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Theorem cvmliftlem10 30981
Description: Lemma for cvmlift 30986. 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 30977, cvmliftlem7 30978 (to show it is a function and a lift), cvmliftlem8 30979 (to show it is continuous), and cvmliftlem9 30980 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 21008 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 11667 . . . 4 ℕ = (ℤ‘1)
31, 2syl6eleq 2708 . . 3 (𝜑𝑁 ∈ (ℤ‘1))
4 eluzfz2 12291 . . 3 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
53, 4syl 17 . 2 (𝜑𝑁 ∈ (1...𝑁))
6 eleq1 2686 . . . . . 6 (𝑦 = 1 → (𝑦 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
7 oveq2 6612 . . . . . . . . . . 11 (𝑦 = 1 → (1...𝑦) = (1...1))
8 1z 11351 . . . . . . . . . . . 12 1 ∈ ℤ
9 fzsn 12325 . . . . . . . . . . . 12 (1 ∈ ℤ → (1...1) = {1})
108, 9ax-mp 5 . . . . . . . . . . 11 (1...1) = {1}
117, 10syl6eq 2671 . . . . . . . . . 10 (𝑦 = 1 → (1...𝑦) = {1})
1211iuneq1d 4511 . . . . . . . . 9 (𝑦 = 1 → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ {1} (𝑄𝑘))
13 1ex 9979 . . . . . . . . . 10 1 ∈ V
14 fveq2 6148 . . . . . . . . . 10 (𝑘 = 1 → (𝑄𝑘) = (𝑄‘1))
1513, 14iunxsn 4569 . . . . . . . . 9 𝑘 ∈ {1} (𝑄𝑘) = (𝑄‘1)
1612, 15syl6eq 2671 . . . . . . . 8 (𝑦 = 1 → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = (𝑄‘1))
17 oveq1 6611 . . . . . . . . . . 11 (𝑦 = 1 → (𝑦 / 𝑁) = (1 / 𝑁))
1817oveq2d 6620 . . . . . . . . . 10 (𝑦 = 1 → (0[,](𝑦 / 𝑁)) = (0[,](1 / 𝑁)))
1918oveq2d 6620 . . . . . . . . 9 (𝑦 = 1 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](1 / 𝑁))))
2019oveq1d 6619 . . . . . . . 8 (𝑦 = 1 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶))
2116, 20eleq12d 2692 . . . . . . 7 (𝑦 = 1 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ (𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶)))
2216coeq2d 5244 . . . . . . . 8 (𝑦 = 1 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 ∘ (𝑄‘1)))
2318reseq2d 5356 . . . . . . . 8 (𝑦 = 1 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](1 / 𝑁))))
2422, 23eqeq12d 2636 . . . . . . 7 (𝑦 = 1 → ((𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁)))))
2521, 24anbi12d 746 . . . . . 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 334 . . . . 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 330 . . . 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 2686 . . . . . 6 (𝑦 = 𝑛 → (𝑦 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
29 oveq2 6612 . . . . . . . . 9 (𝑦 = 𝑛 → (1...𝑦) = (1...𝑛))
3029iuneq1d 4511 . . . . . . . 8 (𝑦 = 𝑛 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
31 oveq1 6611 . . . . . . . . . . 11 (𝑦 = 𝑛 → (𝑦 / 𝑁) = (𝑛 / 𝑁))
3231oveq2d 6620 . . . . . . . . . 10 (𝑦 = 𝑛 → (0[,](𝑦 / 𝑁)) = (0[,](𝑛 / 𝑁)))
3332oveq2d 6620 . . . . . . . . 9 (𝑦 = 𝑛 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁))))
3433oveq1d 6619 . . . . . . . 8 (𝑦 = 𝑛 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶))
3530, 34eleq12d 2692 . . . . . . 7 (𝑦 = 𝑛 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶)))
3630coeq2d 5244 . . . . . . . 8 (𝑦 = 𝑛 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)))
3732reseq2d 5356 . . . . . . . 8 (𝑦 = 𝑛 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))
3836, 37eqeq12d 2636 . . . . . . 7 (𝑦 = 𝑛 → ((𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
3935, 38anbi12d 746 . . . . . 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 334 . . . . 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 330 . . . 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 2686 . . . . . 6 (𝑦 = (𝑛 + 1) → (𝑦 ∈ (1...𝑁) ↔ (𝑛 + 1) ∈ (1...𝑁)))
43 oveq2 6612 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (1...𝑦) = (1...(𝑛 + 1)))
4443iuneq1d 4511 . . . . . . . 8 (𝑦 = (𝑛 + 1) → 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘))
45 oveq1 6611 . . . . . . . . . . 11 (𝑦 = (𝑛 + 1) → (𝑦 / 𝑁) = ((𝑛 + 1) / 𝑁))
4645oveq2d 6620 . . . . . . . . . 10 (𝑦 = (𝑛 + 1) → (0[,](𝑦 / 𝑁)) = (0[,]((𝑛 + 1) / 𝑁)))
4746oveq2d 6620 . . . . . . . . 9 (𝑦 = (𝑛 + 1) → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
4847oveq1d 6619 . . . . . . . 8 (𝑦 = (𝑛 + 1) → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
4944, 48eleq12d 2692 . . . . . . 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 5244 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)))
