fourierdlem80 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem80 47196

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 7428	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → (𝑋 + 𝑠) = (𝑋 + 𝑡))
3	2	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑡)))
4	3	oveq1d 7435	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) = ((𝐹‘(𝑋 + 𝑡)) − 𝐶))
5		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → (𝑠 / 2) = (𝑡 / 2))
6	5	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (sin‘(𝑠 / 2)) = (sin‘(𝑡 / 2)))
7	6	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝑡 / 2))))
8	4, 7	oveq12d 7438	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) = (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))
9	8	cbvmptv 5208	. . . . . . . . 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 2784	. . . . . . . 8 ⊢ (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2))))) = 𝑂
11	10	oveq2i 7431	. . . . . . 7 ⊢ (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = (ℝ D 𝑂)
12	11	dmeqi 5885	. . . . . 6 ⊢ dom (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑡)) − 𝐶) / (2 · (sin‘(𝑡 / 2)))))) = dom (ℝ D 𝑂)
13	12	ineq2i 4114	. . . . 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 4594	. . . 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 4163	. . 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 9071	. . . . 5 ⊢ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∈ Fin
17		fzofi 14117	. . . . . 6 ⊢ (0..^𝑁) ∈ Fin
18		eqid 2760	. . . . . . 7 ⊢ (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
19	18	rnmptfi 46186	. . . . . 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 9186	. . . . 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 705	. . . 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 2864	. 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 23	. . . 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 4878	. . . 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 2870	. . 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 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → 𝜑)
29		uniun 4889	. . . . . . . . 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 2852	. . . . . . . 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 4152	. . . . . . . 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 222	. . . . . . 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 487	. . . . . 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	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
35		ovex 7453	. . . . . . . . . . . 12 ⊢ (0...𝑁) ∈ V
36	35	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝑁) ∈ V)
37	34, 36	fexd 7233	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ V)
38		rnexg 7914	. . . . . . . . . 10 ⊢ (𝑆 ∈ V → ran 𝑆 ∈ V)
39		inex1g 5278	. . . . . . . . . 10 ⊢ (ran 𝑆 ∈ V → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V)
40		unisng 4884	. . . . . . . . . 10 ⊢ ((ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ V → ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
41	37, 38, 39, 40	4syl 20	. . . . . . . . 9 ⊢ (𝜑 → ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
42	41	eleq2d 2846	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ↔ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))))
43	42	adantr 486	. . . . . . 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))))
44	43	orbi1d 930	. . . . . 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)))))))
45	33, 44	mpbid 235	. . . . 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))))))
46		dvf 26227	. . . . . . . . 9 ⊢ (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ
47	46	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ)
48		elinel2 4099	. . . . . . . 8 ⊢ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → 𝑠 ∈ dom (ℝ D 𝑂))
49	47, 48	ffvelcdmd 7085	. . . . . . 7 ⊢ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
50	49	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
51		ovex 7453	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ∈ V
52	51	dfiun3 5951	. . . . . . . . . 10 ⊢ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) = ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
53	52	eleq2i 2852	. . . . . . . . 9 ⊢ (𝑠 ∈ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
54	53	bilanri 512	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → 𝑠 ∈ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
55		eliun 4954	. . . . . . . 8 ⊢ (𝑠 ∈ ∪ 𝑗 ∈ (0..^𝑁)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
56	54, 55	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
57		nfv 1946	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
58		nfmpt1 5203	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
59	58	nfrn 5933	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
60	59	nfuni 4873	. . . . . . . . . 10 ⊢ Ⅎ𝑗∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
61	60	nfcri 2914	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
62	57, 61	nfan 1931	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
63		nfv 1946	. . . . . . . 8 ⊢ Ⅎ𝑗((ℝ D 𝑂)‘𝑠) ∈ ℂ
64	46	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (ℝ D 𝑂):dom (ℝ D 𝑂)⟶ℂ)
65		fourierdlem80.y	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑌 = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
66	1	reseq1i 5965	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
67		ioossicc 13564	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1)))
68		fourierdlem80.sjss	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
69	67, 68	sstrid 3919	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
70	69	resmptd 6031	. . . . . . . . . . . . . . . . . . . 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))))))
