fourierdlem92 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem92 45399

Description: The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by its period 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem92.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem92.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem92.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem92.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem92.t	⊢ (𝜑 → 𝑇 ∈ ℝ)
fourierdlem92.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem92.fper	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem92.s	⊢ 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇))
fourierdlem92.h	⊢ 𝐻 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝‘𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem92.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
fourierdlem92.cncf	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem92.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
fourierdlem92.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))

**Assertion**
Ref	Expression
fourierdlem92	⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Distinct variable groups: 𝐴,𝑖,𝑚,𝑝 𝑥,𝐴,𝑖 𝐵,𝑖,𝑚,𝑝 𝑥,𝐵 𝑖,𝐹,𝑥 𝑥,𝐿 𝑖,𝑀,𝑥 𝑚,𝑀,𝑝 𝑄,𝑖,𝑥 𝑄,𝑝 𝑥,𝑅 𝑆,𝑖,𝑥 𝑆,𝑝 𝑇,𝑖,𝑥 𝑇,𝑚,𝑝 𝜑,𝑖,𝑥 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝑃(𝑥,𝑖,𝑚,𝑝) 𝑄(𝑚) 𝑅(𝑖,𝑚,𝑝) 𝑆(𝑚) 𝐹(𝑚,𝑝) 𝐻(𝑥,𝑖,𝑚,𝑝) 𝐿(𝑖,𝑚,𝑝)

Proof of Theorem fourierdlem92 Dummy variables 𝑦 𝑤 𝑧 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem92.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝐴 ∈ ℝ)
3		fourierdlem92.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝐵 ∈ ℝ)
5		fourierdlem92.p	. . 3 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
6		fourierdlem92.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝑀 ∈ ℕ)
8		fourierdlem92.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝑇 ∈ ℝ)
10		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑇) → 0 < 𝑇)
11	9, 10	elrpd 13010	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝑇 ∈ ℝ⁺)
12		fourierdlem92.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝑄 ∈ (𝑃‘𝑀))
14		fourierdlem92.fper	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
15	14	adantlr 712	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑇) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
16		fveq2 6881	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖))
17	16	oveq1d 7416	. . . 4 ⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
18	17	cbvmptv 5251	. . 3 ⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇))
19		fourierdlem92.f	. . . 4 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
20	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝐹:ℝ⟶ℂ)
21		fourierdlem92.cncf	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
22	21	adantlr 712	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
23		fourierdlem92.r	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
24	23	adantlr 712	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
25		fourierdlem92.l	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
26	25	adantlr 712	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
27		eqeq1 2728	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑄‘𝑖) ↔ 𝑥 = (𝑄‘𝑖)))
28		eqeq1 2728	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑄‘(𝑖 + 1)) ↔ 𝑥 = (𝑄‘(𝑖 + 1))))
29		fveq2 6881	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
30	28, 29	ifbieq2d 4546	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦)) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
31	27, 30	ifbieq2d 4546	. . . 4 ⊢ (𝑦 = 𝑥 → if(𝑦 = (𝑄‘𝑖), 𝑅, if(𝑦 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦))) = if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))))
32	31	cbvmptv 5251	. . 3 ⊢ (𝑦 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑦 = (𝑄‘𝑖), 𝑅, if(𝑦 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦)))) = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))))
33		eqid 2724	. . 3 ⊢ (𝑥 ∈ (((𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))‘𝑖)[,]((𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))‘(𝑖 + 1))) ↦ ((𝑦 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑦 = (𝑄‘𝑖), 𝑅, if(𝑦 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦))))‘(𝑥 − 𝑇))) = (𝑥 ∈ (((𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))‘𝑖)[,]((𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))‘(𝑖 + 1))) ↦ ((𝑦 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑦 = (𝑄‘𝑖), 𝑅, if(𝑦 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦))))‘(𝑥 − 𝑇)))
34	2, 4, 5, 7, 11, 13, 15, 18, 20, 22, 24, 26, 32, 33	fourierdlem81 45388	. 2 ⊢ ((𝜑 ∧ 0 < 𝑇) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
35		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝑇 = 0)
36	35	oveq2d 7417	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 𝑇) = (𝐴 + 0))
37	1	recnd 11239	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
38	37	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝐴 ∈ ℂ)
39	38	addridd 11411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 0) = 𝐴)
40	36, 39	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 𝑇) = 𝐴)
41	35	oveq2d 7417	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 𝑇) = (𝐵 + 0))
42	3	recnd 11239	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	42	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝐵 ∈ ℂ)
44	43	addridd 11411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 0) = 𝐵)
45	41, 44	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 𝑇) = 𝐵)