5146reseq2d 5356 . . . . . . . 8 (𝑦 = (𝑛 + 1) → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))
5250, 51eqeq12d 2636 . . . . . . 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 746 . . . . . 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 334 . . . . 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 330 . . . 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 2686 . . . . . 6 (𝑦 = 𝑁 → (𝑦 ∈ (1...𝑁) ↔ 𝑁 ∈ (1...𝑁)))
57 oveq2 6612 . . . . . . . . . 10 (𝑦 = 𝑁 → (1...𝑦) = (1...𝑁))
5857iuneq1d 4511 . . . . . . . . 9 (𝑦 = 𝑁 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝑘 ∈ (1...𝑁)(𝑄𝑘))
59 cvmliftlem.k . . . . . . . . 9 𝐾 = 𝑘 ∈ (1...𝑁)(𝑄𝑘)
6058, 59syl6eqr 2673 . . . . . . . 8 (𝑦 = 𝑁 𝑘 ∈ (1...𝑦)(𝑄𝑘) = 𝐾)
61 oveq1 6611 . . . . . . . . . . 11 (𝑦 = 𝑁 → (𝑦 / 𝑁) = (𝑁 / 𝑁))
6261oveq2d 6620 . . . . . . . . . 10 (𝑦 = 𝑁 → (0[,](𝑦 / 𝑁)) = (0[,](𝑁 / 𝑁)))
6362oveq2d 6620 . . . . . . . . 9 (𝑦 = 𝑁 → (𝐿t (0[,](𝑦 / 𝑁))) = (𝐿t (0[,](𝑁 / 𝑁))))
6463oveq1d 6619 . . . . . . . 8 (𝑦 = 𝑁 → ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶))
6560, 64eleq12d 2692 . . . . . . 7 (𝑦 = 𝑁 → ( 𝑘 ∈ (1...𝑦)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑦 / 𝑁))) Cn 𝐶) ↔ 𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶)))
6660coeq2d 5244 . . . . . . . 8 (𝑦 = 𝑁 → (𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐹𝐾))
6762reseq2d 5356 . . . . . . . 8 (𝑦 = 𝑁 → (𝐺 ↾ (0[,](𝑦 / 𝑁))) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))
6866, 67eqeq12d 2636 . . . . . . 7 (𝑦 = 𝑁 → ((𝐹 𝑘 ∈ (1...𝑦)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑦 / 𝑁))) ↔ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))
6965, 68anbi12d 746 . . . . . 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 334 . . . . 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 330 . . . 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 12290 . . . . . . . . 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 2621 . . . . . . . . 9 (((1 − 1) / 𝑁)[,](1 / 𝑁)) = (((1 − 1) / 𝑁)[,](1 / 𝑁))
8674, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85cvmliftlem8 30979 . . . . . . . 8 ((𝜑 ∧ 1 ∈ (1...𝑁)) → (𝑄‘1) ∈ ((𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶))
8773, 86mpdan 701 . . . . . . 7 (𝜑 → (𝑄‘1) ∈ ((𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶))
88 1m1e0 11033 . . . . . . . . . . . 12 (1 − 1) = 0
8988oveq1i 6614 . . . . . . . . . . 11 ((1 − 1) / 𝑁) = (0 / 𝑁)
901nncnd 10980 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
911nnne0d 11009 . . . . . . . . . . . 12 (𝜑𝑁 ≠ 0)
9290, 91div0d 10744 . . . . . . . . . . 11 (𝜑 → (0 / 𝑁) = 0)
9389, 92syl5eq 2667 . . . . . . . . . 10 (𝜑 → ((1 − 1) / 𝑁) = 0)
9493oveq1d 6619 . . . . . . . . 9 (𝜑 → (((1 − 1) / 𝑁)[,](1 / 𝑁)) = (0[,](1 / 𝑁)))
9594oveq2d 6620 . . . . . . . 8 (𝜑 → (𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) = (𝐿t (0[,](1 / 𝑁))))
9695oveq1d 6619 . . . . . . 7 (𝜑 → ((𝐿t (((1 − 1) / 𝑁)[,](1 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶))
9787, 96eleqtrd 2700 . . . . . 6 (𝜑 → (𝑄‘1) ∈ ((𝐿t (0[,](1 / 𝑁))) Cn 𝐶))
98 simpr 477 . . . . . . . . . 10 ((𝜑 ∧ 1 ∈ (1...𝑁)) → 1 ∈ (1...𝑁))
9974, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 85cvmliftlem7 30978 . . . . . . . . . 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 30977 . . . . . . . . 9 ((𝜑 ∧ 1 ∈ (1...𝑁)) → ((𝑄‘1):(((1 − 1) / 𝑁)[,](1 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁)))))
10173, 100mpdan 701 . . . . . . . 8 (𝜑 → ((𝑄‘1):(((1 − 1) / 𝑁)[,](1 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁)))))
102101simprd 479 . . . . . . 7 (𝜑 → (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁))))
10394reseq2d 5356 . . . . . . 7 (𝜑 → (𝐺 ↾ (((1 − 1) / 𝑁)[,](1 / 𝑁))) = (𝐺 ↾ (0[,](1 / 𝑁))))
104102, 103eqtrd 2655 . . . . . 6 (𝜑 → (𝐹 ∘ (𝑄‘1)) = (𝐺 ↾ (0[,](1 / 𝑁))))
10597, 104jca 554 . . . . 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 11668 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
108107biimpi 206 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
109108adantl 482 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
110 peano2fzr 12296 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘1) ∧ (𝑛 + 1) ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
111110ex 450 . . . . . . . . 9 (𝑛 ∈ (ℤ‘1) → ((𝑛 + 1) ∈ (1...𝑁) → 𝑛 ∈ (1...𝑁)))
