fourierdlem80 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem80 43617

Description: The derivative of 𝑂 is bounded on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem80.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem80.xre	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem80.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem80.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem80.ab	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
fourierdlem80.n0	⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
fourierdlem80.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem80.o	⊢ 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
fourierdlem80.i	⊢ 𝐼 = ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))
fourierdlem80.fbdioo	⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤)
fourierdlem80.fdvbdioo	⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)
fourierdlem80.sf	⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
fourierdlem80.slt	⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))
fourierdlem80.sjss	⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
fourierdlem80.relioo	⊢ (((𝜑 ∧ 𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
fdv	⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ 𝐼)):𝐼⟶ℝ)
fourierdlem80.y	⊢ 𝑌 = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
fourierdlem80.ch	⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧))

**Assertion**
Ref	Expression
fourierdlem80	⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏)

Distinct variable groups: 𝐴,𝑏,𝑟,𝑠,𝑡 𝐵,𝑏,𝑟,𝑠,𝑡 𝐶,𝑏,𝑟,𝑠,𝑡 𝐹,𝑏,𝑟,𝑠,𝑡 𝑤,𝐹,𝑧,𝑠,𝑡 𝑤,𝐼,𝑧 𝑁,𝑏,𝑗,𝑟,𝑠 𝑘,𝑁,𝑗,𝑟 𝑤,𝑁,𝑧,𝑗 𝑂,𝑏,𝑗,𝑟 𝑤,𝑂,𝑧 𝑆,𝑏,𝑗,𝑟,𝑠,𝑡 𝑆,𝑘 𝑤,𝑆,𝑧 𝑋,𝑏,𝑟,𝑠,𝑡 𝑌,𝑠 𝜑,𝑏,𝑗,𝑟,𝑠 𝜒,𝑠,𝑡 𝜑,𝑤,𝑧 Allowed substitution hints: 𝜑(𝑡,𝑘) 𝜒(𝑧,𝑤,𝑗,𝑘,𝑟,𝑏) 𝐴(𝑧,𝑤,𝑗,𝑘) 𝐵(𝑧,𝑤,𝑗,𝑘) 𝐶(𝑧,𝑤,𝑗,𝑘) 𝐹(𝑗,𝑘) 𝐼(𝑡,𝑗,𝑘,𝑠,𝑟,𝑏) 𝑁(𝑡) 𝑂(𝑡,𝑘,𝑠) 𝑋(𝑧,𝑤,𝑗,𝑘) 𝑌(𝑧,𝑤,𝑡,𝑗,𝑘,𝑟,𝑏)

Proof of Theorem fourierdlem80 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem80.o	. . . . . . . . 9 ⊢ 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
2		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
3	2	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
4	3	oveq1d 7270	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) = ((𝐹‘(𝑋 + 𝑡)) − 𝐶))
5		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → (𝑠 / 2) = (𝑡 / 2))
6	5	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (sin‘(𝑠 / 2)) = (sin‘(𝑡 / 2)))
7	6	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝑡 / 2))))
8	4, 7	oveq12d 7273	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) = (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))
9	8	cbvmptv 5183	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))
10	1, 9	eqtr2i 2767	. . . . . . . 8 ⊢ (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))) = 𝑂
11	10	oveq2i 7266	. . . . . . 7 ⊢ (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = (ℝ D 𝑂)
12	11	dmeqi 5802	. . . . . 6 ⊢ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = dom (ℝ D 𝑂)
13	12	ineq2i 4140	. . . . 5 ⊢ (ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))))) = (ran 𝑆 ∩ dom (ℝ D 𝑂))
14	13	sneqi 4569	. . . 4 ⊢ {(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} = {(ran 𝑆 ∩ dom (ℝ D 𝑂))}
15	14	uneq1i 4089	. . 3 ⊢ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
16		snfi 8788	. . . . 5 ⊢ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∈ Fin
17		fzofi 13622	. . . . . 6 ⊢ (0..^𝑁) ∈ Fin
18		eqid 2738	. . . . . . 7 ⊢ (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
19	18	rnmptfi 42596	. . . . . 6 ⊢ ((0..^𝑁) ∈ Fin → ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ Fin)
20	17, 19	ax-mp 5	. . . . 5 ⊢ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ Fin
21		unfi 8917	. . . . 5 ⊢ (({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∈ Fin ∧ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ Fin) → ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ∈ Fin)
22	16, 20, 21	mp2an 688	. . . 4 ⊢ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ∈ Fin
23	22	a1i 11	. . 3 ⊢ (𝜑 → ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ∈ Fin)
24	15, 23	eqeltrid 2843	. 2 ⊢ (𝜑 → ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ∈ Fin)
25		id 22	. . . 4 ⊢ (𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
26	15	unieqi 4849	. . . 4 ⊢ ∪ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
27	25, 26	eleqtrdi 2849	. . 3 ⊢ (𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
28		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → 𝜑)
29		uniun 4861	. . . . . . . . 9 ⊢ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = (∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
30	29	eleq2i 2830	. . . . . . . 8 ⊢ (𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ 𝑠 ∈ (∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
31		elun 4079	. . . . . . . 8 ⊢ (𝑠 ∈ (∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ (𝑠 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
32	30, 31	sylbb 218	. . . . . . 7 ⊢ (𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → (𝑠 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
33	32	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (𝑠 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
34		fourierdlem80.sf	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
35		ovex 7288	. . . . . . . . . . . . . 14 ⊢ (0...𝑁) ∈ V
36	35	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...𝑁) ∈ V)
37	34, 36	fexd 7085	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ V)
38		rnexg 7725	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ V → ran 𝑆 ∈ V)
39	37, 38	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑆 ∈ V)
40		inex1g 5238	. . . . . . . . . . 11 ⊢ (ran 𝑆 ∈ V → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V)
41	39, 40	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V)
42		unisng 4857	. . . . . . . . . 10 ⊢ ((ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V → ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
43	41, 42	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
