fourierdlem92 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem92 46154

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 13072	. . 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 715	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑇) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
16		fveq2 6907	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖))
17	16	oveq1d 7446	. . . 4 ⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
18	17	cbvmptv 5261	. . 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 715	. . 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 715	. . 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 715	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
27		eqeq1 2739	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑄‘𝑖) ↔ 𝑥 = (𝑄‘𝑖)))
28		eqeq1 2739	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑄‘(𝑖 + 1)) ↔ 𝑥 = (𝑄‘(𝑖 + 1))))
29		fveq2 6907	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
30	28, 29	ifbieq2d 4557	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦)) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
31	27, 30	ifbieq2d 4557	. . . 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 5261	. . 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 2735	. . 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 46143	. 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 7447	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 𝑇) = (𝐴 + 0))
37	1	recnd 11287	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
38	37	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝐴 ∈ ℂ)
39	38	addridd 11459	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 0) = 𝐴)
40	36, 39	eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 𝑇) = 𝐴)
41	35	oveq2d 7447	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 𝑇) = (𝐵 + 0))
42	3	recnd 11287	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	42	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝐵 ∈ ℂ)
44	43	addridd 11459	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 0) = 𝐵)
45	41, 44	eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 𝑇) = 𝐵)
46	40, 45	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 = 0) → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = (𝐴[,]𝐵))
47	46	itgeq1d 45913	. . . 4 ⊢ ((𝜑 ∧ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
48	47	adantlr 715	. . 3 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
49		simpll 767	. . . 4 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 𝜑)
50		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ¬ 𝑇 = 0)
51		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ¬ 0 < 𝑇)
52		ioran 985	. . . . . . 7 ⊢ (¬ (𝑇 = 0 ∨ 0 < 𝑇) ↔ (¬ 𝑇 = 0 ∧ ¬ 0 < 𝑇))
53	50, 51, 52	sylanbrc 583	. . . . . 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 11262	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 0 ∈ ℝ)
56	54, 55	lttrid 11397	. . . . . 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 11830	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → (𝑇 < 0 ↔ 0 < -𝑇))
59	57, 58	mpbid 232	. . . 4 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 0 < -𝑇)
60	1, 8	readdcld 11288	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
61	60	recnd 11287	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℂ)
62	8	recnd 11287	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
63	61, 62	negsubd 11624	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝑇) + -𝑇) = ((𝐴 + 𝑇) − 𝑇))
64	37, 62	pncand 11619	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
65	63, 64	eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝑇) + -𝑇) = 𝐴)
66	3, 8	readdcld 11288	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
67	66	recnd 11287	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℂ)
68	67, 62	negsubd 11624	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝑇) + -𝑇) = ((𝐵 + 𝑇) − 𝑇))
69	42, 62	pncand 11619	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
70	68, 69	eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 + 𝑇) + -𝑇) = 𝐵)
71	65, 70	oveq12d 7449	. . . . . . . 8 ⊢ (𝜑 → (((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇)) = (𝐴[,]𝐵))
72	71	eqcomd 2741	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) = (((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇)))
73	72	itgeq1d 45913	. . . . . 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 11288	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → (𝐴 + 𝑇) ∈ ℝ)
78	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝐵 ∈ ℝ)
79	78, 76	readdcld 11288	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → (𝐵 + 𝑇) ∈ ℝ)
80		eqid 2735	. . . . . 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 11688	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → -𝑇 ∈ ℝ)
83		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 0 < -𝑇)
84	82, 83	elrpd 13072	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → -𝑇 ∈ ℝ⁺)
85	5	fourierdlem2 46065	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 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 8888	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
90	88, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
91	90	ffvelcdmda 7104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
92	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑇 ∈ ℝ)
93	91, 92	readdcld 11288	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
94		fourierdlem92.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇))
95	93, 94	fmptd 7134	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:(0...𝑀)⟶ℝ)
96		reex 11244	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
97	96	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ V)
98		ovex 7464	. . . . . . . . . . . . 13 ⊢ (0...𝑀) ∈ V
99	98	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ V)
100	97, 99	elmapd 8879	. . . . . . . . . . 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 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
104	103	oveq1d 7446	. . . . . . . . . . . . . 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 12623	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℤ)
107	6	nnzd 12638	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
108		0le0 12365	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 0
109	108	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 0)
110		nnnn0 12531	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
111	110	nn0ge0d 12588	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ≤ 𝑀)
112	6, 111	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 𝑀)
