fourierdlem92 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem92		Structured version Visualization version GIF version

Theorem fourierdlem92 41915

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 473	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝐴 ∈ ℝ)
3		fourierdlem92.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	adantr 473	. . 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 473	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝑀 ∈ ℕ)
8		fourierdlem92.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
9	8	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝑇 ∈ ℝ)
10		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑇) → 0 < 𝑇)
11	9, 10	elrpd 12248	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝑇 ∈ ℝ⁺)
12		fourierdlem92.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
13	12	adantr 473	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑇) → 𝑄 ∈ (𝑃‘𝑀))
14		fourierdlem92.fper	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
15	14	adantlr 702	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑇) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
16		fveq2 6501	. . . . 5 ⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖))
17	16	oveq1d 6993	. . . 4 ⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘𝑖) + 𝑇))
18	17	cbvmptv 5029	. . 3 ⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇))
19		fourierdlem92.f	. . . 4 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
20	19	adantr 473	. . 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 702	. . 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 702	. . 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 702	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
27		eqeq1 2782	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑄‘𝑖) ↔ 𝑥 = (𝑄‘𝑖)))
28		eqeq1 2782	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑄‘(𝑖 + 1)) ↔ 𝑥 = (𝑄‘(𝑖 + 1))))
29		fveq2 6501	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
30	28, 29	ifbieq2d 4376	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦)) = if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
31	27, 30	ifbieq2d 4376	. . . 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 5029	. . 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 2778	. . 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 41904	. 2 ⊢ ((𝜑 ∧ 0 < 𝑇) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
35		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝑇 = 0)
36	35	oveq2d 6994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 𝑇) = (𝐴 + 0))
37	1	recnd 10470	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
38	37	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝐴 ∈ ℂ)
39	38	addid1d 10642	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 0) = 𝐴)
40	36, 39	eqtrd 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐴 + 𝑇) = 𝐴)
41	35	oveq2d 6994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 𝑇) = (𝐵 + 0))
42	3	recnd 10470	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	42	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝐵 ∈ ℂ)
44	43	addid1d 10642	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 0) = 𝐵)
45	41, 44	eqtrd 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 = 0) → (𝐵 + 𝑇) = 𝐵)
46	40, 45	oveq12d 6996	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 = 0) → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = (𝐴[,]𝐵))
47	46	itgeq1d 41673	. . . 4 ⊢ ((𝜑 ∧ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
48	47	adantlr 702	. . 3 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
49		simpll 754	. . . 4 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 𝜑)
50		simpr 477	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ¬ 𝑇 = 0)
51		simplr 756	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ¬ 0 < 𝑇)
52		ioran 966	. . . . . . 7 ⊢ (¬ (𝑇 = 0 ∨ 0 < 𝑇) ↔ (¬ 𝑇 = 0 ∧ ¬ 0 < 𝑇))
53	50, 51, 52	sylanbrc 575	. . . . . 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 10445	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 0 ∈ ℝ)
56	54, 55	lttrid 10580	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → (𝑇 < 0 ↔ ¬ (𝑇 = 0 ∨ 0 < 𝑇)))
57	53, 56	mpbird 249	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 𝑇 < 0)
58	54	lt0neg1d 11012	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → (𝑇 < 0 ↔ 0 < -𝑇))
59	57, 58	mpbid 224	. . . 4 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → 0 < -𝑇)
60	1, 8	readdcld 10471	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
61	60	recnd 10470	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℂ)
62	8	recnd 10470	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
63	61, 62	negsubd 10806	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝑇) + -𝑇) = ((𝐴 + 𝑇) − 𝑇))
64	37, 62	pncand 10801	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
65	63, 64	eqtrd 2814	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 + 𝑇) + -𝑇) = 𝐴)
66	3, 8	readdcld 10471	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
67	66	recnd 10470	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℂ)
68	67, 62	negsubd 10806	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝑇) + -𝑇) = ((𝐵 + 𝑇) − 𝑇))
69	42, 62	pncand 10801	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
70	68, 69	eqtrd 2814	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 + 𝑇) + -𝑇) = 𝐵)
71	65, 70	oveq12d 6996	. . . . . . . 8 ⊢ (𝜑 → (((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇)) = (𝐴[,]𝐵))
72	71	eqcomd 2784	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) = (((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇)))
73	72	itgeq1d 41673	. . . . . 6 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇))(𝐹‘𝑥) d𝑥)