46	40, 45	oveq12d 7419	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 = 0) → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = (𝐴[,]𝐵))
47	46	itgeq1d 45158	. . . 4 ⊢ ((𝜑 ∧ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
48	47	adantlr 712	. . 3 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
49		simpll 764	. . . 4 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 𝜑)
50		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ¬ 𝑇 = 0)
51		simplr 766	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ¬ 0 < 𝑇)
52		ioran 980	. . . . . . 7 ⊢ (¬ (𝑇 = 0 ∨ 0 < 𝑇) ↔ (¬ 𝑇 = 0 ∧ ¬ 0 < 𝑇))
53	50, 51, 52	sylanbrc 582	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ¬ (𝑇 = 0 ∨ 0 < 𝑇))
54	49, 8	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 𝑇 ∈ ℝ)
55		0red 11214	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 0 ∈ ℝ)
56	54, 55	lttrid 11349	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → (𝑇 < 0 ↔ ¬ (𝑇 = 0 ∨ 0 < 𝑇)))
57	53, 56	mpbird 257	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 𝑇 < 0)
58	54	lt0neg1d 11780	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → (𝑇 < 0 ↔ 0 < -𝑇))
59	57, 58	mpbid 231	. . . 4 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 0 < -𝑇)
60	1, 8	readdcld 11240	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
61	60	recnd 11239	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℂ)
62	8	recnd 11239	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
63	61, 62	negsubd 11574	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝑇) + -𝑇) = ((𝐴 + 𝑇) − 𝑇))
64	37, 62	pncand 11569	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
65	63, 64	eqtrd 2764	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝑇) + -𝑇) = 𝐴)
66	3, 8	readdcld 11240	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
67	66	recnd 11239	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℂ)
68	67, 62	negsubd 11574	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝑇) + -𝑇) = ((𝐵 + 𝑇) − 𝑇))
69	42, 62	pncand 11569	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
70	68, 69	eqtrd 2764	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 + 𝑇) + -𝑇) = 𝐵)
71	65, 70	oveq12d 7419	. . . . . . . 8 ⊢ (𝜑 → (((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇)) = (𝐴[,]𝐵))
72	71	eqcomd 2730	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) = (((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇)))
73	72	itgeq1d 45158	. . . . . 6 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇))(𝐹‘𝑥) d𝑥)
74	73	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0 < -𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇))(𝐹‘𝑥) d𝑥)
75	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝐴 ∈ ℝ)
76	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝑇 ∈ ℝ)
77	75, 76	readdcld 11240	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → (𝐴 + 𝑇) ∈ ℝ)
78	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝐵 ∈ ℝ)
79	78, 76	readdcld 11240	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → (𝐵 + 𝑇) ∈ ℝ)
80		eqid 2724	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝‘𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝‘𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
81	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝑀 ∈ ℕ)
82	76	renegcld 11638	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → -𝑇 ∈ ℝ)
83		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 0 < -𝑇)
84	82, 83	elrpd 13010	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → -𝑇 ∈ ℝ⁺)
85	5	fourierdlem2 45310	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
86	6, 85	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
87	12, 86	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
88	87	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
89		elmapi 8839	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
90	88, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
91	90	ffvelcdmda 7076	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
92	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑇 ∈ ℝ)
93	91, 92	readdcld 11240	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
94		fourierdlem92.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇))
95	93, 94	fmptd 7105	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:(0...𝑀)⟶ℝ)
96		reex 11197	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
97	96	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ V)
98		ovex 7434	. . . . . . . . . . . . 13 ⊢ (0...𝑀) ∈ V
99	98	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ V)
100	97, 99	elmapd 8830	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑆:(0...𝑀)⟶ℝ))
101	95, 100	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_m (0...𝑀)))
102	94	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇)))
103		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
104	103	oveq1d 7416	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
105	104	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
106		0zd 12567	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℤ)
107	6	nnzd 12582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
108		0le0 12310	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 0
109	108	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 0)
110		nnnn0 12476	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
111	110	nn0ge0d 12532	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ≤ 𝑀)
112	6, 111	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 𝑀)