71	66, 70	eqtrid 2807	. . . . . . . . . . . . . . . . . . 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))))))
72	65, 71	eqtr4id 2814	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑌 = (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
73	72	oveq2d 7436	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
74		ax-resscn 11257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ⊆ ℂ
75	74	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ℝ ⊆ ℂ)
76		fourierdlem80.f	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
77	76	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℝ)
78		fourierdlem80.xre	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑋 ∈ ℝ)
79	78	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
80		fourierdlem80.a	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐴 ∈ ℝ)
81		fourierdlem80.b	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐵 ∈ ℝ)
82	80, 81	iccssred 13565	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
83	82	sselda 3908	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
84	79, 83	readdcld 11338	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
85	77, 84	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
86	85	recnd 11337	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
87		fourierdlem80.c	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐶 ∈ ℝ)
88	87	recnd 11337	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐶 ∈ ℂ)
89	88	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)
90	86, 89	subcld 11669	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
91		2cnd 12421	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 2 ∈ ℂ)
92	82, 75	sstrd 3918	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
93	92	sselda 3908	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
94	93	halfcld 12591	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑠 / 2) ∈ ℂ)
95	94	sincld 16298	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
96	91, 95	mulcld 11329	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
97		2ne0 12449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ≠ 0
98	97	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 2 ≠ 0)
99		fourierdlem80.ab	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ( -π[,]π))
100	99	sselda 3908	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ( -π[,]π))
101		eqcom 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑠 = 0 ↔ 0 = 𝑠)
102	101	bilani 510	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 = 𝑠)
103		simpl 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 𝑠 ∈ (𝐴[,]𝐵))
104	102, 103	eqeltrd 2860	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑠 ∈ (𝐴[,]𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
105	104	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴[,]𝐵))
106		fourierdlem80.n0	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
107	106	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴[,]𝐵))
108	105, 107	pm2.65da 829	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ¬ 𝑠 = 0)
109	108	neqned 2962	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ≠ 0)
110		fourierdlem44 47161	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ ( -π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
111	100, 109, 110	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
112	91, 95, 98, 111	mulne0d 11968	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
113	90, 96, 112	divcld 12093	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
114	113, 1	fmptd 7114	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑂:(𝐴[,]𝐵)⟶ℂ)
115		ioossre 13538	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ
116	115	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ℝ)
117		eqid 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
118		tgioo4 25124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
119	117, 118	dvres 26231	. . . . . . . . . . . . . . . . . . . 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))))))
120	75, 114, 82, 116, 119	syl22anc 852	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
121		ioontr 46523	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))
122	121	reseq2i 5966	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
123	120, 122	eqtrdi 2811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
124	123	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) = ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
125	73, 124	eqtr2d 2796	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (ℝ D 𝑌))
126	125	dmeqd 5886	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = dom (ℝ D 𝑌))
127	76	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℝ)
128	78	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑋 ∈ ℝ)
129	82	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ ℝ)
130	34	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
131		elfzofz 13810	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ (0...𝑁))
132	131	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0...𝑁))
133	130, 132	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ (𝐴[,]𝐵))
134	129, 133	sseldd 3909	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) ∈ ℝ)
135		fzofzp1 13899	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0..^𝑁) → (𝑗 + 1) ∈ (0...𝑁))
136	135	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 + 1) ∈ (0...𝑁))
137	130, 136	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ (𝐴[,]𝐵))
138	129, 137	sseldd 3909	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘(𝑗 + 1)) ∈ ℝ)
139		fdv	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ 𝐼)):𝐼⟶ℝ)
140		fourierdlem80.i	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐼 = ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))