44	43	eleq2d 2824	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ↔ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))))
45	44	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (𝑠 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ↔ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))))
46	45	orbi1d 913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((𝑠 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∨ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))))
47	33, 46	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∨ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
48		dvf 24976	. . . . . . . . 9 ⊢ (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ
49	48	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ)
50		elinel2 4126	. . . . . . . 8 ⊢ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → 𝑠 ∈ dom (ℝ D 𝑂))
51	49, 50	ffvelrnd 6944	. . . . . . 7 ⊢ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
52	51	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
53		ovex 7288	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ∈ V
54	53	dfiun3 5864	. . . . . . . . . . 11 ⊢ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) = ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
55	54	eleq2i 2830	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
56	55	biimpri 227	. . . . . . . . 9 ⊢ (𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
57	56	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 ∈ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
58		eliun 4925	. . . . . . . 8 ⊢ (𝑠 ∈ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
59	57, 58	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
60		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
61		nfmpt1 5178	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
62	61	nfrn 5850	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
63	62	nfuni 4843	. . . . . . . . . 10 ⊢ Ⅎ𝑗∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
64	63	nfcri 2893	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
65	60, 64	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
66		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑗((ℝ D 𝑂)‘𝑠) ∈ ℂ
67	48	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ)
68		fourierdlem80.y	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑌 = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
69	1	reseq1i 5876	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
70		ioossicc 13094	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1)))
71		fourierdlem80.sjss	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
72	70, 71	sstrid 3928	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
73	72	resmptd 5937	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
74	69, 73	syl5eq 2791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
75	68, 74	eqtr4id 2798	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑌 = (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
76	75	oveq2d 7271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
77		ax-resscn 10859	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ⊆ ℂ
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ℝ ⊆ ℂ)
79		fourierdlem80.f	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
80	79	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ)
81		fourierdlem80.xre	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑋 ∈ ℝ)
82	81	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
83		fourierdlem80.a	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐴 ∈ ℝ)
84		fourierdlem80.b	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐵 ∈ ℝ)
85	83, 84	iccssred 13095	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
86	85	sselda 3917	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
87	82, 86	readdcld 10935	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
88	80, 87	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
89	88	recnd 10934	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
90		fourierdlem80.c	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐶 ∈ ℝ)
91	90	recnd 10934	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐶 ∈ ℂ)
92	91	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)
93	89, 92	subcld 11262	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
94		2cnd 11981	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 2 ∈ ℂ)
95	85, 78	sstrd 3927	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
96	95	sselda 3917	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
97	96	halfcld 12148	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑠 / 2) ∈ ℂ)
98	97	sincld 15767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
99	94, 98	mulcld 10926	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
100		2ne0 12007	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ≠ 0
101	100	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 2 ≠ 0)
102		fourierdlem80.ab	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
103	102	sselda 3917	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ (-π[,]π))
104		eqcom 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑠 = 0 ↔ 0 = 𝑠)
105	104	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑠 = 0 → 0 = 𝑠)
106	105	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 = 𝑠)
107		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 𝑠 ∈ (𝐴[,]𝐵))
108	106, 107	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
109	108	adantll 710	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
110		fourierdlem80.n0	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
111	110	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴[,]𝐵))
112	109, 111	pm2.65da 813	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ¬ 𝑠 = 0)
113	112	neqned 2949	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ≠ 0)
114		fourierdlem44 43582	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
115	103, 113, 114	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
116	94, 98, 101, 115	mulne0d 11557	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
117	93, 99, 116	divcld 11681	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