113	106, 107, 106, 109, 112	elfzd 13489	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
114	90, 113	ffvelcdmd 7077	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
115	114, 8	readdcld 11240	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) ∈ ℝ)
116	102, 105, 113, 115	fvmptd 6995	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘0) = ((𝑄‘0) + 𝑇))
117		simprll 776	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))) → (𝑄‘0) = 𝐴)
118	87, 117	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
119	118	oveq1d 7416	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) = (𝐴 + 𝑇))
120	116, 119	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘0) = (𝐴 + 𝑇))
121		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
122	121	oveq1d 7416	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑀 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
123	122	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
124	6	nnnn0d 12529	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
125		nn0uz 12861	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
126	124, 125	eleqtrdi 2835	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
127		eluzfz2 13506	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
128	126, 127	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
129	90, 128	ffvelcdmd 7077	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
130	129, 8	readdcld 11240	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) ∈ ℝ)
131	102, 123, 128, 130	fvmptd 6995	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝑀) = ((𝑄‘𝑀) + 𝑇))
132		simprlr 777	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))) → (𝑄‘𝑀) = 𝐵)
133	87, 132	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
134	133	oveq1d 7416	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) = (𝐵 + 𝑇))
135	131, 134	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝑀) = (𝐵 + 𝑇))
136	120, 135	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘0) = (𝐴 + 𝑇) ∧ (𝑆‘𝑀) = (𝐵 + 𝑇)))
137	90	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
138		elfzofz 13645	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
139	138	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
140	137, 139	ffvelcdmd 7077	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
141		fzofzp1 13726	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
142	141	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
143	137, 142	ffvelcdmd 7077	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
144	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ)
145	87	simprrd 771	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
146	145	r19.21bi 3240	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
147	140, 143, 144, 146	ltadd1dd 11822	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
148	140, 144	readdcld 11240	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
149	94	fvmpt2 6999	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑄‘𝑖) + 𝑇) ∈ ℝ) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
150	139, 148, 149	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
151	94, 18	eqtr4i 2755	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))
152	151	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇)))
153		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑖 + 1) → (𝑄‘𝑗) = (𝑄‘(𝑖 + 1)))
154	153	oveq1d 7416	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑖 + 1) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
155	154	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
156	143, 144	readdcld 11240	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
157	152, 155, 142, 156	fvmptd 6995	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) + 𝑇))
158	147, 150, 157	3brtr4d 5170	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
159	158	ralrimiva 3138	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
160	101, 136, 159	jca32 515	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑆‘0) = (𝐴 + 𝑇) ∧ (𝑆‘𝑀) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
161		fourierdlem92.h	. . . . . . . . . . 11 ⊢ 𝐻 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝‘𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
162	161	fourierdlem2 45310	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑆 ∈ (𝐻‘𝑀) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑆‘0) = (𝐴 + 𝑇) ∧ (𝑆‘𝑀) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
163	6, 162	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (𝐻‘𝑀) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑆‘0) = (𝐴 + 𝑇) ∧ (𝑆‘𝑀) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
164	160, 163	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (𝐻‘𝑀))
165	161	fveq1i 6882	. . . . . . . 8 ⊢ (𝐻‘𝑀) = ((𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝‘𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑀)
166	164, 165	eleqtrdi 2835	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ((𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝‘𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑀))
167	166	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝑆 ∈ ((𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝‘𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})‘𝑀))
168	60	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
169	66	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ)
170		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
171		eliccre 44703	. . . . . . . . . . . 12 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
