ((𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
295292, 294eleqtrd 2841 . . . . . . . . . 10 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
296275reseq1d 5999 . . . . . . . . . . 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 6781 . . . . . . . . . . . 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 1370 . . . . . . . . . . 11 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
299296, 298eqtr3d 2777 . . . . . . . . . 10 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
300 restabs 23189 . . . . . . . . . . . 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 1370 . . . . . . . . . . 11 ((𝜑𝜒) → ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
302301oveq1d 7446 . . . . . . . . . 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 2854 . . . . . . . . 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 23318 . . . . . . . 8 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
305152reseq2d 6000 . . . . . . . . 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 6000 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
309307, 308eqtrd 2775 . . . . . . . . . . 11 ((𝜑𝜒) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
310306, 309uneq12d 4179 . . . . . . . . . 10 ((𝜑𝜒) → ((𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) ∪ (𝐹 ∘ (𝑄‘(𝑛 + 1)))) = ((𝐺 ↾ (0[,](𝑛 / 𝑁))) ∪ (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
311 coundi 6269 . . . . . . . . . 10 (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = ((𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) ∪ (𝐹 ∘ (𝑄‘(𝑛 + 1))))
312 resundi 6014 . . . . . . . . . 10 (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝐺 ↾ (0[,](𝑛 / 𝑁))) ∪ (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
313310, 311, 3123eqtr4g 2800 . . . . . . . . 9 ((𝜑𝜒) → (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
314275coeq2d 5876 . . . . . . . . 9 ((𝜑𝜒) → (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)))
315305, 313, 3143eqtr2rd 2782 . . . . . . . 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 12282 . . 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 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605  {csn 4631  cop 4637   cuni 4912   ciun 4996   class class class wbr 5148  cmpt 5231   I cid 5582   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  ccom 5693  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  crio 7387  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  cz 12611  cuz 12876  (,)cioo 13384  [,]cicc 13387  ...cfz 13544  seqcseq 14039  t crest 17467  topGenctg 17484  Topctop 22915  Clsdccld 23040   Cn ccn 23248  Homeochmeo 23777  IIcii 24915   CovMap ccvm 35240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-icc 13391  df-fz 13545  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-rest 17469  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-cn 23251  df-hmeo 23779  df-ii 24917  df-cvm 35241
This theorem is referenced by:  cvmliftlem11  35280
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