74	73	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 0 < -𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇))(𝐹‘𝑥) d𝑥)
75	1	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝐴 ∈ ℝ)
76	8	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝑇 ∈ ℝ)
77	75, 76	readdcld 10471	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → (𝐴 + 𝑇) ∈ ℝ)
78	3	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝐵 ∈ ℝ)
79	78, 76	readdcld 10471	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → (𝐵 + 𝑇) ∈ ℝ)
80		eqid 2778	. . . . . 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 473	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝑀 ∈ ℕ)
82	76	renegcld 10870	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → -𝑇 ∈ ℝ)
83		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < -𝑇) → 0 < -𝑇)
84	82, 83	elrpd 12248	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → -𝑇 ∈ ℝ⁺)
85	5	fourierdlem2 41826	. . . . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
88	87	simpld 487	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)))
89		elmapi 8230	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
90	88, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
91	90	ffvelrnda 6678	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
92	8	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑇 ∈ ℝ)
93	91, 92	readdcld 10471	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
94		fourierdlem92.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇))
95	93, 94	fmptd 6703	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆:(0...𝑀)⟶ℝ)
96		reex 10428	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
97	96	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ V)
98		ovex 7010	. . . . . . . . . . . . 13 ⊢ (0...𝑀) ∈ V
99	98	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ V)
100	97, 99	elmapd 8222	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_𝑚 (0...𝑀)) ↔ 𝑆:(0...𝑀)⟶ℝ))
101	95, 100	mpbird 249	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_𝑚 (0...𝑀)))
102	94	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇)))
103		fveq2 6501	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → (𝑄‘𝑖) = (𝑄‘0))
104	103	oveq1d 6993	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
105	104	adantl 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘0) + 𝑇))
106		0zd 11808	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℤ)
107	6	nnzd 11902	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℤ)
108	106, 107, 106	3jca 1108	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
109		0le0 11551	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ 0
110	109	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 0)
111		nnnn0 11718	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
112	111	nn0ge0d 11773	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 0 ≤ 𝑀)
113	6, 112	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝑀)
114	108, 110, 113	jca32 508	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
115		elfz2 12718	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
116	114, 115	sylibr 226	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
117	90, 116	ffvelrnd 6679	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
118	117, 8	readdcld 10471	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) ∈ ℝ)
119	102, 105, 116, 118	fvmptd 6603	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘0) = ((𝑄‘0) + 𝑇))
120		simprll 766	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))) → (𝑄‘0) = 𝐴)
121	87, 120	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
122	121	oveq1d 6993	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘0) + 𝑇) = (𝐴 + 𝑇))
123	119, 122	eqtrd 2814	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘0) = (𝐴 + 𝑇))
124		fveq2 6501	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑀 → (𝑄‘𝑖) = (𝑄‘𝑀))
125	124	oveq1d 6993	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑀 → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
126	125	adantl 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → ((𝑄‘𝑖) + 𝑇) = ((𝑄‘𝑀) + 𝑇))
127	6	nnnn0d 11770	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
128		nn0uz 12097	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
129	127, 128	syl6eleq 2876	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
130		eluzfz2 12734	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
131	129, 130	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
132	90, 131	ffvelrnd 6679	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
133	132, 8	readdcld 10471	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) ∈ ℝ)
134	102, 126, 131, 133	fvmptd 6603	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝑀) = ((𝑄‘𝑀) + 𝑇))
135		simprlr 767	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))) → (𝑄‘𝑀) = 𝐵)
136	87, 135	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
137	136	oveq1d 6993	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝑀) + 𝑇) = (𝐵 + 𝑇))
138	134, 137	eqtrd 2814	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝑀) = (𝐵 + 𝑇))
139	123, 138	jca 504	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘0) = (𝐴 + 𝑇) ∧ (𝑆‘𝑀) = (𝐵 + 𝑇)))
140	90	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
141		elfzofz 12872	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
142	141	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
143	140, 142	ffvelrnd 6679	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
144		fzofzp1 12952	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
145	144	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
146	140, 145	ffvelrnd 6679	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
147	8	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℝ)
