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Theorem fourierdlem86 38885
Description: Continuity of 𝑂 and its limits with respect to the 𝑆 partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem86.f (𝜑𝐹:ℝ⟶ℝ)
fourierdlem86.xre (𝜑𝑋 ∈ ℝ)
fourierdlem86.p 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem86.m (𝜑𝑀 ∈ ℕ)
fourierdlem86.v (𝜑𝑉 ∈ (𝑃𝑀))
fourierdlem86.fcn ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
fourierdlem86.r ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))
fourierdlem86.l ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))
fourierdlem86.a (𝜑𝐴 ∈ ℝ)
fourierdlem86.b (𝜑𝐵 ∈ ℝ)
fourierdlem86.altb (𝜑𝐴 < 𝐵)
fourierdlem86.ab (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
fourierdlem86.n0 (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
fourierdlem86.c (𝜑𝐶 ∈ ℝ)
fourierdlem86.o 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem86.q 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
fourierdlem86.t 𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵)))
fourierdlem86.n 𝑁 = ((#‘𝑇) − 1)
fourierdlem86.s 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
fourierdlem86.d 𝐷 = (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))
fourierdlem86.e 𝐸 = (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2)))))
fourierdlem86.u 𝑈 = (𝑖 ∈ (0..^𝑀)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
Assertion
Ref Expression
fourierdlem86 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝐷 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ 𝐸 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
Distinct variable groups:   𝐴,𝑠   𝐵,𝑠   𝐶,𝑖,𝑠   𝑖,𝐹,𝑠   𝐿,𝑠   𝑖,𝑀,𝑗   𝑚,𝑀,𝑝,𝑖   𝑀,𝑠,𝑗   𝑓,𝑁   𝑖,𝑁,𝑠   𝑖,𝑂   𝑄,𝑖,𝑠   𝑅,𝑠   𝑆,𝑓   𝑆,𝑖,𝑠   𝑇,𝑓   𝑈,𝑖   𝑖,𝑉,𝑗   𝑉,𝑝   𝑉,𝑠   𝑖,𝑋,𝑗   𝑚,𝑋,𝑝   𝑋,𝑠   𝑓,𝑗,𝜑   𝜑,𝑖,𝑠
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝐴(𝑓,𝑖,𝑗,𝑚,𝑝)   𝐵(𝑓,𝑖,𝑗,𝑚,𝑝)   𝐶(𝑓,𝑗,𝑚,𝑝)   𝐷(𝑓,𝑖,𝑗,𝑚,𝑠,𝑝)   𝑃(𝑓,𝑖,𝑗,𝑚,𝑠,𝑝)   𝑄(𝑓,𝑗,𝑚,𝑝)   𝑅(𝑓,𝑖,𝑗,𝑚,𝑝)   𝑆(𝑗,𝑚,𝑝)   𝑇(𝑖,𝑗,𝑚,𝑠,𝑝)   𝑈(𝑓,𝑗,𝑚,𝑠,𝑝)   𝐸(𝑓,𝑖,𝑗,𝑚,𝑠,𝑝)   𝐹(𝑓,𝑗,𝑚,𝑝)   𝐿(𝑓,𝑖,𝑗,𝑚,𝑝)   𝑀(𝑓)   𝑁(𝑗,𝑚,𝑝)   𝑂(𝑓,𝑗,𝑚,𝑠,𝑝)   𝑉(𝑓,𝑚)   𝑋(𝑓)

Proof of Theorem fourierdlem86
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fourierdlem86.d . . 3 𝐷 = (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))
2 fourierdlem86.xre . . . . . . . . 9 (𝜑𝑋 ∈ ℝ)
32adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑋 ∈ ℝ)
4 fourierdlem86.p . . . . . . . 8 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
5 fourierdlem86.m . . . . . . . . 9 (𝜑𝑀 ∈ ℕ)
65adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
7 fourierdlem86.v . . . . . . . . 9 (𝜑𝑉 ∈ (𝑃𝑀))
87adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑉 ∈ (𝑃𝑀))
9 fourierdlem86.a . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
109adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ)
11 fourierdlem86.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ)
1211adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ℝ)
13 fourierdlem86.altb . . . . . . . . 9 (𝜑𝐴 < 𝐵)
1413adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐴 < 𝐵)
15 fourierdlem86.ab . . . . . . . . 9 (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
1615adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ (-π[,]π))
17 fourierdlem86.q . . . . . . . 8 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
18 fourierdlem86.t . . . . . . . 8 𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵)))
19 fourierdlem86.n . . . . . . . 8 𝑁 = ((#‘𝑇) − 1)
20 fourierdlem86.s . . . . . . . 8 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
21 simpr 475 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
22 fourierdlem86.u . . . . . . . 8 𝑈 = (𝑖 ∈ (0..^𝑀)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
23 biid 249 . . . . . . . 8 ((((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑦)(,)(𝑄‘(𝑦 + 1)))) ↔ (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑦)(,)(𝑄‘(𝑦 + 1)))))