141	140	feq2i 6701	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℝ D (𝐹 ↾ 𝐼)):𝐼⟶ℝ ↔ (ℝ D (𝐹 ↾ 𝐼)):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
142	139, 141	sylib 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ 𝐼)):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
143	140	reseq2i 5966	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ↾ 𝐼) = (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
144	143	oveq2i 7431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ D (𝐹 ↾ 𝐼)) = (ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))
145	144	feq1i 6700	. . . . . . . . . . . . . . . . . . . 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)
146	142, 145	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
147	99	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴[,]𝐵) ⊆ ( -π[,]π))
148	69, 147	sstrd 3918	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ( -π[,]π))
149	106	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ (𝐴[,]𝐵))
150	69, 149	ssneldd 3911	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
151	87	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐶 ∈ ℝ)
152	127, 128, 134, 138, 146, 148, 150, 151, 65	fourierdlem57 47173	. . . . . . . . . . . . . . . . . 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))))
153	152	simpli 489	. . . . . . . . . . . . . . . . 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)))))
154	153	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D 𝑌):((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ)
155		fdm 6719	. . . . . . . . . . . . . . . 16 ⊢ ((ℝ D 𝑌):((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))⟶ℝ → dom (ℝ D 𝑌) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
156	154, 155	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → dom (ℝ D 𝑌) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
157	126, 156	eqtr2d 2796	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) = dom ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
158		resss 5991	. . . . . . . . . . . . . . 15 ⊢ ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ (ℝ D 𝑂)
159		dmss 5883	. . . . . . . . . . . . . . 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))
160	158, 159	mp1i 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → dom ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ⊆ dom (ℝ D 𝑂))
161	157, 160	eqsstrd 3942	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
162	161	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ dom (ℝ D 𝑂))
163		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
164	162, 163	sseldd 3909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → 𝑠 ∈ dom (ℝ D 𝑂))
165	64, 164	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
166	165	3exp 1137	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)))
167	166	adantr 486	. . . . . . . 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)))
168	62, 63, 167	rexlimd 3269	. . . . . . 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))
169	56, 168	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
170	50, 169	jaodan 972	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∨ 𝑠 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
171	28, 45, 170	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
172	171	abscld 15606	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
173	27, 172	sylan2 605	. 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)
174		id 23	. . . 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))))))
175	174, 15	eleqtrdi 2870	. . 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))))))
176		elsni 4600	. . . . . 6 ⊢ (𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
177		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
178		fzfid 14116	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝑁) ∈ Fin)
179		rnffi 46190	. . . . . . . . . . 11 ⊢ ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (0...𝑁) ∈ Fin) → ran 𝑆 ∈ Fin)
180	34, 178, 179	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑆 ∈ Fin)
181		infi 9261	. . . . . . . . . 10 ⊢ (ran 𝑆 ∈ Fin → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
182	180, 181	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
183	182	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∈ Fin)
184	177, 183	eqeltrd 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → 𝑟 ∈ Fin)
185		nfv 1946	. . . . . . . . 9 ⊢ Ⅎ𝑠𝜑
186		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑠ran 𝑆
187		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑠ℝ
188		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑠 D
189		nfmpt1 5203	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑠(𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
190	1, 189	nfcxfr 2920	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑠𝑂
191	187, 188, 190	nfov 7450	. . . . . . . . . . . 12 ⊢ Ⅎ𝑠(ℝ D 𝑂)
192	191	nfdm 5932	. . . . . . . . . . 11 ⊢ Ⅎ𝑠dom (ℝ D 𝑂)
193	186, 192	nfin 4121	. . . . . . . . . 10 ⊢ Ⅎ𝑠(ran 𝑆 ∩ dom (ℝ D 𝑂))
194	193	nfeq2 2939	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))
195	185, 194	nfan 1931	. . . . . . . 8 ⊢ Ⅎ𝑠(𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
196		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ 𝑟)
197		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠 ∈ 𝑟) → 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)))
198	196, 197	eleqtrd 2862	. . . . . . . . . . . 12 ⊢ ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
199	198, 48	syl 18	. . . . . . . . . . 11 ⊢ ((𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂)) ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
200	199	adantll 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ dom (ℝ D 𝑂))
201	46	ffvelcdmi 7083	. . . . . . . . . . 11 ⊢ (𝑠 ∈ dom (ℝ D 𝑂) → ((ℝ D 𝑂)‘𝑠) ∈ ℂ)