118	117, 1	fmptd 6970	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑂:(𝐴[,]𝐵)⟶ℂ)
119		ioossre 13069	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ
120	119	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ)
121		eqid 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
122	121	tgioo2 23872	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
123	121, 122	dvres 24980	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℝ ⊆ ℂ ∧ 𝑂:(𝐴[,]𝐵)⟶ℂ) ∧ ((𝐴[,]𝐵) ⊆ ℝ ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ)) → (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
124	78, 118, 85, 120, 123	syl22anc 835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
125		ioontr 42939	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))
126	125	reseq2i 5877	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
127	124, 126	eqtrdi 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
128	127	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
129	76, 128	eqtr2d 2779	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (ℝ D 𝑌))
130	129	dmeqd 5803	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = dom (ℝ D 𝑌))
131	79	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
132	81	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑋 ∈ ℝ)
133	85	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ ℝ)
134	34	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
135		elfzofz 13331	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ (0...𝑁))
136	135	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0...𝑁))
137	134, 136	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ (𝐴[,]𝐵))
138	133, 137	sseldd 3918	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ ℝ)
139		fzofzp1 13412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0..^𝑁) → (𝑗 + 1) ∈ (0...𝑁))
140	139	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 + 1) ∈ (0...𝑁))
141	134, 140	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ (𝐴[,]𝐵))
142	133, 141	sseldd 3918	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ ℝ)
143		fdv	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ 𝐼)):𝐼⟶ℝ)
144		fourierdlem80.i	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐼 = ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))
145	144	feq2i 6576	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℝ D (𝐹 ↾ 𝐼)):𝐼⟶ℝ ↔ (ℝ D (𝐹 ↾ 𝐼)):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
146	143, 145	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ 𝐼)):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
147	144	reseq2i 5877	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ↾ 𝐼) = (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
148	147	oveq2i 7266	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ D (𝐹 ↾ 𝐼)) = (ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))
149	148	feq1i 6575	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℝ D (𝐹 ↾ 𝐼)):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ ↔ (ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
150	146, 149	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
151	102	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ (-π[,]π))
152	72, 151	sstrd 3927	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
153	110	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ (𝐴[,]𝐵))
154	72, 153	ssneldd 3920	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
155	90	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ)
156	131, 132, 138, 142, 150, 152, 154, 155, 68	fourierdlem57 43594	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑌):((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ ∧ (ℝ D 𝑌) = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘(𝑋 + 𝑠)) · (2 · (sin‘(𝑠 / 2)))) − ((cos‘(𝑠 / 2)) · ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) / ((2 · (sin‘(𝑠 / 2)))↑2))))) ∧ (ℝ D (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (2 · (sin‘(𝑠 / 2))))) = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (cos‘(𝑠 / 2))))
157	156	simpli 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑌):((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ ∧ (ℝ D 𝑌) = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘(𝑋 + 𝑠)) · (2 · (sin‘(𝑠 / 2)))) − ((cos‘(𝑠 / 2)) · ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) / ((2 · (sin‘(𝑠 / 2)))↑2)))))
158	157	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌):((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ)
159		fdm 6593	. . . . . . . . . . . . . . . 16 ⊢ ((ℝ D 𝑌):((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ → dom (ℝ D 𝑌) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
160	158, 159	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → dom (ℝ D 𝑌) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
161	130, 160	eqtr2d 2779	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) = dom ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
162		resss 5905	. . . . . . . . . . . . . . 15 ⊢ ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ (ℝ D 𝑂)
163		dmss 5800	. . . . . . . . . . . . . . 15 ⊢ (((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ (ℝ D 𝑂) → dom ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ dom (ℝ D 𝑂))
164	162, 163	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ dom (ℝ D 𝑂))
165	161, 164	eqsstrd 3955	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
166	165	3adant3 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
167		simp3 1136	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
168	166, 167	sseldd 3918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ dom (ℝ D 𝑂))
169	67, 168	ffvelrnd 6944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
170	169	3exp 1117	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)))
171	170	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → (𝑗 ∈ (0..^𝑁) → (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)))
172	65, 66, 171	rexlimd 3245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → (∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ))
173	59, 172	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