148	87	simprrd 761	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
149	148	r19.21bi 3158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
150	143, 146, 147, 149	ltadd1dd 11054	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) < ((𝑄‘(𝑖 + 1)) + 𝑇))
151	143, 147	readdcld 10471	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
152	94	fvmpt2 6607	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑄‘𝑖) + 𝑇) ∈ ℝ) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
153	142, 151, 152	syl2anc 576	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) = ((𝑄‘𝑖) + 𝑇))
154	94, 18	eqtr4i 2805	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇))
155	154	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑆 = (𝑗 ∈ (0...𝑀) ↦ ((𝑄‘𝑗) + 𝑇)))
156		fveq2 6501	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑖 + 1) → (𝑄‘𝑗) = (𝑄‘(𝑖 + 1)))
157	156	oveq1d 6993	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑖 + 1) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
158	157	adantl 474	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑄‘𝑗) + 𝑇) = ((𝑄‘(𝑖 + 1)) + 𝑇))
159	146, 147	readdcld 10471	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
160	155, 158, 145, 159	fvmptd 6603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘(𝑖 + 1)) = ((𝑄‘(𝑖 + 1)) + 𝑇))
161	150, 153, 160	3brtr4d 4962	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
162	161	ralrimiva 3132	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
163	101, 139, 162	jca32 508	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑆‘0) = (𝐴 + 𝑇) ∧ (𝑆‘𝑀) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
164		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)))})
165	164	fourierdlem2 41826	. . . . . . . . . 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))))))
166	6, 165	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))))))
167	163, 166	mpbird 249	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (𝐻‘𝑀))
168	164	fveq1i 6502	. . . . . . . 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)
169	167, 168	syl6eleq 2876	. . . . . . 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))
170	169	adantr 473	. . . . . 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))
171	60	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
172	66	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ)
173		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
174		eliccre 41213	. . . . . . . . . . . 12 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
175	171, 172, 173, 174	syl3anc 1351	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
176	175	recnd 10470	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℂ)
177	62	negcld 10787	. . . . . . . . . . 11 ⊢ (𝜑 → -𝑇 ∈ ℂ)
178	177	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → -𝑇 ∈ ℂ)
179	176, 178	addcld 10461	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ∈ ℂ)
180		simpl 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝜑)
181	1	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ∈ ℝ)
182	3	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐵 ∈ ℝ)
183	8	renegcld 10870	. . . . . . . . . . . . 13 ⊢ (𝜑 → -𝑇 ∈ ℝ)
184	183	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → -𝑇 ∈ ℝ)
185	175, 184	readdcld 10471	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ∈ ℝ)
186	63, 64	eqtr2d 2815	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) + -𝑇))
187	186	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) + -𝑇))
188	171	rexrd 10492	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ^*)
189	172	rexrd 10492	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ^*)
190		iccgelb 12612	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝑇) ∈ ℝ^* ∧ (𝐵 + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
191	188, 189, 173, 190	syl3anc 1351	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
192	171, 175, 184, 191	leadd1dd 11057	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) + -𝑇) ≤ (𝑥 + -𝑇))
193	187, 192	eqbrtrd 4952	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ≤ (𝑥 + -𝑇))
194		iccleub 12611	. . . . . . . . . . . . . 14 ⊢ (((𝐴 + 𝑇) ∈ ℝ^* ∧ (𝐵 + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
195	188, 189, 173, 194	syl3anc 1351	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
196	175, 172, 184, 195	leadd1dd 11057	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ≤ ((𝐵 + 𝑇) + -𝑇))
197	172	recnd 10470	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℂ)
198	62	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℂ)
199	197, 198	negsubd 10806	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) + -𝑇) = ((𝐵 + 𝑇) − 𝑇))
200	69	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
201	199, 200	eqtrd 2814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) + -𝑇) = 𝐵)
202	196, 201	breqtrd 4956	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ≤ 𝐵)
203	181, 182, 185, 193, 202	eliccd 41211	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) ∈ (𝐴[,]𝐵))
204	180, 203	jca 504	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝜑 ∧ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵)))
205		eleq1 2853	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵)))
206	205	anbi2d 619	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 + -𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵))))
207		oveq1 6985	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝑦 + 𝑇) = ((𝑥 + -𝑇) + 𝑇))
208	207	fveq2d 6505	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 + -𝑇) + 𝑇)))
209		fveq2 6501	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + -𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 + -𝑇)))