243, 4, 6, 8, 10, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23fourierdlem50 38849 . . . . . . 7 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑈 ∈ (0..^𝑀) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))))
2524simpld 473 . . . . . 6 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑈 ∈ (0..^𝑀))
26 id 22 . . . . . . 7 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝜑𝑗 ∈ (0..^𝑁)))
2724simprd 477 . . . . . . 7 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1))))
2826, 25, 27jca31 554 . . . . . 6 ((𝜑𝑗 ∈ (0..^𝑁)) → (((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))))
29 nfv 1828 . . . . . . . 8 𝑖(((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1))))
30 nfv 1828 . . . . . . . . . . . . . . 15 𝑖(𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1))
31 nfcsb1v 3510 . . . . . . . . . . . . . . 15 𝑖𝑈 / 𝑖𝐿
32 nfcv 2746 . . . . . . . . . . . . . . 15 𝑖(𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))
3330, 31, 32nfif 4060 . . . . . . . . . . . . . 14 𝑖if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1)))))
34 nfcv 2746 . . . . . . . . . . . . . 14 𝑖
35 nfcv 2746 . . . . . . . . . . . . . 14 𝑖𝐶
3633, 34, 35nfov 6549 . . . . . . . . . . . . 13 𝑖(if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶)
37 nfcv 2746 . . . . . . . . . . . . 13 𝑖 /
38 nfcv 2746 . . . . . . . . . . . . 13 𝑖(𝑆‘(𝑗 + 1))
3936, 37, 38nfov 6549 . . . . . . . . . . . 12 𝑖((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1)))
40 nfcv 2746 . . . . . . . . . . . 12 𝑖 ·
41 nfcv 2746 . . . . . . . . . . . 12 𝑖((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))
4239, 40, 41nfov 6549 . . . . . . . . . . 11 𝑖(((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))
4342nfel1 2760 . . . . . . . . . 10 𝑖(((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1)))
44 nfv 1828 . . . . . . . . . . . . . . 15 𝑖(𝑆𝑗) = (𝑄𝑈)
45 nfcsb1v 3510 . . . . . . . . . . . . . . 15 𝑖𝑈 / 𝑖𝑅
46 nfcv 2746 . . . . . . . . . . . . . . 15 𝑖(𝐹‘(𝑋 + (𝑆𝑗)))
4744, 45, 46nfif 4060 . . . . . . . . . . . . . 14 𝑖if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗))))
4847, 34, 35nfov 6549 . . . . . . . . . . . . 13 𝑖(if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶)
49 nfcv 2746 . . . . . . . . . . . . 13 𝑖(𝑆𝑗)
5048, 37, 49nfov 6549 . . . . . . . . . . . 12 𝑖((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗))
51 nfcv 2746 . . . . . . . . . . . 12 𝑖((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))
5250, 40, 51nfov 6549 . . . . . . . . . . 11 𝑖(((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2)))))
5352nfel1 2760 . . . . . . . . . 10 𝑖(((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))
5443, 53nfan 1814 . . . . . . . . 9 𝑖((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗)))
55 nfv 1828 . . . . . . . . 9 𝑖(𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)
5654, 55nfan 1814 . . . . . . . 8 𝑖(((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
5729, 56nfim 1811 . . . . . . 7 𝑖((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
58 eleq1 2671 . . . . . . . . . 10 (𝑖 = 𝑈 → (𝑖 ∈ (0..^𝑀) ↔ 𝑈 ∈ (0..^𝑀)))
5958anbi2d 735 . . . . . . . . 9 (𝑖 = 𝑈 → (((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ↔ ((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀))))
60 fveq2 6084 . . . . . . . . . . 11 (𝑖 = 𝑈 → (𝑄𝑖) = (𝑄𝑈))
61 oveq1 6530 . . . . . . . . . . . 12 (𝑖 = 𝑈 → (𝑖 + 1) = (𝑈 + 1))
6261fveq2d 6088 . . . . . . . . . . 11 (𝑖 = 𝑈 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑈 + 1)))
6360, 62oveq12d 6541 . . . . . . . . . 10 (𝑖 = 𝑈 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1))))
6463sseq2d 3591 . . . . . . . . 9 (𝑖 = 𝑈 → (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))))