112109, 111syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝑛 + 1) ∈ (1...𝑁) → 𝑛 ∈ (1...𝑁)))
113112imim1d 82 . . . . . . 7 ((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 . . . . . . . . . . 11 (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 2621 . . . . . . . . . . . . 13 (𝐿t (0[,]((𝑛 + 1) / 𝑁))) = (𝐿t (0[,]((𝑛 + 1) / 𝑁)))
116 0re 9984 . . . . . . . . . . . . . . 15 0 ∈ ℝ
117114simplbi 476 . . . . . . . . . . . . . . . . . . . 20 (𝜒 → (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)))
118117adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)))
119118simprd 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (𝑛 + 1) ∈ (1...𝑁))
120 elfznn 12312 . . . . . . . . . . . . . . . . . 18 ((𝑛 + 1) ∈ (1...𝑁) → (𝑛 + 1) ∈ ℕ)
121119, 120syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → (𝑛 + 1) ∈ ℕ)
122121nnred 10979 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → (𝑛 + 1) ∈ ℝ)
1231adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → 𝑁 ∈ ℕ)
124122, 123nndivred 11013 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ((𝑛 + 1) / 𝑁) ∈ ℝ)
125 iccssre 12197 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ) → (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ)
126116, 124, 125sylancr 694 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ)
127117simpld 475 . . . . . . . . . . . . . . . . . . 19 (𝜒𝑛 ∈ ℕ)
128127adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → 𝑛 ∈ ℕ)
129128nnred 10979 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → 𝑛 ∈ ℝ)
130129, 123nndivred 11013 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ℝ)
131 icccld 22480 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ (𝑛 / 𝑁) ∈ ℝ) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
132116, 130, 131sylancr 694 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
13383fveq2i 6151 . . . . . . . . . . . . . . 15 (Clsd‘𝐿) = (Clsd‘(topGen‘ran (,)))
134132, 133syl6eleqr 2709 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘𝐿))
135 ssun1 3754 . . . . . . . . . . . . . . 15 (0[,](𝑛 / 𝑁)) ⊆ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
136116a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → 0 ∈ ℝ)
137128nnnn0d 11295 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → 𝑛 ∈ ℕ0)
138137nn0ge0d 11298 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → 0 ≤ 𝑛)
139123nnred 10979 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → 𝑁 ∈ ℝ)
140123nngt0d 11008 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → 0 < 𝑁)
141 divge0 10836 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ (𝑛 / 𝑁))
142129, 138, 139, 140, 141syl22anc 1324 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → 0 ≤ (𝑛 / 𝑁))
143129ltp1d 10898 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → 𝑛 < (𝑛 + 1))
144 ltdiv1 10831 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
145129, 122, 139, 140, 144syl112anc 1327 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (𝑛 < (𝑛 + 1) ↔ (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁)))
146143, 145mpbid 222 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (𝑛 / 𝑁) < ((𝑛 + 1) / 𝑁))
147130, 124, 146ltled 10129 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))
148 elicc2 12180 . . . . . . . . . . . . . . . . . 18 ((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 694 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ((𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)) ↔ ((𝑛 / 𝑁) ∈ ℝ ∧ 0 ≤ (𝑛 / 𝑁) ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁))))
150130, 142, 147, 149mpbir3and 1243 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (0[,]((𝑛 + 1) / 𝑁)))
151 iccsplit 12247 . . . . . . . . . . . . . . . 16 ((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 1323 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) = ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
153135, 152syl5sseqr 3633 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)))
154 uniretop 22476 . . . . . . . . . . . . . . . 16 ℝ = (topGen‘ran (,))
15583unieqi 4411 . . . . . . . . . . . . . . . 16 𝐿 = (topGen‘ran (,))
156154, 155eqtr4i 2646 . . . . . . . . . . . . . . 15 ℝ = 𝐿
157156restcldi 20887 . . . . . . . . . . . . . 14 (((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 1323 . . . . . . . . . . . . 13 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ∈ (Clsd‘(𝐿t (0[,]((𝑛 + 1) / 𝑁)))))