210	208, 209	eqeq12d 2793	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 + -𝑇) → ((𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦) ↔ (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘(𝑥 + -𝑇))))
211	206, 210	imbi12d 337	. . . . . . . . . 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)))))
212		eleq1 2853	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
213	212	anbi2d 619	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵))))
214		oveq1 6985	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑇) = (𝑦 + 𝑇))
215	214	fveq2d 6505	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑦 + 𝑇)))
216		fveq2 6501	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
217	215, 216	eqeq12d 2793	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦)))
218	213, 217	imbi12d 337	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))))
219	218, 14	chvarv 2327	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))
220	211, 219	vtoclg 3486	. . . . . . . . 9 ⊢ ((𝑥 + -𝑇) ∈ ℂ → ((𝜑 ∧ (𝑥 + -𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘(𝑥 + -𝑇))))
221	179, 204, 220	sylc 65	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘(𝑥 + -𝑇)))
222	176, 198	negsubd 10806	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 + -𝑇) = (𝑥 − 𝑇))
223	222	oveq1d 6993	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 + -𝑇) + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
224	176, 198	npcand 10804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
225	223, 224	eqtrd 2814	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 + -𝑇) + 𝑇) = 𝑥)
226	225	fveq2d 6505	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘((𝑥 + -𝑇) + 𝑇)) = (𝐹‘𝑥))
227	221, 226	eqtr3d 2816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘(𝑥 + -𝑇)) = (𝐹‘𝑥))
228	227	adantlr 702	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘(𝑥 + -𝑇)) = (𝐹‘𝑥))
229		fveq2 6501	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝑆‘𝑗) = (𝑆‘𝑖))
230	229	oveq1d 6993	. . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝑆‘𝑗) + -𝑇) = ((𝑆‘𝑖) + -𝑇))
231	230	cbvmptv 5029	. . . . . 6 ⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑆‘𝑗) + -𝑇)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑆‘𝑖) + -𝑇))
232	19	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 0 < -𝑇) → 𝐹:ℝ⟶ℂ)
233	19	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:ℝ⟶ℂ)
234		ioossre 12617	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ℝ
235	234	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ⊆ ℝ)
236	233, 235	feqresmpt 6565	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) = (𝑥 ∈ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) ↦ (𝐹‘𝑥)))
237	153, 160	oveq12d 6996	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) = (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)))
238	143, 146, 147	iooshift 41230	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)})
239	237, 238	eqtrd 2814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)})
240	239	mpteq1d 5017	. . . . . . . . 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)))
241		simpll 754	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝜑)
242		simplr 756	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝑖 ∈ (0..^𝑀))
243	238	eleq2d 2851	. . . . . . . . . . . . . 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)}))
244	243	biimpar 470	. . . . . . . . . . . . 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)))
245	143	rexrd 10492	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ^*)
246	245	3adant3 1112	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) ∈ ℝ^*)
247	146	rexrd 10492	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
248	247	3adant3 1112	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ^*)
249		elioore 12587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)) → 𝑥 ∈ ℝ)
250	249	adantl 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
251	8	adantr 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑇 ∈ ℝ)
252	250, 251	resubcld 10871	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
253	252	3adant2 1111	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
254	143	recnd 10470	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℂ)
255	62	adantr 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑇 ∈ ℂ)
256	254, 255	pncand 10801	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘𝑖) + 𝑇) − 𝑇) = (𝑄‘𝑖))
257	256	eqcomd 2784	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
258	257	3adant3 1112	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) = (((𝑄‘𝑖) + 𝑇) − 𝑇))
259	151	3adant3 1112	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ)
260	250	3adant2 1111	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ ℝ)
261	8	3ad2ant1 1113	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑇 ∈ ℝ)
262	151	rexrd 10492	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
263	262	3adant3 1112	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) ∈ ℝ^*)
264	159	rexrd 10492	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
265	264	3adant3 1112	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^*)
266		simp3 1118	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇)))
267		ioogtlb 41202	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑄‘𝑖) + 𝑇) ∈ ℝ^* ∧ ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) < 𝑥)
268	263, 265, 266, 267	syl3anc 1351	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘𝑖) + 𝑇) < 𝑥)
269	259, 260, 261, 268	ltsub1dd 11055	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘𝑖) + 𝑇) − 𝑇) < (𝑥 − 𝑇))
270	258, 269	eqbrtrd 4952	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘𝑖) < (𝑥 − 𝑇))
271	159	3adant3 1112	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ)