6559, 64anbi12d 742 . . . . . . . 8 (𝑖 = 𝑈 → ((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ↔ (((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1))))))
6662eqeq2d 2615 . . . . . . . . . . . . . . 15 (𝑖 = 𝑈 → ((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)) ↔ (𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1))))
67 csbeq1a 3503 . . . . . . . . . . . . . . 15 (𝑖 = 𝑈𝐿 = 𝑈 / 𝑖𝐿)
6866, 67ifbieq1d 4054 . . . . . . . . . . . . . 14 (𝑖 = 𝑈 → if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) = if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))))
6968oveq1d 6538 . . . . . . . . . . . . 13 (𝑖 = 𝑈 → (if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) = (if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶))
7069oveq1d 6538 . . . . . . . . . . . 12 (𝑖 = 𝑈 → ((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) = ((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))))
7170oveq1d 6538 . . . . . . . . . . 11 (𝑖 = 𝑈 → (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) = (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))))
7271eleq1d 2667 . . . . . . . . . 10 (𝑖 = 𝑈 → ((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ↔ (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1)))))
7360eqeq2d 2615 . . . . . . . . . . . . . . 15 (𝑖 = 𝑈 → ((𝑆𝑗) = (𝑄𝑖) ↔ (𝑆𝑗) = (𝑄𝑈)))
74 csbeq1a 3503 . . . . . . . . . . . . . . 15 (𝑖 = 𝑈𝑅 = 𝑈 / 𝑖𝑅)
7573, 74ifbieq1d 4054 . . . . . . . . . . . . . 14 (𝑖 = 𝑈 → if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) = if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))))
7675oveq1d 6538 . . . . . . . . . . . . 13 (𝑖 = 𝑈 → (if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) = (if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶))
7776oveq1d 6538 . . . . . . . . . . . 12 (𝑖 = 𝑈 → ((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) = ((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)))
7877oveq1d 6538 . . . . . . . . . . 11 (𝑖 = 𝑈 → (((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) = (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))))
7978eleq1d 2667 . . . . . . . . . 10 (𝑖 = 𝑈 → ((((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗)) ↔ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))))
8072, 79anbi12d 742 . . . . . . . . 9 (𝑖 = 𝑈 → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ↔ ((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗)))))
8180anbi1d 736 . . . . . . . 8 (𝑖 = 𝑈 → ((((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)) ↔ (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))))
8265, 81imbi12d 332 . . . . . . 7 (𝑖 = 𝑈 → (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))) ↔ ((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))))
83 fourierdlem86.f . . . . . . . 8 (𝜑𝐹:ℝ⟶ℝ)
84 fourierdlem86.fcn . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
85 fourierdlem86.r . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))
86 fourierdlem86.l . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))
87 fourierdlem86.n0 . . . . . . . 8 (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
88 fourierdlem86.c . . . . . . . 8 (𝜑𝐶 ∈ ℝ)
89 fourierdlem86.o . . . . . . . 8 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))))
90 eqid 2605 . . . . . . . 8 (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) = (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))
91 eqid 2605 . . . . . . . 8 (((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) = (((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2)))))
92 biid 249 . . . . . . . 8 ((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ↔ (((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))
9383, 2, 4, 5, 7, 84, 85, 86, 9, 11, 13, 15, 87, 88, 89, 17, 18, 19, 20, 90, 91, 92fourierdlem76 38875 . . . . . . 7 ((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