172	168, 169, 170, 171	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
173	172	recnd 11239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℂ)
174	62	negcld 11555	. . . . . . . . . . 11 ⊢ (𝜑 → -𝑇 ∈ ℂ)
175	174	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → -𝑇 ∈ ℂ)
176	173, 175	addcld 11230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ∈ ℂ)
177		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝜑)
178	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ∈ ℝ)
179	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐵 ∈ ℝ)
180	8	renegcld 11638	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝑇 ∈ ℝ)
181	180	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → -𝑇 ∈ ℝ)
182	172, 181	readdcld 11240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ∈ ℝ)
183	63, 64	eqtr2d 2765	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) + -𝑇))
184	183	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) + -𝑇))
185	168	rexrd 11261	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ^*)
186	169	rexrd 11261	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ^*)
187		iccgelb 13377	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝑇) ∈ ℝ^* ∧ (𝐵 + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
188	185, 186, 170, 187	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
189	168, 172, 181, 188	leadd1dd 11825	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) + -𝑇) ≤ (𝑥 + -𝑇))
190	184, 189	eqbrtrd 5160	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ≤ (𝑥 + -𝑇))
191		iccleub 13376	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝑇) ∈ ℝ^* ∧ (𝐵 + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
192	185, 186, 170, 191	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
193	172, 169, 181, 192	leadd1dd 11825	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ≤ ((𝐵 + 𝑇) + -𝑇))
194	169	recnd 11239	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℂ)
195	62	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℂ)
196	194, 195	negsubd 11574	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) + -𝑇) = ((𝐵 + 𝑇) − 𝑇))
197	69	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
198	196, 197	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) + -𝑇) = 𝐵)
199	193, 198	breqtrd 5164	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ≤ 𝐵)
200	178, 179, 182, 190, 199	eliccd 44702	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ∈ (𝐴[,]𝐵))
201	177, 200	jca 511	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝜑 ∧ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵)))
202		eleq1 2813	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵)))
203	202	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 + -𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵))))
204		oveq1 7408	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝑦 + 𝑇) = ((𝑥 + -𝑇) + 𝑇))
205	204	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 + -𝑇) + 𝑇)))
206		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 + -𝑇)))
207	205, 206	eqeq12d 2740	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 + -𝑇) → ((𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦) ↔ (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘(𝑥 + -𝑇))))
208	203, 207	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 + -𝑇) → (((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)) ↔ ((𝜑 ∧ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘(𝑥 + -𝑇)))))
209		eleq1 2813	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
210	209	anbi2d 628	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵))))
211		oveq1 7408	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
212	211	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
213		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
214	212, 213	eqeq12d 2740	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)))
215	210, 214	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))))
216	215, 14	chvarvv 1994	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
217	208, 216	vtoclg 3535	. . . . . . . . 9 ⊢ ((𝑥 + -𝑇) ∈ ℂ → ((𝜑 ∧ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘(𝑥 + -𝑇))))
218	176, 201, 217	sylc 65	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘(𝑥 + -𝑇)))
219	173, 195	negsubd 11574	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) = (𝑥 − 𝑇))
220	219	oveq1d 7416	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 + -𝑇) + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
221	173, 195	npcand 11572	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
222	220, 221	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 + -𝑇) + 𝑇) = 𝑥)
223	222	fveq2d 6885	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘𝑥))
224	218, 223	eqtr3d 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘(𝑥 + -𝑇)) = (𝐹‘𝑥))
225	224	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘(𝑥 + -𝑇)) = (𝐹‘𝑥))
226		fveq2 6881	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝑆‘𝑗) = (𝑆‘𝑖))
227	226	oveq1d 7416	. . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝑆‘𝑗) + -𝑇) = ((𝑆‘𝑖) + -𝑇))
228	227	cbvmptv 5251	. . . . . 6 ⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑆‘𝑗) + -𝑇)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑆‘𝑖) + -𝑇))
229	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝐹:ℝ⟶ℂ)
230	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
231		ioossre 13382	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ℝ