113	106, 107, 106, 109, 112	elfzd 13552	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
114	90, 113	ffvelcdmd 7105	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
115	114, 8	readdcld 11288	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) ∈ ℝ)
116	102, 105, 113, 115	fvmptd 7023	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘0) = ((𝑄‘0) + 𝑇))
117		simprll 779	. . . . . . . . . . . . . 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 7446	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) = (𝐴 + 𝑇))
120	116, 119	eqtrd 2775	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘0) = (𝐴 + 𝑇))
121		fveq2 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
122	121	oveq1d 7446	. . . . . . . . . . . . . 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 12585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
125		nn0uz 12918	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
126	124, 125	eleqtrdi 2849	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
127		eluzfz2 13569	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
128	126, 127	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
129	90, 128	ffvelcdmd 7105	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
130	129, 8	readdcld 11288	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) ∈ ℝ)
131	102, 123, 128, 130	fvmptd 7023	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝑀) = ((𝑄‘𝑀) + 𝑇))
132		simprlr 780	. . . . . . . . . . . . . 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 7446	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) = (𝐵 + 𝑇))
135	131, 134	eqtrd 2775	. . . . . . . . . . 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 13712	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
139	138	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
140	137, 139	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
141		fzofzp1 13800	. . . . . . . . . . . . . . 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 7105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
144	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ)
145	87	simprrd 774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
146	145	r19.21bi 3249	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
147	140, 143, 144, 146	ltadd1dd 11872	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
148	140, 144	readdcld 11288	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
149	94	fvmpt2 7027	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑄‘𝑖) + 𝑇) ∈ ℝ) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
150	139, 148, 149	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
151	94, 18	eqtr4i 2766	. . . . . . . . . . . . . 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 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑖 + 1) → (𝑄‘𝑗) = (𝑄‘(𝑖 + 1)))
154	153	oveq1d 7446	. . . . . . . . . . . . . 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 11288	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
157	152, 155, 142, 156	fvmptd 7023	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) + 𝑇))
158	147, 150, 157	3brtr4d 5180	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
159	158	ralrimiva 3144	. . . . . . . . . 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 46065	. . . . . . . . . 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 6908	. . . . . . . 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 2849	. . . . . . 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 45458	. . . . . . . . . . . 12 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
172	168, 169, 170, 171	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
173	172	recnd 11287	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℂ)
174	62	negcld 11605	. . . . . . . . . . 11 ⊢ (𝜑 → -𝑇 ∈ ℂ)
175	174	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → -𝑇 ∈ ℂ)
176	173, 175	addcld 11278	. . . . . . . . 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 11688	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝑇 ∈ ℝ)
181	180	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → -𝑇 ∈ ℝ)
182	172, 181	readdcld 11288	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ∈ ℝ)
183	63, 64	eqtr2d 2776	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) + -𝑇))
184	183	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) + -𝑇))
185	168	rexrd 11309	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ^*)
186	169	rexrd 11309	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ^*)
187		iccgelb 13440	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝑇) ∈ ℝ^* ∧ (𝐵 + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
188	185, 186, 170, 187	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
189	168, 172, 181, 188	leadd1dd 11875	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) + -𝑇) ≤ (𝑥 + -𝑇))
190	184, 189	eqbrtrd 5170	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ≤ (𝑥 + -𝑇))
191		iccleub 13439	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝑇) ∈ ℝ^* ∧ (𝐵 + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
192	185, 186, 170, 191	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
193	172, 169, 181, 192	leadd1dd 11875	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ≤ ((𝐵 + 𝑇) + -𝑇))
194	169	recnd 11287	. . . . . . . . . . . . . 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 11624	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) + -𝑇) = 𝐵)
199	193, 198	breqtrd 5174	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ≤ 𝐵)
200	178, 179, 182, 190, 199	eliccd 45457	. . . . . . . . . 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 2827	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵)))
203	202	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 + -𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵))))
204		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝑦 + 𝑇) = ((𝑥 + -𝑇) + 𝑇))
205	204	fveq2d 6911	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 + -𝑇) + 𝑇)))
206		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 + -𝑇)))
207	205, 206	eqeq12d 2751	. . . . . . . . . . 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 2827	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
210	209	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵))))
211		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
212	211	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
213		fveq2 6907	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