174	52, 173	jaodan 954	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∨ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
175	28, 47, 174	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
176	175	abscld 15076	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
177	27, 176	sylan2 592	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
178		id 22	. . . 4 ⊢ (𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
179	178, 15	eleqtrdi 2849	. . 3 ⊢ (𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
180		elsni 4575	. . . . . 6 ⊢ (𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
181		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
182		fzfid 13621	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝑁) ∈ Fin)
183		rnffi 42600	. . . . . . . . . . 11 ⊢ ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (0...𝑁) ∈ Fin) → ran 𝑆 ∈ Fin)
184	34, 182, 183	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑆 ∈ Fin)
185		infi 8972	. . . . . . . . . 10 ⊢ (ran 𝑆 ∈ Fin → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
186	184, 185	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
187	186	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
188	181, 187	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 ∈ Fin)
189		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑠𝜑
190		nfcv 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑠ran 𝑆
191		nfcv 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑠ℝ
192		nfcv 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑠 D
193		nfmpt1 5178	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑠(𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
194	1, 193	nfcxfr 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑠𝑂
195	191, 192, 194	nfov 7285	. . . . . . . . . . . 12 ⊢ Ⅎ𝑠(ℝ D 𝑂)
196	195	nfdm 5849	. . . . . . . . . . 11 ⊢ Ⅎ𝑠dom (ℝ D 𝑂)
197	190, 196	nfin 4147	. . . . . . . . . 10 ⊢ Ⅎ𝑠(ran 𝑆 ∩ dom (ℝ D 𝑂))
198	197	nfeq2 2923	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))
199	189, 198	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑠(𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
200		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ 𝑟)
201		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠 ∈ 𝑟) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
202	200, 201	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
203	202, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
204	203	adantll 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
205	48	ffvelrni 6942	. . . . . . . . . . 11 ⊢ (𝑠 ∈ dom (ℝ D 𝑂) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
206	205	abscld 15076	. . . . . . . . . 10 ⊢ (𝑠 ∈ dom (ℝ D 𝑂) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
207	204, 206	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠 ∈ 𝑟) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
208	207	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (𝑠 ∈ 𝑟 → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ))
209	199, 208	ralrimi 3139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
210		fimaxre3 11851	. . . . . . 7 ⊢ ((𝑟 ∈ Fin ∧ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
211	188, 209, 210	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
212	180, 211	sylan2 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
213	212	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
214		simpll 763	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → 𝜑)
215		elunnel1 4080	. . . . . 6 ⊢ ((𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
216	215	adantll 710	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
217		vex 3426	. . . . . . . . 9 ⊢ 𝑟 ∈ V
218	18	elrnmpt 5854	. . . . . . . . 9 ⊢ (𝑟 ∈ V → (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
219	217, 218	ax-mp 5	. . . . . . . 8 ⊢ (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
220	219	biimpi 215	. . . . . . 7 ⊢ (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
221	220	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
222	62	nfcri 2893	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
223	60, 222	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
224		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑗∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦
225		fourierdlem80.fbdioo	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤)
226		fourierdlem80.fdvbdioo	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)
227		reeanv 3292	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧) ↔ (∃𝑤 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∃𝑧 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧))
228	225, 226, 227	sylanbrc 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧))
229		simp1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → (𝜑 ∧ 𝑗 ∈ (0..^𝑁)))
230		simp2l 1197	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → 𝑤 ∈ ℝ)
231		simp2r 1198	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → 𝑧 ∈ ℝ)
232	229, 230, 231	jca31 514	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ))
233		simp3l 1199	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤)
234		simp3r 1200	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)
235	232, 233, 234	jca31 514	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧))
236		fourierdlem80.ch	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧))
237	235, 236	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → 𝜒)
238	236	biimpi 215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜒 → (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧))
239		simp-5l 781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧) → 𝜑)
240	238, 239	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → 𝜑)
241	240, 79	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝐹:ℝ⟶ℝ)
242	240, 81	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝑋 ∈ ℝ)
243		simp-4l 779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧) → (𝜑 ∧ 𝑗 ∈ (0..^𝑁)))
244	238, 243	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → (𝜑 ∧ 𝑗 ∈ (0..^𝑁)))
245	244, 138	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (𝑆‘𝑗) ∈ ℝ)
246	244, 142	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (𝑆‘(𝑗 + 1)) ∈ ℝ)