232	231	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ℝ)
233	230, 232	feqresmpt 6951	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
234	150, 157	oveq12d 7419	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) = (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)))
235	140, 143, 144	iooshift 44720	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)})
236	234, 235	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)})
237	236	mpteq1d 5233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘𝑥)))
238		simpll 764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝜑)
239		simplr 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝑖 ∈ (0..^𝑀))
240	235	eleq2d 2811	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)) ↔ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}))
241	240	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)))
242	140	rexrd 11261	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
243	242	3adant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ∈ ℝ^*)
244	143	rexrd 11261	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
245	244	3adant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
246		elioore 13351	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)) → 𝑥 ∈ ℝ)
247	246	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
248	8	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑇 ∈ ℝ)
249	247, 248	resubcld 11639	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
250	249	3adant2 1128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
251	140	recnd 11239	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
252	62	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℂ)
253	251, 252	pncand 11569	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇) − 𝑇) = (𝑄‘𝑖))
254	253	eqcomd 2730	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
255	254	3adant3 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
256	148	3adant3 1129	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
257	247	3adant2 1128	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
258	8	3ad2ant1 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑇 ∈ ℝ)
259	148	rexrd 11261	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
260	259	3adant3 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
261	156	rexrd 11261	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
262	261	3adant3 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
263		simp3 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)))
264		ioogtlb 44693	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑄‘𝑖) + 𝑇) ∈ ℝ^* ∧ ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) < 𝑥)
265	260, 262, 263, 264	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) < 𝑥)
266	256, 257, 258, 265	ltsub1dd 11823	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘𝑖) + 𝑇) − 𝑇) < (𝑥 − 𝑇))
267	255, 266	eqbrtrd 5160	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) < (𝑥 − 𝑇))
268	156	3adant3 1129	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
269		iooltub 44708	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑄‘𝑖) + 𝑇) ∈ ℝ^* ∧ ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 < ((𝑄‘(𝑖 + 1)) + 𝑇))
270	260, 262, 263, 269	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 < ((𝑄‘(𝑖 + 1)) + 𝑇))
271	257, 268, 258, 270	ltsub1dd 11823	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) < (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
272	143	recnd 11239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
273	272, 252	pncand 11569	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
274	273	3adant3 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
275	271, 274	breqtrd 5164	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
276	243, 245, 250, 267, 275	eliood 44696	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
277	238, 239, 241, 276	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
278		fvres 6900	. . . . . . . . . . . 12 ⊢ ((𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
279	277, 278	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
280	238, 241, 249	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝑥 − 𝑇) ∈ ℝ)
281	1	3ad2ant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 ∈ ℝ)
282	3	3ad2ant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐵 ∈ ℝ)
283	64	eqcomd 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
284	283	3ad2ant1 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
285	60	3ad2ant1 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
286	1	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ)
287	1	rexrd 11261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
288	287	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
289	3	rexrd 11261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
290	289	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
291	5, 6, 12	fourierdlem15 45323	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
292	291	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
293	292, 139	ffvelcdmd 7077	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
294		iccgelb 13377	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑄‘𝑖) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝑖))
295	288, 290, 293, 294	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ≤ (𝑄‘𝑖))
296	286, 140, 144, 295	leadd1dd 11825	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐴 + 𝑇) ≤ ((𝑄‘𝑖) + 𝑇))
297	296	3adant3 1129	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐴 + 𝑇) ≤ ((𝑄‘𝑖) + 𝑇))