214	212, 213	eqeq12d 2751	. . . . . . . . . . . 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 1996	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
217	208, 216	vtoclg 3554	. . . . . . . . 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 11624	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) = (𝑥 − 𝑇))
220	219	oveq1d 7446	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 + -𝑇) + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
221	173, 195	npcand 11622	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
222	220, 221	eqtrd 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 + -𝑇) + 𝑇) = 𝑥)
223	222	fveq2d 6911	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘𝑥))
224	218, 223	eqtr3d 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘(𝑥 + -𝑇)) = (𝐹‘𝑥))
225	224	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘(𝑥 + -𝑇)) = (𝐹‘𝑥))
226		fveq2 6907	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝑆‘𝑗) = (𝑆‘𝑖))
227	226	oveq1d 7446	. . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝑆‘𝑗) + -𝑇) = ((𝑆‘𝑖) + -𝑇))
228	227	cbvmptv 5261	. . . . . 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 13445	. . . . . . . . . . 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 6978	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
234	150, 157	oveq12d 7449	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) = (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)))
235	140, 143, 144	iooshift 45475	. . . . . . . . . . 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 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)})
237	236	mpteq1d 5243	. . . . . . . . 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 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝜑)
239		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝑖 ∈ (0..^𝑀))
240	235	eleq2d 2825	. . . . . . . . . . . . . 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 11309	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
243	242	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ∈ ℝ^*)
244	143	rexrd 11309	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
245	244	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
246		elioore 13414	. . . . . . . . . . . . . . . . 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 11689	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
250	249	3adant2 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
251	140	recnd 11287	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
252	62	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℂ)
253	251, 252	pncand 11619	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇) − 𝑇) = (𝑄‘𝑖))
254	253	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
255	254	3adant3 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
256	148	3adant3 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
257	247	3adant2 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
258	8	3ad2ant1 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑇 ∈ ℝ)
259	148	rexrd 11309	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
260	259	3adant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
261	156	rexrd 11309	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
262	261	3adant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
263		simp3 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)))
264		ioogtlb 45448	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) < 𝑥)
266	256, 257, 258, 265	ltsub1dd 11873	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘𝑖) + 𝑇) − 𝑇) < (𝑥 − 𝑇))
267	255, 266	eqbrtrd 5170	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) < (𝑥 − 𝑇))
268	156	3adant3 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
269		iooltub 45463	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 < ((𝑄‘(𝑖 + 1)) + 𝑇))
271	257, 268, 258, 270	ltsub1dd 11873	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) < (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
272	143	recnd 11287	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
273	272, 252	pncand 11619	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
274	273	3adant3 1131	. . . . . . . . . . . . . . 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 5174	. . . . . . . . . . . . . 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 45451	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 6926	. . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝑥 − 𝑇) ∈ ℝ)
281	1	3ad2ant1 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 ∈ ℝ)
282	3	3ad2ant1 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐵 ∈ ℝ)
283	64	eqcomd 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
284	283	3ad2ant1 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
285	60	3ad2ant1 1132	. . . . . . . . . . . . . . . . . 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 11309	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
288	287	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
289	3	rexrd 11309	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
290	289	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
291	5, 6, 12	fourierdlem15 46078	. . . . . . . . . . . . . . . . . . . . . . . 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 7105	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
294		iccgelb 13440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑄‘𝑖) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝑖))
295	288, 290, 293, 294	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ≤ (𝑄‘𝑖))
296	286, 140, 144, 295	leadd1dd 11875	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐴 + 𝑇) ≤ ((𝑄‘𝑖) + 𝑇))
297	296	3adant3 1131	. . . . . . . . . . . . . . . . . . 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 11417	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐴 + 𝑇) < 𝑥)
299	285, 257, 258, 298	ltsub1dd 11873	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) < (𝑥 − 𝑇))
300	284, 299	eqbrtrd 5170	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 < (𝑥 − 𝑇))
301	281, 250, 300	ltled 11407	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 ≤ (𝑥 − 𝑇))
302	143	3adant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
303	292, 142	ffvelcdmd 7105	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (𝐴[,]𝐵))
304		iccleub 13439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ (𝐴[,]𝐵)) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
305	288, 290, 303, 304	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
306	305	3adant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