202	201	abscld 15606	. . . . . . . . . 10 ⊢ (𝑠 ∈ dom (ℝ D 𝑂) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
203	200, 202	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) ∧ 𝑠 ∈ 𝑟) → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
204	203	ex 418	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → (𝑠 ∈ 𝑟 → (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ))
205	195, 204	ralrimi 3260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ)
206		fimaxre3 12263	. . . . . . 7 ⊢ ((𝑟 ∈ Fin ∧ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
207	184, 205, 206	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = (ran 𝑆 ∩ dom (ℝ D 𝑂))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
208	176, 207	sylan2 605	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ {(ran 𝑆 ∩ dom (ℝ D 𝑂))}) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
209	208	adantlr 728	. . . 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)
210		simpll 779	. . . . 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)
211		elunnel1 4153	. . . . . 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)))))
212	211	adantll 727	. . . . 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)))))
213		vex 3454	. . . . . . . 8 ⊢ 𝑟 ∈ V
214	18	elrnmpt 5939	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
215	213, 214	ax-mp 5	. . . . . . 7 ⊢ (𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ↔ ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
216	215	bilani 510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑗 ∈ (0..^𝑁)𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
217	59	nfcri 2914	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
218	57, 217	nfan 1931	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
219		nfv 1946	. . . . . . 7 ⊢ Ⅎ𝑗∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦
220		fourierdlem80.fbdioo	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤)
221		fourierdlem80.fdvbdioo	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)
222		reeanv 3234	. . . . . . . . . . . . 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))
223	220, 221, 222	sylanbrc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∃𝑧 ∈ ℝ (∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧))
224		simp1 1154	. . . . . . . . . . . . . . . . . 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)))
225		simp2l 1218	. . . . . . . . . . . . . . . . . 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)
226		simp2r 1219	. . . . . . . . . . . . . . . . . 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)
227	224, 225, 226	jca31 524	. . . . . . . . . . . . . . . . 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))
228		simp3l 1220	. . . . . . . . . . . . . . . . 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)
229		simp3r 1221	. . . . . . . . . . . . . . . . 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)
230	227, 228, 229	jca31 524	. . . . . . . . . . . . . . . 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))
231		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))
232	230, 231	sylibr 237	. . . . . . . . . . . . . . 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)
233	231	biimpi 219	. . . . . . . . . . . . . . . . . . . . 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))
234		simp-5l 797	. . . . . . . . . . . . . . . . . . . . 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)
235	233, 234	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → 𝜑)
236	235, 76	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝐹:ℝ⟶ℝ)
237	235, 78	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝑋 ∈ ℝ)
238		simp-4l 795	. . . . . . . . . . . . . . . . . . . . 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)))
239	233, 238	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → (𝜑 ∧ 𝑗 ∈ (0..^𝑁)))
240	239, 134	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (𝑆‘𝑗) ∈ ℝ)
241	239, 138	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (𝑆‘(𝑗 + 1)) ∈ ℝ)
242		fourierdlem80.slt	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))
243	239, 242	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1)))
244	68, 147	sstrd 3918	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ ( -π[,]π))
245	239, 244	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ ( -π[,]π))
246	68, 149	ssneldd 3911	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ¬ 0 ∈ ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))))
247	239, 246	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → ¬ 0 ∈ ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))))
248	239, 146	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → (ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))):((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))⟶ℝ)
249		simp-4r 796	. . . . . . . . . . . . . . . . . . . 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)
250	233, 249	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝑤 ∈ ℝ)
251	233	simplrd 782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤)
252		id 23	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))))
253	252, 140	eleqtrrdi 2871	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) → 𝑡 ∈ 𝐼)
254		rspa 3251	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤 ∧ 𝑡 ∈ 𝐼) → (abs‘(𝐹‘𝑡)) ≤ 𝑤)
255	251, 253, 254	syl2an 608	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜒 ∧ 𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘(𝐹‘𝑡)) ≤ 𝑤)
256		simpllr 788	. . . . . . . . . . . . . . . . . . . 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)
257	233, 256	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝑧 ∈ ℝ)
258	144	fveq1i 6886	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ D (𝐹 ↾ 𝐼))‘𝑡) = ((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)