247		fourierdlem80.slt	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))
248	244, 247	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))
249	71, 151	sstrd 3927	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
250	244, 249	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (-π[,]π))
251	71, 153	ssneldd 3920	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))))
252	244, 251	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → ¬ 0 ∈ ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))))
253	244, 150	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
254		simp-4r 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧) → 𝑤 ∈ ℝ)
255	238, 254	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝑤 ∈ ℝ)
256	238	simplrd 766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤)
257		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
258	257, 144	eleqtrrdi 2850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡 ∈ 𝐼)
259		rspa 3130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ 𝑡 ∈ 𝐼) → (abs‘(𝐹‘𝑡)) ≤ 𝑤)
260	256, 258, 259	syl2an 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜒 ∧ 𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘(𝐹‘𝑡)) ≤ 𝑤)
261		simpllr 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧) → 𝑧 ∈ ℝ)
262	238, 261	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝑧 ∈ ℝ)
263	148	fveq1i 6757	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ D (𝐹 ↾ 𝐼))‘𝑡) = ((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)
264	263	fveq2i 6759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) = (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡))
265	238	simprd 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜒 → ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)
266	265	r19.21bi 3132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜒 ∧ 𝑡 ∈ 𝐼) → (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)
267	264, 266	eqbrtrrid 5106	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜒 ∧ 𝑡 ∈ 𝐼) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
268	258, 267	sylan2 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜒 ∧ 𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
269	240, 90	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝐶 ∈ ℝ)
270	241, 242, 245, 246, 248, 250, 252, 253, 255, 260, 262, 268, 269, 68	fourierdlem68 43605	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜒 → (dom (ℝ D 𝑌) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ∧ ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
271	270	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
272	270	simpld 494	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → dom (ℝ D 𝑌) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
273	272	raleqdv 3339	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜒 → (∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
274	273	rexbidv 3225	. . . . . . . . . . . . . . . . 17 ⊢ (𝜒 → (∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
275	271, 274	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
276	125	eqcomi 2747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) = ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
277	276	reseq2i 5877	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
278	277	fveq1i 6757	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠)
279		fvres 6775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → (((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
280	279	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
281	244, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜒 → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
282	281	resmptd 5937	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜒 → ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
283	69, 282	syl5eq 2791	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜒 → (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
284	68, 283	eqtr4id 2798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜒 → 𝑌 = (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
285	284	oveq2d 7271	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜒 → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
286	285	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜒 → ((ℝ D 𝑌)‘𝑠) = ((ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
287	124	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
288	240, 287	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜒 → ((ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
289	286, 288	eqtr2d 2779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜒 → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
290	289	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
291	278, 280, 290	3eqtr3a 2803	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) = ((ℝ D 𝑌)‘𝑠))
292	291	fveq2d 6760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (abs‘((ℝ D 𝑂)‘𝑠)) = (abs‘((ℝ D 𝑌)‘𝑠)))
293	292	breq1d 5080	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ (abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
294	293	ralbidva 3119	. . . . . . . . . . . . . . . . 17 ⊢ (𝜒 → (∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
295	294	rexbidv 3225	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 → (∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
296	275, 295	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
297	237, 296	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ (𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
298	297	3exp 1117	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
299	298	rexlimdvv 3221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
300	228, 299	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
301	300	3adant3 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
302		raleq 3333	. . . . . . . . . . . 12 ⊢ (𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → (∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
303	302	3ad2ant3 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
304	303	rexbidv 3225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