272		iooltub 41218	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑄‘𝑖) + 𝑇) ∈ ℝ^* ∧ ((𝑄‘(𝑖 + 1)) + 𝑇) ∈ ℝ^* ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 < ((𝑄‘(𝑖 + 1)) + 𝑇))
273	263, 265, 266, 272	syl3anc 1351	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝑥 < ((𝑄‘(𝑖 + 1)) + 𝑇))
274	260, 271, 261, 273	ltsub1dd 11055	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) < (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇))
275	146	recnd 10470	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℂ)
276	275, 255	pncand 10801	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
277	276	3adant3 1112	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (((𝑄‘(𝑖 + 1)) + 𝑇) − 𝑇) = (𝑄‘(𝑖 + 1)))
278	274, 277	breqtrd 4956	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) < (𝑄‘(𝑖 + 1)))
279	246, 248, 253, 270, 278	eliood 41205	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
280	241, 242, 244, 279	syl3anc 1351	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
281		fvres 6520	. . . . . . . . . . . 12 ⊢ ((𝑥 − 𝑇) ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘(𝑥 − 𝑇)) = (𝐹‘(𝑥 − 𝑇)))
282	280, 281	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)))
283	241, 244, 252	syl2anc 576	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝑥 − 𝑇) ∈ ℝ)
284	1	3ad2ant1 1113	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 ∈ ℝ)
285	3	3ad2ant1 1113	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐵 ∈ ℝ)
286	64	eqcomd 2784	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
287	286	3ad2ant1 1113	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
288	60	3ad2ant1 1113	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
289	1	adantr 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ)
290	1	rexrd 10492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
291	290	adantr 473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
292	3	rexrd 10492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
293	292	adantr 473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
294	5, 6, 12	fourierdlem15 41839	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
295	294	adantr 473	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
296	295, 142	ffvelrnd 6679	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
297		iccgelb 12612	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑄‘𝑖) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝑖))
298	291, 293, 296, 297	syl3anc 1351	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ≤ (𝑄‘𝑖))
299	289, 143, 147, 298	leadd1dd 11057	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐴 + 𝑇) ≤ ((𝑄‘𝑖) + 𝑇))
300	299	3adant3 1112	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐴 + 𝑇) ≤ ((𝑄‘𝑖) + 𝑇))
301	288, 259, 260, 300, 268	lelttrd 10600	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝐴 + 𝑇) < 𝑥)
302	288, 260, 261, 301	ltsub1dd 11055	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) < (𝑥 − 𝑇))
303	287, 302	eqbrtrd 4952	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 < (𝑥 − 𝑇))
304	284, 253, 303	ltled 10590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → 𝐴 ≤ (𝑥 − 𝑇))
305	146	3adant3 1112	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
306	295, 145	ffvelrnd 6679	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ (𝐴[,]𝐵))
307		iccleub 12611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑄‘(𝑖 + 1)) ∈ (𝐴[,]𝐵)) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
308	291, 293, 306, 307	syl3anc 1351	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
309	308	3adant3 1112	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑄‘(𝑖 + 1)) ≤ 𝐵)
310	253, 305, 285, 278, 309	ltletrd 10602	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) < 𝐵)
311	253, 285, 310	ltled 10590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
312	284, 285, 253, 304, 311	eliccd 41211	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ (((𝑄‘𝑖) + 𝑇)(,)((𝑄‘(𝑖 + 1)) + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
313	241, 242, 244, 312	syl3anc 1351	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
314	241, 313	jca 504	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
315		eleq1 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)))
316	315	anbi2d 619	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))))
317		oveq1 6985	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝑦 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
318	317	fveq2d 6505	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘((𝑥 − 𝑇) + 𝑇)))
319		fveq2 6501	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 − 𝑇) → (𝐹‘𝑦) = (𝐹‘(𝑥 − 𝑇)))
320	318, 319	eqeq12d 2793	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 − 𝑇) → ((𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦) ↔ (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
321	316, 320	imbi12d 337	. . . . . . . . . . . . 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)))))
322	321, 219	vtoclg 3486	. . . . . . . . . . . 12 ⊢ ((𝑥 − 𝑇) ∈ ℝ → ((𝜑 ∧ (𝑥 − 𝑇) ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘(𝑥 − 𝑇))))
323	283, 314, 322	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)))
324	244, 249	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → 𝑥 ∈ ℝ)
325		recn 10427	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
326	325	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
327	62	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑇 ∈ ℂ)
328	326, 327	npcand 10804	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
329	328	fveq2d 6505	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘𝑥))
330	241, 324, 329	syl2anc 576	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}) → (𝐹‘((𝑥 − 𝑇) + 𝑇)) = (𝐹‘𝑥))