159 icccld 22480 . . . . . . . . . . . . . . . 16 (((𝑛 / 𝑁) ∈ ℝ ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
160130, 124, 159syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(topGen‘ran (,))))
161160, 133syl6eleqr 2709 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘𝐿))
162 ssun2 3755 . . . . . . . . . . . . . . 15 ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
163162, 152syl5sseqr 3633 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ⊆ (0[,]((𝑛 + 1) / 𝑁)))
164156restcldi 20887 . . . . . . . . . . . . . 14 (((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 1323 . . . . . . . . . . . . 13 ((𝜑𝜒) → ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)) ∈ (Clsd‘(𝐿t (0[,]((𝑛 + 1) / 𝑁)))))
166 retop 22475 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) ∈ Top
16783, 166eqeltri 2694 . . . . . . . . . . . . . . 15 𝐿 ∈ Top
168156restuni 20876 . . . . . . . . . . . . . . 15 ((𝐿 ∈ Top ∧ (0[,]((𝑛 + 1) / 𝑁)) ⊆ ℝ) → (0[,]((𝑛 + 1) / 𝑁)) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
169167, 126, 168sylancr 694 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
170152, 169eqtr3d 2657 . . . . . . . . . . . . 13 ((𝜑𝜒) → ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t (0[,]((𝑛 + 1) / 𝑁))))
171114simprbi 480 . . . . . . . . . . . . . . . . . . . 20 (𝜒 → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
172171adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))
173172simpld 475 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶))
174 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (𝐿t (0[,](𝑛 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁)))
175174, 75cnf 20960 . . . . . . . . . . . . . . . . . 18 ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) → 𝑘 ∈ (1...𝑛)(𝑄𝑘): (𝐿t (0[,](𝑛 / 𝑁)))⟶𝐵)
176173, 175syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘): (𝐿t (0[,](𝑛 / 𝑁)))⟶𝐵)
177 iccssre 12197 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ ∧ (𝑛 / 𝑁) ∈ ℝ) → (0[,](𝑛 / 𝑁)) ⊆ ℝ)
178116, 130, 177sylancr 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) ⊆ ℝ)
179156restuni 20876 . . . . . . . . . . . . . . . . . . 19 ((𝐿 ∈ Top ∧ (0[,](𝑛 / 𝑁)) ⊆ ℝ) → (0[,](𝑛 / 𝑁)) = (𝐿t (0[,](𝑛 / 𝑁))))
180167, 178, 179sylancr 694 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (0[,](𝑛 / 𝑁)) = (𝐿t (0[,](𝑛 / 𝑁))))
181180feq2d 5988 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 𝑘 ∈ (1...𝑛)(𝑄𝑘): (𝐿t (0[,](𝑛 / 𝑁)))⟶𝐵))
182176, 181mpbird 247 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵)
183 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))
184 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → (𝑛 + 1) ∈ (1...𝑁))
18574, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 183cvmliftlem7 30978 . . . . . . . . . . . . . . . . . . . 20 ((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 30977 . . . . . . . . . . . . . . . . . . 19 ((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 487 . . . . . . . . . . . . . . . . . 18 ((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 475 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
189128nncnd 10980 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → 𝑛 ∈ ℂ)
190 ax-1cn 9938 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℂ
191 pncan 10231 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
192189, 190, 191sylancl 693 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → ((𝑛 + 1) − 1) = 𝑛)
193192oveq1d 6619 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (((𝑛 + 1) − 1) / 𝑁) = (𝑛 / 𝑁))
194193oveq1d 6619 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁)) = ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
195194feq2d 5988 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1)):((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 ↔ (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵))
196188, 195mpbid 222 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵)
197 ffun 6005 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 → Fun 𝑘 ∈ (1...𝑛)(𝑄𝑘))
198182, 197syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → Fun 𝑘 ∈ (1...𝑛)(𝑄𝑘))
199128, 108syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝜒) → 𝑛 ∈ (ℤ‘1))
200 eluzfz2 12291 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ (1...𝑛))