305	301, 304	mpbird 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
306	305	3exp 1117	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
307	306	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → (𝑗 ∈ (0..^𝑁) → (𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
308	223, 224, 307	rexlimd 3245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → (∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦))
309	221, 308	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
310	214, 216, 309	syl2anc 583	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) ∧ ¬ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
311	213, 310	pm2.61dan 809	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
312	179, 311	sylan2 592	. 2 ⊢ ((𝜑 ∧ 𝑟 ∈ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
313		pm3.22 459	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → (𝑟 ∈ ran 𝑆 ∧ 𝑟 ∈ dom (ℝ D 𝑂)))
314		elin 3899	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ↔ (𝑟 ∈ ran 𝑆 ∧ 𝑟 ∈ dom (ℝ D 𝑂)))
315	313, 314	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
316	315	adantll 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
317	43	eqcomd 2744	. . . . . . . . . . 11 ⊢ (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
318	317	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
319	316, 318	eleqtrd 2841	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
320	319	orcd 869	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → (𝑟 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
321		simpll 763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝜑)
322	77	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → ℝ ⊆ ℂ)
323	118	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑂:(𝐴[,]𝐵)⟶ℂ)
324	83	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝐴 ∈ ℝ)
325	84	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝐵 ∈ ℝ)
326	324, 325	iccssred 13095	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → (𝐴[,]𝐵) ⊆ ℝ)
327	322, 323, 326	dvbss 24970	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → dom (ℝ D 𝑂) ⊆ (𝐴[,]𝐵))
328		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ dom (ℝ D 𝑂))
329	327, 328	sseldd 3918	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ (𝐴[,]𝐵))
330	329	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (𝐴[,]𝐵))
331		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ¬ 𝑟 ∈ ran 𝑆)
332		fourierdlem80.relioo	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
333		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑘 → (𝑆‘𝑗) = (𝑆‘𝑘))
334		oveq1 7262	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
335	334	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑘 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝑘 + 1)))
336	333, 335	oveq12d 7273	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) = ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
337		ovex 7288	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))) ∈ V
338	336, 18, 337	fvmpt 6857	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) = ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
339	338	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ 𝑟 ∈ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1)))))
340	339	rexbiia 3176	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
341	332, 340	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
342	53, 18	dmmpti 6561	. . . . . . . . . . . . 13 ⊢ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (0..^𝑁)
343	342	rexeqi 3338	. . . . . . . . . . . 12 ⊢ (∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
344	341, 343	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
345	321, 330, 331, 344	syl21anc 834	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
346		funmpt 6456	. . . . . . . . . . 11 ⊢ Fun (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
347		elunirn 7106	. . . . . . . . . . 11 ⊢ (Fun (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘)))
348	346, 347	mp1i 13	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → (𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑘 ∈ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘)))
349	345, 348	mpbird 256	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
350	349	olcd 870	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → (𝑟 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
351	320, 350	pm2.61dan 809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → (𝑟 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
352		elun 4079	. . . . . . 7 ⊢ (𝑟 ∈ (∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ (𝑟 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
353	351, 352	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ (∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
354	353, 29	eleqtrrdi 2850	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
355	354	ralrimiva 3107	. . . 4 ⊢ (𝜑 → ∀𝑟 ∈ dom (ℝ D 𝑂)𝑟 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
356		dfss3 3905	. . . 4 ⊢ (dom (ℝ D 𝑂) ⊆ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) ↔ ∀𝑟 ∈ dom (ℝ D 𝑂)𝑟 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
357	355, 356	sylibr 233	. . 3 ⊢ (𝜑 → dom (ℝ D 𝑂) ⊆ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
358	357, 26	sseqtrrdi 3968	. 2 ⊢ (𝜑 → dom (ℝ D 𝑂) ⊆ ∪ ({(ran 𝑆 ∩ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
359	24, 177, 312, 358	ssfiunibd 42738	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 {csn 4558 ∪ cuni 4836 ∪ ciun 4921 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ran crn 5581 ↾ cres 5582 Fun wfun 6412 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 2c2 11958 (,)cioo 13008 [,]cicc 13011 ...cfz 13168 ..^cfzo 13311 ↑cexp 13710 abscabs 14873 sincsin 15701 cosccos 15702 πcpi 15704 TopOpenctopn 17049 topGenctg 17065 ℂ_fldccnfld 20510 intcnt 22076 D cdv 24932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-t1 22373 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936

This theorem is referenced by: fourierdlem103 43640 fourierdlem104 43641

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