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Theorem fourierdlem86 40727
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,𝑚,𝑝   𝑗,𝑀,𝑠,𝑖   𝑓,𝑁   𝑖,𝑁,𝑠   𝑖,𝑂   𝑄,𝑖,𝑠   𝑅,𝑠   𝑆,𝑓   𝑆,𝑖,𝑠   𝑇,𝑓   𝑈,𝑖   𝑖,𝑉,𝑝   𝑗,𝑉,𝑠   𝑖,𝑋,𝑚,𝑝   𝑗,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 480 . . . . . . . 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 480 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
7 fourierdlem86.v . . . . . . . . 9 (𝜑𝑉 ∈ (𝑃𝑀))
87adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑉 ∈ (𝑃𝑀))
9 fourierdlem86.a . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
109adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ)
11 fourierdlem86.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ)
1211adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ℝ)
13 fourierdlem86.altb . . . . . . . . 9 (𝜑𝐴 < 𝐵)
1413adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐴 < 𝐵)
15 fourierdlem86.ab . . . . . . . . 9 (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
1615adantr 480 . . . . . . . 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 476 . . . . . . . 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 251 . . . . . . . 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 40691 . . . . . . 7 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑈 ∈ (0..^𝑀) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))))
2524simpld 474 . . . . . 6 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝑈 ∈ (0..^𝑀))
26 id 22 . . . . . . 7 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝜑𝑗 ∈ (0..^𝑁)))
2724simprd 478 . . . . . . 7 ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1))))
2826, 25, 27jca31 556 . . . . . 6 ((𝜑𝑗 ∈ (0..^𝑁)) → (((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))))
29 nfv 1883 . . . . . . . 8 𝑖(((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1))))
30 nfv 1883 . . . . . . . . . . . . . . 15 𝑖(𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1))
31 nfcsb1v 3582 . . . . . . . . . . . . . . 15 𝑖𝑈 / 𝑖𝐿
32 nfcv 2793 . . . . . . . . . . . . . . 15 𝑖(𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))
3330, 31, 32nfif 4148 . . . . . . . . . . . . . 14 𝑖if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1)))))
34 nfcv 2793 . . . . . . . . . . . . . 14 𝑖
35 nfcv 2793 . . . . . . . . . . . . . 14 𝑖𝐶
3633, 34, 35nfov 6716 . . . . . . . . . . . . 13 𝑖(if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶)
37 nfcv 2793 . . . . . . . . . . . . 13 𝑖 /
38 nfcv 2793 . . . . . . . . . . . . 13 𝑖(𝑆‘(𝑗 + 1))
3936, 37, 38nfov 6716 . . . . . . . . . . . 12 𝑖((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1)))
40 nfcv 2793 . . . . . . . . . . . 12 𝑖 ·
41 nfcv 2793 . . . . . . . . . . . 12 𝑖((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))
4239, 40, 41nfov 6716 . . . . . . . . . . 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 2808 . . . . . . . . . 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 1883 . . . . . . . . . . . . . . 15 𝑖(𝑆𝑗) = (𝑄𝑈)
45 nfcsb1v 3582 . . . . . . . . . . . . . . 15 𝑖𝑈 / 𝑖𝑅
46 nfcv 2793 . . . . . . . . . . . . . . 15 𝑖(𝐹‘(𝑋 + (𝑆𝑗)))
4744, 45, 46nfif 4148 . . . . . . . . . . . . . 14 𝑖if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗))))
4847, 34, 35nfov 6716 . . . . . . . . . . . . 13 𝑖(if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶)
49 nfcv 2793 . . . . . . . . . . . . 13 𝑖(𝑆𝑗)
5048, 37, 49nfov 6716 . . . . . . . . . . . 12 𝑖((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗))
51 nfcv 2793 . . . . . . . . . . . 12 𝑖((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2))))
5250, 40, 51nfov 6716 . . . . . . . . . . 11 𝑖(((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2)))))
5352nfel1 2808 . . . . . . . . . 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 1868 . . . . . . . . 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 1883 . . . . . . . . 9 𝑖(𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)
5654, 55nfan 1868 . . . . . . . 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 1865 . . . . . . 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 2718 . . . . . . . . . 10 (𝑖 = 𝑈 → (𝑖 ∈ (0..^𝑀) ↔ 𝑈 ∈ (0..^𝑀)))
5958anbi2d 740 . . . . . . . . 9 (𝑖 = 𝑈 → (((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ↔ ((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀))))
60 fveq2 6229 . . . . . . . . . . 11 (𝑖 = 𝑈 → (𝑄𝑖) = (𝑄𝑈))
61 oveq1 6697 . . . . . . . . . . . 12 (𝑖 = 𝑈 → (𝑖 + 1) = (𝑈 + 1))
6261fveq2d 6233 . . . . . . . . . . 11 (𝑖 = 𝑈 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑈 + 1)))
6360, 62oveq12d 6708 . . . . . . . . . 10 (𝑖 = 𝑈 → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1))))
6463sseq2d 3666 . . . . . . . . 9 (𝑖 = 𝑈 → (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))))
6559, 64anbi12d 747 . . . . . . . 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 2661 . . . . . . . . . . . . . . 15 (𝑖 = 𝑈 → ((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)) ↔ (𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1))))
67 csbeq1a 3575 . . . . . . . . . . . . . . 15 (𝑖 = 𝑈𝐿 = 𝑈 / 𝑖𝐿)
6866, 67ifbieq1d 4142 . . . . . . . . . . . . . 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 6705 . . . . . . . . . . . . 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 6705 . . . . . . . . . . . 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 6705 . . . . . . . . . . 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 2715 . . . . . . . . . 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 2661 . . . . . . . . . . . . . . 15 (𝑖 = 𝑈 → ((𝑆𝑗) = (𝑄𝑖) ↔ (𝑆𝑗) = (𝑄𝑈)))
74 csbeq1a 3575 . . . . . . . . . . . . . . 15 (𝑖 = 𝑈𝑅 = 𝑈 / 𝑖𝑅)
7573, 74ifbieq1d 4142 . . . . . . . . . . . . . 14 (𝑖 = 𝑈 → if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) = if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))))
7675oveq1d 6705 . . . . . . . . . . . . 13 (𝑖 = 𝑈 → (if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) = (if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶))
7776oveq1d 6705 . . . . . . . . . . . 12 (𝑖 = 𝑈 → ((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) = ((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)))
7877oveq1d 6705 . . . . . . . . . . 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 2715 . . . . . . . . . 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 747 . . . . . . . . 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 741 . . . . . . . 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 333 . . . . . . 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 2651 . . . . . . . 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 2651 . . . . . . . 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 251 . . . . . . . 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 40717 . . . . . . 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 3296 . . . . . 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 65 . . . . 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 474 . . . 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 474 . . 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 2734 . 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 478 . . 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 2734 . 2 ((𝜑𝑗 ∈ (0..^𝑁)) → 𝐸 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗)))
10295simprd 478 . 2 ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
10398, 101, 102jca31 556 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 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  csb 3566  cun 3605  cin 3606  wss 3607  ifcif 4119  {cpr 4212   class class class wbr 4685  cmpt 4762  ran crn 5144  cres 5145  cio 5887  wf 5922  cfv 5926   Isom wiso 5927  crio 6650  (class class class)co 6690  𝑚 cmap 7899  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cmin 10304  -cneg 10305   / cdiv 10722  cn 11058  2c2 11108  (,)cioo 12213  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504  #chash 13157  sincsin 14838  πcpi 14841  cnccncf 22726   lim climc 23671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676
This theorem is referenced by:  fourierdlem103  40744  fourierdlem104  40745
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