201199, 200syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝜒) → 𝑛 ∈ (1...𝑛))
202 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑛 → (𝑄𝑘) = (𝑄𝑛))
203202ssiun2s 4530 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑛) → (𝑄𝑛) ⊆ 𝑘 ∈ (1...𝑛)(𝑄𝑘))
204201, 203syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → (𝑄𝑛) ⊆ 𝑘 ∈ (1...𝑛)(𝑄𝑘))
205 peano2rem 10292 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
206129, 205syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝜒) → (𝑛 − 1) ∈ ℝ)
207206, 123nndivred 11013 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ∈ ℝ)
208207rexrd 10033 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ∈ ℝ*)
209130rexrd 10033 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ℝ*)
210129ltm1d 10900 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝜒) → (𝑛 − 1) < 𝑛)
211 ltdiv1 10831 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑛 − 1) < 𝑛 ↔ ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁)))
212206, 129, 139, 140, 211syl112anc 1327 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝜒) → ((𝑛 − 1) < 𝑛 ↔ ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁)))
213210, 212mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) < (𝑛 / 𝑁))
214207, 130, 213ltled 10129 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝜒) → ((𝑛 − 1) / 𝑁) ≤ (𝑛 / 𝑁))
215 ubicc2 12231 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑛 − 1) / 𝑁) ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 − 1) / 𝑁) ≤ (𝑛 / 𝑁)) → (𝑛 / 𝑁) ∈ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
216208, 209, 214, 215syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
217199, 119, 110syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝜒) → 𝑛 ∈ (1...𝑁))
218 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))
219 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
22074, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 218cvmliftlem7 30978 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝑄‘(𝑛 − 1))‘((𝑛 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑛 − 1) / 𝑁))}))
22174, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 218, 219, 220cvmliftlem6 30977 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝑄𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄𝑛)) = (𝐺 ↾ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))))
222217, 221syldan 487 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝜒) → ((𝑄𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵 ∧ (𝐹 ∘ (𝑄𝑛)) = (𝐺 ↾ (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))))
223222simpld 475 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝜒) → (𝑄𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵)
224 fdm 6008 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑄𝑛):(((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁))⟶𝐵 → dom (𝑄𝑛) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
225223, 224syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝜒) → dom (𝑄𝑛) = (((𝑛 − 1) / 𝑁)[,](𝑛 / 𝑁)))
226216, 225eleqtrrd 2701 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ dom (𝑄𝑛))
227 funssfv 6166 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∧ (𝑄𝑛) ⊆ 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∧ (𝑛 / 𝑁) ∈ dom (𝑄𝑛)) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
228198, 204, 226, 227syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
229192fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → (𝑄‘((𝑛 + 1) − 1)) = (𝑄𝑛))
230229, 193fveq12d 6154 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄𝑛)‘(𝑛 / 𝑁)))
23174, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84cvmliftlem9 30980 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)))
232119, 231syldan 487 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)))
233193fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
234232, 233eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝜒) → ((𝑄‘((𝑛 + 1) − 1))‘(((𝑛 + 1) − 1) / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
235228, 230, 2343eqtr2d 2661 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁)) = ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁)))
236235opeq2d 4377 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → ⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩ = ⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩)