307	250, 302, 282, 275, 306	ltletrd 11419	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) < 𝐵)
308	250, 282, 307	ltled 11407	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
309	281, 282, 250, 301, 308	eliccd 45457	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
310	238, 239, 241, 309	syl3anc 1370	. . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
313	312	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))))
314		oveq1 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
315	314	fveq2d 6911	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
316		fveq2 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 − 𝑇)))
317	315, 316	eqeq12d 2751	. . . . . . . . . . . . . 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 3554	. . . . . . . . . . . 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 11243	. . . . . . . . . . . . . . 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 11622	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
326	325	fveq2d 6911	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘𝑥))
327	238, 321, 326	syl2anc 584	. . . . . . . . . . 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 2782	. . . . . . . . . 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 5248	. . . . . . . . 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 2779	. . . . . . . 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 13446	. . . . . . . . . . 11 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
332	331	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
333		eqeq1 2739	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
334	333	rexbidv 3177	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)))
335		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
336	335	eqeq2d 2746	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
337	336	cbvrexvw 3236	. . . . . . . . . . . 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 3444	. . . . . . . . . 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 2735	. . . . . . . . . 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 45830	. . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
343	342	oveq1d 7446	. . . . . . . . 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 2841	. . . . . . . 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 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
346	345	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
347		ffdm 6766	. . . . . . . . . . . 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 13445	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
352		fdm 6746	. . . . . . . . . . 11 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
353	230, 352	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = ℝ)
354	351, 353	sseqtrrid 4049	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
355	339	eqcomi 2744	. . . . . . . . 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 4043	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ⊆ dom 𝐹)
357	339, 356	eqsstrrid 4045	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)} ⊆ dom 𝐹)
358		simpll 767	. . . . . . . . . 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 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
363		ioossicc 13470	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
364	363	sseli 3991	. . . . . . . . . . . 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 46064	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ (𝐴[,]𝐵))
367		eleq1 2827	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑧 ∈ (𝐴[,]𝐵)))
368	367	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵))))
369		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 + 𝑇) = (𝑧 + 𝑇))
370	369	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑧 + 𝑇)))
371		fveq2 6907	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
372	370, 371	eqeq12d 2751	. . . . . . . . . . . 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 1996	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
375	358, 366, 374	syl2anc 584	. . . . . . . . 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 45584	. . . . . . . 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 2787	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
378	377	reseq2d 6000	. . . . . . . . 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 2741	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) = (𝑆‘𝑖))
380	378, 379	oveq12d 7449	. . . . . . . 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 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘𝑖)))
382	381	adantlr 715	. . . . . 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 45584	. . . . . . . 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 2741	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) = (𝑆‘(𝑖 + 1)))
385	378, 384	oveq12d 7449	. . . . . . . 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 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘(𝑖 + 1))))
387	386	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘(𝑖 + 1))))
388		eqeq1 2739	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑆‘𝑖) ↔ 𝑥 = (𝑆‘𝑖)))
389		eqeq1 2739	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑆‘(𝑖 + 1)) ↔ 𝑥 = (𝑆‘(𝑖 + 1))))
390	389, 29	ifbieq2d 4557	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → if(𝑦 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦)) = if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
391	388, 390	ifbieq2d 4557	. . . . . . 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 5261	. . . . . 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 2735	. . . . . 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 46143	. . . . 5 ⊢ ((𝜑 ∧ 0 < -𝑇) → ∫(((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥)
395	74, 394	eqtr2d 2776	. . . 4 ⊢ ((𝜑 ∧ 0 < -𝑇) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
396	49, 59, 395	syl2anc 584	. . 3 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
397	48, 396	pm2.61dan 813	. 2 ⊢ ((𝜑 ∧ ¬ 0 < 𝑇) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
398	34, 397	pm2.61dan 813	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {crab 3433 Vcvv 3478 ⊆ wss 3963 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 ℕcn 12264 ℕ₀cn0 12524 ℤ_≥cuz 12876 (,)cioo 13384 [,]cicc 13387 ...cfz 13544 ..^cfzo 13691 –cn→ccncf 24916 ∫citg 25667 lim_ℂ climc 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 df-ditg 25897 df-limc 25916 df-dv 25917

This theorem is referenced by: fourierdlem107 46169

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