259	258	fveq2i 6888	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) = (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡))
260	233	simprd 501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜒 → ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)
261	260	r19.21bi 3254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜒 ∧ 𝑡 ∈ 𝐼) → (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)
262	259, 261	eqbrtrrid 5140	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜒 ∧ 𝑡 ∈ 𝐼) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
263	253, 262	sylan2 605	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜒 ∧ 𝑡 ∈ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))))‘𝑡)) ≤ 𝑧)
264	235, 87	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → 𝐶 ∈ ℝ)
265	236, 237, 240, 241, 243, 245, 247, 248, 250, 255, 257, 263, 264, 65	fourierdlem68 47184	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜒 → (dom (ℝ D 𝑌) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ∧ ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
266	265	simprd 501	. . . . . . . . . . . . . . . . 17 ⊢ (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑌)(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
267	265	simpld 500	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 → dom (ℝ D 𝑌) = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
268	267	raleqdv 3319	. . . . . . . . . . . . . . . . . 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))
269	268	rexbidv 3186	. . . . . . . . . . . . . . . . 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))
270	266, 269	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦)
271	121	eqcomi 2769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) = ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
272	271	reseq2i 5966	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = ((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
273	272	fveq1i 6886	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠)
274		fvres 6904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → (((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
275	274	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑠) = ((ℝ D 𝑂)‘𝑠))
276	239, 69	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜒 → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))
277	276	resmptd 6031	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
278	66, 277	eqtrid 2807	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜒 → (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
279	65, 278	eqtr4id 2814	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜒 → 𝑌 = (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
280	279	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜒 → (ℝ D 𝑌) = (ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
281	280	fveq1d 6887	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜒 → ((ℝ D 𝑌)‘𝑠) = ((ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
282	120	fveq1d 6887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
283	235, 282	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜒 → ((ℝ D (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠))
284	281, 283	eqtr2d 2796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜒 → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
285	284	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (((ℝ D 𝑂) ↾ ((int‘(topGen‘ran (,)))‘((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))‘𝑠) = ((ℝ D 𝑌)‘𝑠))
286	273, 275, 285	3eqtr3a 2819	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((ℝ D 𝑂)‘𝑠) = ((ℝ D 𝑌)‘𝑠))
287	286	fveq2d 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → (abs‘((ℝ D 𝑂)‘𝑠)) = (abs‘((ℝ D 𝑌)‘𝑠)))
288	287	breq1d 5112	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜒 ∧ 𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ((abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦 ↔ (abs‘((ℝ D 𝑌)‘𝑠)) ≤ 𝑦))
289	288	ralbidva 3183	. . . . . . . . . . . . . . . . 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))
290	289	rexbidv 3186	. . . . . . . . . . . . . . . 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))
291	270, 290	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
292	232, 291	syl 18	. . . . . . . . . . . . . 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)
293	292	3exp 1137	. . . . . . . . . . . . 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)))
294	293	rexlimdvv 3218	. . . . . . . . . . . 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))
295	223, 294	mpd 16	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
296	295	3adant3 1150	. . . . . . . . . 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)
297		raleq 3316	. . . . . . . . . . . 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))
298	297	3ad2ant3 1153	. . . . . . . . . . 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))
299	298	rexbidv 3186	. . . . . . . . . 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))
300	296, 299	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁) ∧ 𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
301	300	3exp 1137	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ (0..^𝑁) → (𝑟 = ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)))
302	301	adantr 486	. . . . . . 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)))
303	218, 219, 302	rexlimd 3269	. . . . . 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))
304	216, 303	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))) → ∃𝑦 ∈ ℝ ∀𝑠 ∈ 𝑟 (abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑦)
305	210, 212, 304	syl2anc 596	. . . 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)
306	209, 305	pm2.61dan 825	. . 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)