237236sneqd 4160 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → {⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩} = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
238 ffn 6002 . . . . . . . . . . . . . . . . . . . 20 ( 𝑘 ∈ (1...𝑛)(𝑄𝑘):(0[,](𝑛 / 𝑁))⟶𝐵 𝑘 ∈ (1...𝑛)(𝑄𝑘) Fn (0[,](𝑛 / 𝑁)))
239182, 238syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → 𝑘 ∈ (1...𝑛)(𝑄𝑘) Fn (0[,](𝑛 / 𝑁)))
240 0xr 10030 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ*
241240a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → 0 ∈ ℝ*)
242 ubicc2 12231 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ* ∧ 0 ≤ (𝑛 / 𝑁)) → (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁)))
243241, 209, 142, 242syl3anc 1323 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ (0[,](𝑛 / 𝑁)))
244 fnressn 6379 . . . . . . . . . . . . . . . . . . 19 (( 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))⟩})
245239, 243, 244syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), ( 𝑘 ∈ (1...𝑛)(𝑄𝑘)‘(𝑛 / 𝑁))⟩})
246 ffn 6002 . . . . . . . . . . . . . . . . . . . 20 ((𝑄‘(𝑛 + 1)):((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))⟶𝐵 → (𝑄‘(𝑛 + 1)) Fn ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
247196, 246syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)) Fn ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
248124rexrd 10033 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → ((𝑛 + 1) / 𝑁) ∈ ℝ*)
249 lbicc2 12230 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 / 𝑁) ∈ ℝ* ∧ ((𝑛 + 1) / 𝑁) ∈ ℝ* ∧ (𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁)) → (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
250209, 248, 147, 249syl3anc 1323 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (𝑛 / 𝑁) ∈ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))
251 fnressn 6379 . . . . . . . . . . . . . . . . . . 19 (((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))⟩})
252247, 250, 251syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}) = {⟨(𝑛 / 𝑁), ((𝑄‘(𝑛 + 1))‘(𝑛 / 𝑁))⟩})
253237, 245, 2523eqtr4d 2665 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}) = ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}))
254 df-icc 12124 . . . . . . . . . . . . . . . . . . . . 21 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
255 xrmaxle 11957 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ* ∧ (𝑛 / 𝑁) ∈ ℝ*𝑧 ∈ ℝ*) → (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0) ≤ 𝑧 ↔ (0 ≤ 𝑧 ∧ (𝑛 / 𝑁) ≤ 𝑧)))
256 xrlemin 11958 . . . . . . . . . . . . . . . . . . . . 21 ((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))))
257254, 255, 256ixxin 12134 . . . . . . . . . . . . . . . . . . . 20 (((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))))
258241, 209, 209, 248, 257syl22anc 1324 . . . . . . . . . . . . . . . . . . 19 ((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))))
259142iftrued 4066 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0) = (𝑛 / 𝑁))
260147iftrued 4066 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝜒) → if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁)) = (𝑛 / 𝑁))
261259, 260oveq12d 6622 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → (if(0 ≤ (𝑛 / 𝑁), (𝑛 / 𝑁), 0)[,]if((𝑛 / 𝑁) ≤ ((𝑛 + 1) / 𝑁), (𝑛 / 𝑁), ((𝑛 + 1) / 𝑁))) = ((𝑛 / 𝑁)[,](𝑛 / 𝑁)))
262 iccid 12162 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 / 𝑁) ∈ ℝ* → ((𝑛 / 𝑁)[,](𝑛 / 𝑁)) = {(𝑛 / 𝑁)})
263209, 262syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝜒) → ((𝑛 / 𝑁)[,](𝑛 / 𝑁)) = {(𝑛 / 𝑁)})
264258, 261, 2633eqtrd 2659 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = {(𝑛 / 𝑁)})
265264reseq2d 5356 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ {(𝑛 / 𝑁)}))
266264reseq2d 5356 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ {(𝑛 / 𝑁)}))
267253, 265, 2663eqtr4d 2665 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝑄‘(𝑛 + 1)) ↾ ((0[,](𝑛 / 𝑁)) ∩ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
268 fresaun 6032 . . . . . . . . . . . . . . . 16 (( 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)
269182, 196, 267, 268syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
270 fzsuc 12330 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (ℤ‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
271199, 270syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝜒) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