298	285, 256, 257, 297, 265	lelttrd 11369	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐴 + 𝑇) < 𝑥)
299	285, 257, 258, 298	ltsub1dd 11823	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) < (𝑥 − 𝑇))
300	284, 299	eqbrtrd 5160	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 < (𝑥 − 𝑇))
301	281, 250, 300	ltled 11359	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 ≤ (𝑥 − 𝑇))
302	143	3adant3 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
303	292, 142	ffvelcdmd 7077	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (𝐴[,]𝐵))
304		iccleub 13376	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ (𝐴[,]𝐵)) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
305	288, 290, 303, 304	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
306	305	3adant3 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
307	250, 302, 282, 275, 306	ltletrd 11371	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) < 𝐵)
308	250, 282, 307	ltled 11359	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
309	281, 282, 250, 301, 308	eliccd 44702	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
310	238, 239, 241, 309	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
311	238, 310	jca 511	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
312		eleq1 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
313	312	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))))
314		oveq1 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
315	314	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
316		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 − 𝑇)))
317	315, 316	eqeq12d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦) ↔ (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
318	313, 317	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 − 𝑇) → (((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)) ↔ ((𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇)))))
319	318, 216	vtoclg 3535	. . . . . . . . . . . 12 ⊢ ((𝑥 − 𝑇) ∈ ℝ → ((𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
320	280, 311, 319	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
321	241, 246	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝑥 ∈ ℝ)
322		recn 11196	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
323	322	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
324	62	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℂ)
325	323, 324	npcand 11572	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
326	325	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘𝑥))
327	238, 321, 326	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘𝑥))
328	279, 320, 327	3eqtr2rd 2771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝐹‘𝑥) = ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
329	328	mpteq2dva 5238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))))
330	233, 237, 329	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))))
331		ioosscn 13383	. . . . . . . . . . 11 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
332	331	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
333		eqeq1 2728	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
334	333	rexbidv 3170	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)))
335		oveq1 7408	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
336	335	eqeq2d 2735	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
337	336	cbvrexvw 3227	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇))
338	334, 337	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)))
339	338	cbvrabv 3434	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}
340		eqid 2724	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)))
341	332, 252, 339, 21, 340	cncfshift 45075	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))) ∈ ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}–cn→ℂ))
342	236	eqcomd 2730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
343	342	oveq1d 7416	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}–cn→ℂ) = (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
344	341, 343	eleqtrd 2827	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ↦ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇))) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
345	330, 344	eqeltrd 2825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
346	345	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
347		ffdm 6737	. . . . . . . . . . . 12 ⊢ (𝐹:ℝ⟶ℂ → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
348	19, 347	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
349	348	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
350	349	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:dom 𝐹⟶ℂ)
351		ioossre 13382	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
352		fdm 6716	. . . . . . . . . . 11 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
353	230, 352	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = ℝ)
354	351, 353	sseqtrrid 4027	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
355	339	eqcomi 2733	. . . . . . . . 9 ⊢ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)} = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}
356	232, 342, 353	3sstr4d 4021	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ⊆ dom 𝐹)
357	339, 356	eqsstrrid 4023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)} ⊆ dom 𝐹)
358		simpll 764	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝜑)
359	358, 287	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐴 ∈ ℝ^*)