307	175, 306	sylan2 605	. 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)
308		pm3.22 465	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → (𝑟 ∈ ran 𝑆 ∧ 𝑟 ∈ dom (ℝ D 𝑂)))
309		elin 4093	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)) ↔ (𝑟 ∈ ran 𝑆 ∧ 𝑟 ∈ dom (ℝ D 𝑂)))
310	308, 309	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ dom (ℝ D 𝑂) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
311	310	adantll 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (ran 𝑆 ∩ dom (ℝ D 𝑂)))
312	41	eqcomd 2766	. . . . . . . . . . 11 ⊢ (𝜑 → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
313	312	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → (ran 𝑆 ∩ dom (ℝ D 𝑂)) = ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
314	311, 313	eleqtrd 2862	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))})
315	314	orcd 887	. . . . . . . 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))))))
316		simpll 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝜑)
317	74	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → ℝ ⊆ ℂ)
318	114	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑂:(𝐴[,]𝐵)⟶ℂ)
319	80	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝐴 ∈ ℝ)
320	81	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝐵 ∈ ℝ)
321	319, 320	iccssred 13565	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → (𝐴[,]𝐵) ⊆ ℝ)
322	317, 318, 321	dvbss 26221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → dom (ℝ D 𝑂) ⊆ (𝐴[,]𝐵))
323		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ dom (ℝ D 𝑂))
324	322, 323	sseldd 3909	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ (𝐴[,]𝐵))
325	324	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ (𝐴[,]𝐵))
326		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ¬ 𝑟 ∈ ran 𝑆)
327		fourierdlem80.relioo	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
328		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑘 → (𝑆‘𝑗) = (𝑆‘𝑘))
329		oveq1 7427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
330	329	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑘 → (𝑆‘(𝑗 + 1)) = (𝑆‘(𝑘 + 1)))
331	328, 330	oveq12d 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) = ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
332		ovex 7453	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))) ∈ V
333	331, 18, 332	fvmpt 6993	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) = ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
334	333	eleq2d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0..^𝑁) → (𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ 𝑟 ∈ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1)))))
335	334	rexbiia 3107	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘) ↔ ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1))))
336	327, 335	sylibr 237	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))‘𝑘))
337	51, 18	dmmpti 6683	. . . . . . . . . . . . 13 ⊢ dom (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) = (0..^𝑁)
338	337	rexeqi 3318	. . . . . . . . . . . 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))
339	336, 338	sylibr 237	. . . . . . . . . . 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))
340	316, 325, 326, 339	syl21anc 851	. . . . . . . . . 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))
341		funmpt 6578	. . . . . . . . . . 11 ⊢ Fun (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))
342		elunirn 7255	. . . . . . . . . . 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)))
343	341, 342	mp1i 14	. . . . . . . . . 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)))
344	340, 343	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) ∧ ¬ 𝑟 ∈ ran 𝑆) → 𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))))
345	344	olcd 888	. . . . . . . 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))))))
346	315, 345	pm2.61dan 825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → (𝑟 ∈ ∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∨ 𝑟 ∈ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
347		elun 4152	. . . . . . 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))))))
348	346, 347	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ (∪ {(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
349	348, 29	eleqtrrdi 2871	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ dom (ℝ D 𝑂)) → 𝑟 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
350	349	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑟 ∈ dom (ℝ D 𝑂)𝑟 ∈ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
351		dfss3 3897	. . . 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))))))
352	350, 351	sylibr 237	. . 3 ⊢ (𝜑 → dom (ℝ D 𝑂) ⊆ ∪ ({(ran 𝑆 ∩ dom (ℝ D 𝑂))} ∪ ran (𝑗 ∈ (0..^𝑁) ↦ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))))))
353	352, 26	sseqtrrdi 3949	. 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))))))
354	24, 173, 307, 353	ssfiunibd 46325	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 ∩ cin 4089 ∪ cun 4149 {csn 4583 ∪ cuni 4866 ∪ ciun 4950 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5650 ran crn 5651 ↾ cres 5652 Fun wfun 6532 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 Fincfn 8973 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 < clt 11343 ≤ cle 11344 − cmin 11541 -cneg 11542 / cdiv 11973 2c2 12397 (,)cioo 13476 [,]cicc 13479 ...cfz 13639 ..^cfzo 13788 ↑cexp 14204 abscabs 15401 sincsin 16229 cosccos 16230 πcpi 16232 TopOpenctopn 17592 topGenctg 17608 ℂ_fldccnfld 21678 intcnt 23335 D cdv 26183

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-t1 23632 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187

This theorem is used by: fourierdlem103 47219 fourierdlem104 47220

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