272271iuneq1d 4511 . . . . . . . . . . . . . . . . 17 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) = 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘))
273 iunxun 4571 . . . . . . . . . . . . . . . . . 18 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘))
274 ovex 6632 . . . . . . . . . . . . . . . . . . . 20 (𝑛 + 1) ∈ V
275 fveq2 6148 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑛 + 1) → (𝑄𝑘) = (𝑄‘(𝑛 + 1)))
276274, 275iunxsn 4569 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘) = (𝑄‘(𝑛 + 1))
277276uneq2i 3742 . . . . . . . . . . . . . . . . . 18 ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ 𝑘 ∈ {(𝑛 + 1)} (𝑄𝑘)) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))
278273, 277eqtri 2643 . . . . . . . . . . . . . . . . 17 𝑘 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})(𝑄𝑘) = ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))
279272, 278syl6req 2672 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) = 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘))
280279feq1d 5987 . . . . . . . . . . . . . . 15 ((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))
281269, 280mpbid 222 . . . . . . . . . . . . . 14 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘):((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
282170feq2d 5988 . . . . . . . . . . . . . 14 ((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))
283281, 282mpbid 222 . . . . . . . . . . . . 13 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘): (𝐿t (0[,]((𝑛 + 1) / 𝑁)))⟶𝐵)
284279reseq1d 5355 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ (0[,](𝑛 / 𝑁))))
285 fresaunres1 6034 . . . . . . . . . . . . . . . 16 (( 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))
286182, 196, 267, 285syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ (0[,](𝑛 / 𝑁))) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
287284, 286eqtr3d 2657 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ (0[,](𝑛 / 𝑁))) = 𝑘 ∈ (1...𝑛)(𝑄𝑘))
288167a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → 𝐿 ∈ Top)
289 ovex 6632 . . . . . . . . . . . . . . . . 17 (0[,]((𝑛 + 1) / 𝑁)) ∈ V
290289a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → (0[,]((𝑛 + 1) / 𝑁)) ∈ V)
291 restabs 20879 . . . . . . . . . . . . . . . 16 ((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))))
292288, 153, 290, 291syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) = (𝐿t (0[,](𝑛 / 𝑁))))
293292oveq1d 6619 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶) = ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶))
294173, 287, 2933eltr4d 2713 . . . . . . . . . . . . 13 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ (0[,](𝑛 / 𝑁))) ∈ (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶))
29574, 75, 76, 77, 78, 79, 80, 1, 81, 82, 83, 84, 183cvmliftlem8 30979 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 + 1) ∈ (1...𝑁)) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
296119, 295syldan 487 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
297194oveq2d 6620 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → (𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
298297oveq1d 6619 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ((𝐿t ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) = ((𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
299296, 298eleqtrd 2700 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (𝑄‘(𝑛 + 1)) ∈ ((𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
300279reseq1d 5355 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
301 fresaunres2 6033 . . . . . . . . . . . . . . . 16 (( 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)))
302182, 196, 267, 301syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1))) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
303300, 302eqtr3d 2657 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝑄‘(𝑛 + 1)))
304 restabs 20879 . . . . . . . . . . . . . . . 16 ((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))))
305288, 163, 290, 304syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
306305oveq1d 6619 . . . . . . . . . . . . . 14 ((𝜑𝜒) → (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) = ((𝐿t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