360	358, 289	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐵 ∈ ℝ^*)
361	358, 291	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
362		simplr 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
363		ioossicc 13407	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
364	363	sseli 3970	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑧 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
365	364	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
366	359, 360, 361, 362, 365	fourierdlem1 45309	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ (𝐴[,]𝐵))
367		eleq1 2813	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑧 ∈ (𝐴[,]𝐵)))
368	367	anbi2d 628	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵))))
369		oveq1 7408	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 + 𝑇) = (𝑧 + 𝑇))
370	369	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑧 + 𝑇)))
371		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
372	370, 371	eqeq12d 2740	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧)))
373	368, 372	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))))
374	373, 14	chvarvv 1994	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
375	358, 366, 374	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
376	350, 332, 354, 252, 355, 357, 375, 23	limcperiod 44829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}) lim_ℂ ((𝑄‘𝑖) + 𝑇)))
377	355, 342	eqtrid 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
378	377	reseq2d 5971	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}) = (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))))
379	150	eqcomd 2730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) = (𝑆‘𝑖))
380	378, 379	oveq12d 7419	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}) lim_ℂ ((𝑄‘𝑖) + 𝑇)) = ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘𝑖)))
381	376, 380	eleqtrd 2827	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘𝑖)))
382	381	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘𝑖)))
383	350, 332, 354, 252, 355, 357, 375, 25	limcperiod 44829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}) lim_ℂ ((𝑄‘(𝑖 + 1)) + 𝑇)))
384	157	eqcomd 2730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) = (𝑆‘(𝑖 + 1)))
385	378, 384	oveq12d 7419	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}) lim_ℂ ((𝑄‘(𝑖 + 1)) + 𝑇)) = ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘(𝑖 + 1))))
386	383, 385	eleqtrd 2827	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘(𝑖 + 1))))
387	386	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘(𝑖 + 1))))
388		eqeq1 2728	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑆‘𝑖) ↔ 𝑥 = (𝑆‘𝑖)))
389		eqeq1 2728	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑆‘(𝑖 + 1)) ↔ 𝑥 = (𝑆‘(𝑖 + 1))))
390	389, 29	ifbieq2d 4546	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → if(𝑦 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦)) = if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
391	388, 390	ifbieq2d 4546	. . . . . . 7 ⊢ (𝑦 = 𝑥 → if(𝑦 = (𝑆‘𝑖), 𝑅, if(𝑦 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦))) = if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))))
392	391	cbvmptv 5251	. . . . . 6 ⊢ (𝑦 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑦 = (𝑆‘𝑖), 𝑅, if(𝑦 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦)))) = (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑥 = (𝑆‘𝑖), 𝑅, if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥))))
393		eqid 2724	. . . . . 6 ⊢ (𝑥 ∈ (((𝑗 ∈ (0...𝑀) ↦ ((𝑆‘𝑗) + -𝑇))‘𝑖)[,]((𝑗 ∈ (0...𝑀) ↦ ((𝑆‘𝑗) + -𝑇))‘(𝑖 + 1))) ↦ ((𝑦 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑦 = (𝑆‘𝑖), 𝑅, if(𝑦 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦))))‘(𝑥 − -𝑇))) = (𝑥 ∈ (((𝑗 ∈ (0...𝑀) ↦ ((𝑆‘𝑗) + -𝑇))‘𝑖)[,]((𝑗 ∈ (0...𝑀) ↦ ((𝑆‘𝑗) + -𝑇))‘(𝑖 + 1))) ↦ ((𝑦 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ if(𝑦 = (𝑆‘𝑖), 𝑅, if(𝑦 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦))))‘(𝑥 − -𝑇)))
394	77, 79, 80, 81, 84, 167, 225, 228, 229, 346, 382, 387, 392, 393	fourierdlem81 45388	. . . . 5 ⊢ ((𝜑 ∧ 0 < -𝑇) → ∫(((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥)
395	74, 394	eqtr2d 2765	. . . 4 ⊢ ((𝜑 ∧ 0 < -𝑇) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
396	49, 59, 395	syl2anc 583	. . 3 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
397	48, 396	pm2.61dan 810	. 2 ⊢ ((𝜑 ∧ ¬ 0 < 𝑇) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
398	34, 397	pm2.61dan 810	1 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 {crab 3424 Vcvv 3466 ⊆ wss 3940 ifcif 4520 class class class wbr 5138 ↦ cmpt 5221 dom cdm 5666 ↾ cres 5668 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ↑_m cmap 8816 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 − cmin 11441 -cneg 11442 ℕcn 12209 ℕ₀cn0 12469 ℤ_≥cuz 12819 (,)cioo 13321 [,]cicc 13324 ...cfz 13481 ..^cfzo 13624 –cn→ccncf 24718 ∫citg 25469 lim_ℂ climc 25713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-symdif 4234 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-lp 22962 df-perf 22963 df-cn 23053 df-cnp 23054 df-haus 23141 df-cmp 23213 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-xms 24148 df-ms 24149 df-tms 24150 df-cncf 24720 df-ovol 25315 df-vol 25316 df-mbf 25470 df-itg1 25471 df-itg2 25472 df-ibl 25473 df-itg 25474 df-0p 25521 df-ditg 25698 df-limc 25717 df-dv 25718

This theorem is referenced by: fourierdlem107 45414

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