fourierdlem86 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem86 43988

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	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem86.xre	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem86.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem86.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem86.v	⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
fourierdlem86.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
fourierdlem86.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
fourierdlem86.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
fourierdlem86.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem86.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem86.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem86.ab	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
fourierdlem86.n0	⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
fourierdlem86.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem86.o	⊢ 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem86.q	⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 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 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))
fourierdlem86.e	⊢ 𝐸 = (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2)))))
fourierdlem86.u	⊢ 𝑈 = (℩𝑖 ∈ (0..^𝑀)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))

**Assertion**
Ref	Expression
fourierdlem86	⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐷 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ 𝐸 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(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.

Step	Hyp	Ref	Expression
1		fourierdlem86.d	. . 3 ⊢ 𝐷 = (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))
2		fourierdlem86.xre	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ)
3	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑋 ∈ ℝ)
4		fourierdlem86.p	. . . . . . . 8 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
5		fourierdlem86.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
7		fourierdlem86.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
8	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑉 ∈ (𝑃‘𝑀))
9		fourierdlem86.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
10	9	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ)
11		fourierdlem86.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
12	11	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐵 ∈ ℝ)
13		fourierdlem86.altb	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 𝐵)
14	13	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 < 𝐵)
15		fourierdlem86.ab	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
16	15	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ (-π[,]π))
17		fourierdlem86.q	. . . . . . . 8 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
22		fourierdlem86.u	. . . . . . . 8 ⊢ 𝑈 = (℩𝑖 ∈ (0..^𝑀)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
23		biid 260	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑦)(,)(𝑄‘(𝑦 + 1)))) ↔ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑦 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑦)(,)(𝑄‘(𝑦 + 1)))))
24	3, 4, 6, 8, 10, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23	fourierdlem50 43952	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑈 ∈ (0..^𝑀) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1)))))
25	24	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑈 ∈ (0..^𝑀))
26		id 22	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝜑 ∧ 𝑗 ∈ (0..^𝑁)))
27	24	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1))))
28	26, 25, 27	jca31 515	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1)))))
29		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1))))
30		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1))
31		nfcsb1v 3866	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖⦋𝑈 / 𝑖⦌𝐿
32		nfcv 2904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))
33	30, 31, 32	nfif 4500	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1)))))
34		nfcv 2904	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 −
35		nfcv 2904	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝐶
36	33, 34, 35	nfov 7346	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶)
37		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖 /
38		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑆‘(𝑗 + 1))
39	36, 37, 38	nfov 7346	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1)))
40		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 ·
41		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))
42	39, 40, 41	nfov 7346	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))
43	42	nfel1 2920	. . . . . . . . . 10 ⊢ Ⅎ𝑖(((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1)))
44		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝑆‘𝑗) = (𝑄‘𝑈)
45		nfcsb1v 3866	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖⦋𝑈 / 𝑖⦌𝑅
46		nfcv 2904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖(𝐹‘(𝑋 + (𝑆‘𝑗)))
47	44, 45, 46	nfif 4500	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗))))
48	47, 34, 35	nfov 7346	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶)
49		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑆‘𝑗)
50	48, 37, 49	nfov 7346	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗))
51		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))
52	50, 40, 51	nfov 7346	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2)))))
53	52	nfel1 2920	. . . . . . . . . 10 ⊢ Ⅎ𝑖(((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))
54	43, 53	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑖((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
55		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)
56	54, 55	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑖(((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
57	29, 56	nfim 1898	. . . . . . 7 ⊢ Ⅎ𝑖((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1)))) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
58		eleq1 2824	. . . . . . . . . 10 ⊢ (𝑖 = 𝑈 → (𝑖 ∈ (0..^𝑀) ↔ 𝑈 ∈ (0..^𝑀)))
59	58	anbi2d 629	. . . . . . . . 9 ⊢ (𝑖 = 𝑈 → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀))))
60		fveq2 6811	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → (𝑄‘𝑖) = (𝑄‘𝑈))
61		oveq1 7323	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → (𝑖 + 1) = (𝑈 + 1))
62	61	fveq2d 6815	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑈 + 1)))
63	60, 62	oveq12d 7334	. . . . . . . . . 10 ⊢ (𝑖 = 𝑈 → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1))))
64	63	sseq2d 3962	. . . . . . . . 9 ⊢ (𝑖 = 𝑈 → (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ↔ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1)))))
65	59, 64	anbi12d 631	. . . . . . . 8 ⊢ (𝑖 = 𝑈 → ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ↔ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1))))))
66	62	eqeq2d 2747	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → ((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)) ↔ (𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1))))
67		csbeq1a 3855	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → 𝐿 = ⦋𝑈 / 𝑖⦌𝐿)
68	66, 67	ifbieq1d 4494	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) = if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))))
69	68	oveq1d 7331	. . . . . . . . . . . . 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 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶))
70	69	oveq1d 7331	. . . . . . . . . . . 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 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))))
71	70	oveq1d 7331	. . . . . . . . . . 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 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))))
72	71	eleq1d 2821	. . . . . . . . . 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 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ↔ (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1)))))
73	60	eqeq2d 2747	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → ((𝑆‘𝑗) = (𝑄‘𝑖) ↔ (𝑆‘𝑗) = (𝑄‘𝑈)))
74		csbeq1a 3855	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑈 → 𝑅 = ⦋𝑈 / 𝑖⦌𝑅)
75	73, 74	ifbieq1d 4494	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑈 → if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) = if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))))
76	75	oveq1d 7331	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑈 → (if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) = (if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶))
77	76	oveq1d 7331	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑈 → ((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) = ((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)))
78	77	oveq1d 7331	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑈 → (((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) = (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))))
79	78	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑖 = 𝑈 → ((((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)) ↔ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))))
80	72, 79	anbi12d 631	. . . . . . . . 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 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ↔ ((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))))
81	80	anbi1d 630	. . . . . . . 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 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)) ↔ (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))))
82	65, 81	imbi12d 344	. . . . . . 7 ⊢ (𝑖 = 𝑈 → (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(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 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))) ↔ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1)))) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))))
83		fourierdlem86.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
84		fourierdlem86.fcn	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
85		fourierdlem86.r	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
86		fourierdlem86.l	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
87		fourierdlem86.n0	. . . . . . . 8 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
88		fourierdlem86.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
89		fourierdlem86.o	. . . . . . . 8 ⊢ 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))))
90		eqid 2736	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) = (((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2)))))
92		biid 260	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ↔ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
93	83, 2, 4, 5, 7, 84, 85, 86, 9, 11, 13, 15, 87, 88, 89, 17, 18, 19, 20, 90, 91, 92	fourierdlem76 43978	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(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 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
94	57, 82, 93	vtoclg1f 3512	. . . . . 6 ⊢ (𝑈 ∈ (0..^𝑀) → ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑈 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1)))) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))))
95	25, 28, 94	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
96	95	simpld 495	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))))
97	96	simpld 495	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), ⦋𝑈 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))))
98	1, 97	eqeltrid 2841	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐷 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))))
99		fourierdlem86.e	. . 3 ⊢ 𝐸 = (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2)))))
100	96	simprd 496	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
101	99, 100	eqeltrid 2841	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐸 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
102	95	simprd 496	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
103	98, 101, 102	jca31 515	1 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐷 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ 𝐸 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 {crab 3403 ⦋csb 3841 ∪ cun 3894 ∩ cin 3895 ⊆ wss 3896 ifcif 4470 {cpr 4572 class class class wbr 5086 ↦ cmpt 5169 ran crn 5608 ↾ cres 5609 ℩cio 6415 ⟶wf 6461 ‘cfv 6465 Isom wiso 6466 ℩crio 7272 (class class class)co 7316 ↑_m cmap 8664 ℂcc 10948 ℝcr 10949 0cc0 10950 1c1 10951 + caddc 10953 · cmul 10955 < clt 11088 − cmin 11284 -cneg 11285 / cdiv 11711 ℕcn 12052 2c2 12107 (,)cioo 13158 [,]cicc 13161 ...cfz 13318 ..^cfzo 13461 ♯chash 14123 sincsin 15849 πcpi 15852 –cn→ccncf 24119 lim_ℂ climc 25106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-inf2 9476 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027 ax-pre-sup 11028 ax-addf 11029 ax-mulf 11030