9457, 82, 93vtoclg1f 3233 . . . . . 6 (𝑈 ∈ (0..^𝑀) → ((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))))
9525, 28, 94sylc 62 . . . . 5 ((𝜑𝑗 ∈ (0..^𝑁)) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
9695simpld 473 . . . 4 ((𝜑𝑗 ∈ (0..^𝑁)) → ((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))))
9796simpld 473 . . 3 ((𝜑𝑗 ∈ (0..^𝑁)) → (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))))
981, 97syl5eqel 2687 . 2 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐷 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))))
99 fourierdlem86.e . . 3 𝐸 = (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2)))))
10096simprd 477 . . 3 ((𝜑𝑗 ∈ (0..^𝑁)) → (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗)))
10199, 100syl5eqel 2687 . 2 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐸 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗)))
10295simprd 477 . 2 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
10398, 101, 102jca31 554 1 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝐷 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ 𝐸 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  wral 2891  {crab 2895  csb 3494  cun 3533  cin 3534  wss 3535  ifcif 4031  {cpr 4122   class class class wbr 4573  cmpt 4633  ran crn 5025  cres 5026  cio 5748  wf 5782  cfv 5786   Isom wiso 5787  crio 6484  (class class class)co 6523  𝑚 cmap 7717  cc 9786  cr 9787  0cc0 9788  1c1 9789   + caddc 9791   · cmul 9793   < clt 9926  cmin 10113  -cneg 10114   / cdiv 10529  cn 10863  2c2 10913  (,)cioo 11998  [,]cicc 12001  ...cfz 12148  ..^cfzo 12285  #chash 12930  sincsin 14575  πcpi 14578  cnccncf 22414   lim climc 23345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866  ax-addf 9867  ax-mulf 9868
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-of 6768  df-om 6931  df-1st 7032  df-2nd 7033  df-supp 7156  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-ixp 7768  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-fsupp 8132  df-fi 8173  df-sup 8204  df-inf 8205  df-oi 8271  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-uz 11516  df-q 11617  df-rp 11661  df-xneg 11774  df-xadd 11775  df-xmul 11776  df-ioo 12002  df-ioc 12003  df-ico 12004  df-icc 12005  df-fz 12149  df-fzo 12286  df-fl 12406  df-mod 12482  df-seq 12615  df-exp 12674  df-fac 12874  df-bc 12903  df-hash 12931  df-shft 13597  df-cj 13629  df-re 13630  df-im 13631  df-sqrt 13765  df-abs 13766  df-limsup 13992  df-clim 14009  df-rlim 14010  df-sum 14207  df-ef 14579  df-sin 14581  df-cos 14582  df-pi 14584  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-ress 15644  df-plusg 15723  df-mulr 15724  df-starv 15725  df-sca 15726  df-vsca 15727  df-ip 15728  df-tset 15729  df-ple 15730  df-ds 15733  df-unif 15734  df-hom 15735  df-cco 15736  df-rest 15848  df-topn 15849  df-0g 15867  df-gsum 15868  df-topgen 15869  df-pt 15870  df-prds 15873  df-xrs 15927  df-qtop 15932  df-imas 15933  df-xps 15935  df-mre 16011  df-mrc 16012  df-acs 16014  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-submnd 17101  df-mulg 17306  df-cntz 17515  df-cmn 17960  df-psmet 19501  df-xmet 19502  df-met 19503  df-bl 19504  df-mopn 19505  df-fbas 19506  df-fg 19507  df-cnfld 19510  df-top 20459  df-bases 20460  df-topon 20461  df-topsp 20462  df-cld 20571  df-ntr 20572  df-cls 20573  df-nei 20650  df-lp 20688  df-perf 20689  df-cn 20779  df-cnp 20780  df-haus 20867  df-tx 21113  df-hmeo 21306  df-fil 21398  df-fm 21490  df-flim 21491  df-flf 21492  df-xms 21872  df-ms 21873  df-tms 21874  df-cncf 22416  df-limc 23349  df-dv 23350
This theorem is referenced by:  fourierdlem103  38902  fourierdlem104  38903
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