307299, 303, 3063eltr4d 2713 . . . . . . . . . . . . 13 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) ∈ (((𝐿t (0[,]((𝑛 + 1) / 𝑁))) ↾t ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
308115, 75, 158, 165, 170, 283, 294, 307paste 21008 . . . . . . . . . . . 12 ((𝜑𝜒) → 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶))
309152reseq2d 5356 . . . . . . . . . . . . 13 ((𝜑𝜒) → (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
310172simprd 479 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))
311187simprd 479 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))))
312194reseq2d 5356 . . . . . . . . . . . . . . . 16 ((𝜑𝜒) → (𝐺 ↾ ((((𝑛 + 1) − 1) / 𝑁)[,]((𝑛 + 1) / 𝑁))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
313311, 312eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝜑𝜒) → (𝐹 ∘ (𝑄‘(𝑛 + 1))) = (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
314310, 313uneq12d 3746 . . . . . . . . . . . . . 14 ((𝜑𝜒) → ((𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) ∪ (𝐹 ∘ (𝑄‘(𝑛 + 1)))) = ((𝐺 ↾ (0[,](𝑛 / 𝑁))) ∪ (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
315 coundi 5595 . . . . . . . . . . . . . 14 (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = ((𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) ∪ (𝐹 ∘ (𝑄‘(𝑛 + 1))))
316 resundi 5369 . . . . . . . . . . . . . 14 (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))) = ((𝐺 ↾ (0[,](𝑛 / 𝑁))) ∪ (𝐺 ↾ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁))))
317314, 315, 3163eqtr4g 2680 . . . . . . . . . . . . 13 ((𝜑𝜒) → (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = (𝐺 ↾ ((0[,](𝑛 / 𝑁)) ∪ ((𝑛 / 𝑁)[,]((𝑛 + 1) / 𝑁)))))
318279coeq2d 5244 . . . . . . . . . . . . 13 ((𝜑𝜒) → (𝐹 ∘ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∪ (𝑄‘(𝑛 + 1)))) = (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)))
319309, 317, 3183eqtr2rd 2662 . . . . . . . . . . . 12 ((𝜑𝜒) → (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁))))
320308, 319jca 554 . . . . . . . . . . 11 ((𝜑𝜒) → ( 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))
321114, 320sylan2br 493 . . . . . . . . . 10 ((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)))))
322321expr 642 . . . . . . . . 9 ((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))))))
323322expr 642 . . . . . . . 8 ((𝜑𝑛 ∈ 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)))))))
324323a2d 29 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((𝑛 + 1) ∈ (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 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))))
325113, 324syld 47 . . . . . 6 ((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 + 1))(𝑄𝑘) ∈ ((𝐿t (0[,]((𝑛 + 1) / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...(𝑛 + 1))(𝑄𝑘)) = (𝐺 ↾ (0[,]((𝑛 + 1) / 𝑁)))))))
326325expcom 451 . . . . 5 (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))))) → ((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))))))))
327326a2d 29 . . . 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))))))))
32827, 41, 55, 71, 106, 327nnind 10982 . . 3 (𝑁 ∈ ℕ → (𝜑 → (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))))
3291, 328mpcom 38 . 2 (𝜑 → (𝑁 ∈ (1...𝑁) → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))))
3305, 329mpd 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  ifcif 4058  𝒫 cpw 4130  {csn 4148  cop 4154   cuni 4402   ciun 4485   class class class wbr 4613  cmpt 4673   I cid 4984   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  ccom 5078  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  crio 6564  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  cc 9878  cr 9879  0cc0 9880  1c1 9881   + caddc 9883  *cxr 10017   < clt 10018  cle 10019  cmin 10210   / cdiv 10628  cn 10964  cz 11321  cuz 11631  (,)cioo 12117  [,]cicc 12120  ...cfz 12268  seqcseq 12741  t crest 16002  topGenctg 16019  Topctop 20617  Clsdccld 20730   Cn ccn 20938  Homeochmeo 21466  IIcii 22586   CovMap ccvm 30942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-icc 12124  df-fz 12269  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-rest 16004  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-cn 20941  df-hmeo 21468  df-ii 22588  df-cvm 30943
This theorem is referenced by:  cvmliftlem11  30982
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