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-of 7574 df-om 7759 df-1st 7877 df-2nd 7878 df-supp 8026 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-2o 8346 df-er 8547 df-map 8666 df-pm 8667 df-ixp 8735 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-fsupp 9205 df-fi 9246 df-sup 9277 df-inf 9278 df-oi 9345 df-card 9774 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-div 11712 df-nn 12053 df-2 12115 df-3 12116 df-4 12117 df-5 12118 df-6 12119 df-7 12120 df-8 12121 df-9 12122 df-n0 12313 df-z 12399 df-dec 12517 df-uz 12662 df-q 12768 df-rp 12810 df-xneg 12927 df-xadd 12928 df-xmul 12929 df-ioo 13162 df-ioc 13163 df-ico 13164 df-icc 13165 df-fz 13319 df-fzo 13462 df-fl 13591 df-mod 13669 df-seq 13801 df-exp 13862 df-fac 14067 df-bc 14096 df-hash 14124 df-shft 14854 df-cj 14886 df-re 14887 df-im 14888 df-sqrt 15022 df-abs 15023 df-limsup 15256 df-clim 15273 df-rlim 15274 df-sum 15474 df-ef 15853 df-sin 15855 df-cos 15856 df-pi 15858 df-struct 16922 df-sets 16939 df-slot 16957 df-ndx 16969 df-base 16987 df-ress 17016 df-plusg 17049 df-mulr 17050 df-starv 17051 df-sca 17052 df-vsca 17053 df-ip 17054 df-tset 17055 df-ple 17056 df-ds 17058 df-unif 17059 df-hom 17060 df-cco 17061 df-rest 17207 df-topn 17208 df-0g 17226 df-gsum 17227 df-topgen 17228 df-pt 17229 df-prds 17232 df-xrs 17287 df-qtop 17292 df-imas 17293 df-xps 17295 df-mre 17369 df-mrc 17370 df-acs 17372 df-mgm 18400 df-sgrp 18449 df-mnd 18460 df-submnd 18505 df-mulg 18774 df-cntz 18996 df-cmn 19460 df-psmet 20669 df-xmet 20670 df-met 20671 df-bl 20672 df-mopn 20673 df-fbas 20674 df-fg 20675 df-cnfld 20678 df-top 22123 df-topon 22140 df-topsp 22162 df-bases 22176 df-cld 22250 df-ntr 22251 df-cls 22252 df-nei 22329 df-lp 22367 df-perf 22368 df-cn 22458 df-cnp 22459 df-haus 22546 df-tx 22793 df-hmeo 22986 df-fil 23077 df-fm 23169 df-flim 23170 df-flf 23171 df-xms 23553 df-ms 23554 df-tms 23555 df-cncf 24121 df-limc 25110 df-dv 25111

This theorem is referenced by: fourierdlem103 44005 fourierdlem104 44006

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