331	282, 323, 330	3eqtr2rd 2821	. . . . . . . . . 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)))
332	331	mpteq2dva 5023	. . . . . . . . 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))))
333	236, 240, 332	3eqtrd 2818	. . . . . . . 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))))
334		ioosscn 41201	. . . . . . . . . . 11 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ
335	334	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℂ)
336		eqeq1 2782	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
337	336	rexbidv 3242	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇)))
338		oveq1 6985	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
339	338	eqeq2d 2788	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
340	339	cbvrexv 3384	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇))
341	337, 340	syl6bb 279	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)))
342	341	cbvrabv 3412	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}
343		eqid 2778	. . . . . . . . . 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)))
344	335, 255, 342, 21, 343	cncfshift 41588	. . . . . . . . 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))
345	239	eqcomd 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
346	345	oveq1d 6993	. . . . . . . . 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))
347	344, 346	eleqtrd 2868	. . . . . . . 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))
348	333, 347	eqeltrd 2866	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
349	348	adantlr 702	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) ∈ (((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))–cn→ℂ))
350		ffdm 6367	. . . . . . . . . . . 12 ⊢ (𝐹:ℝ⟶ℂ → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
351	19, 350	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
352	351	simpld 487	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
353	352	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐹:dom 𝐹⟶ℂ)
354		ioossre 12617	. . . . . . . . . 10 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ℝ
355		fdm 6354	. . . . . . . . . . 11 ⊢ (𝐹:ℝ⟶ℂ → dom 𝐹 = ℝ)
356	233, 355	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom 𝐹 = ℝ)
357	354, 356	syl5sseqr 3912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
358	342	eqcomi 2787	. . . . . . . . 9 ⊢ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)} = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)}
359	235, 345, 356	3sstr4d 3906	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑤 = (𝑧 + 𝑇)} ⊆ dom 𝐹)
360	342, 359	syl5eqssr 3908	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)} ⊆ dom 𝐹)
361		simpll 754	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝜑)
362	361, 290	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐴 ∈ ℝ^*)
363	361, 292	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝐵 ∈ ℝ^*)
364	361, 294	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
365		simplr 756	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑖 ∈ (0..^𝑀))
366		ioossicc 12641	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
367	366	sseli 3856	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → 𝑧 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
368	367	adantl 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))))
369	362, 363, 364, 365, 368	fourierdlem1 41825	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → 𝑧 ∈ (𝐴[,]𝐵))
370		eleq1 2853	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐴[,]𝐵) ↔ 𝑧 ∈ (𝐴[,]𝐵)))
371	370	anbi2d 619	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ↔ (𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵))))
372		oveq1 6985	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 + 𝑇) = (𝑧 + 𝑇))
373	372	fveq2d 6505	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘(𝑧 + 𝑇)))
374		fveq2 6501	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
375	373, 374	eqeq12d 2793	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥) ↔ (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧)))
376	371, 375	imbi12d 337	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))))
377	376, 14	chvarv 2327	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
378	361, 369, 377	syl2anc 576	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑧 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → (𝐹‘(𝑧 + 𝑇)) = (𝐹‘𝑧))
379	353, 335, 357, 255, 358, 360, 378, 23	limcperiod 41341	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}) lim_ℂ ((𝑄‘𝑖) + 𝑇)))
380	358, 345	syl5eq 2826	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)} = ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1))))
381	380	reseq2d 5696	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}) = (𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))))
382	153	eqcomd 2784	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖) + 𝑇) = (𝑆‘𝑖))
383	381, 382	oveq12d 6996	. . . . . . . 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)))
384	379, 383	eleqtrd 2868	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘𝑖)))
385	384	adantlr 702	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘𝑖)))
386	353, 335, 357, 255, 358, 360, 378, 25	limcperiod 41341	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))𝑥 = (𝑦 + 𝑇)}) lim_ℂ ((𝑄‘(𝑖 + 1)) + 𝑇)))
387	160	eqcomd 2784	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘(𝑖 + 1)) + 𝑇) = (𝑆‘(𝑖 + 1)))
388	381, 387	oveq12d 6996	. . . . . . . 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))))
389	386, 388	eleqtrd 2868	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘(𝑖 + 1))))
390	389	adantlr 702	. . . . . 6 ⊢ (((𝜑 ∧ 0 < -𝑇) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑆‘𝑖)(,)(𝑆‘(𝑖 + 1)))) lim_ℂ (𝑆‘(𝑖 + 1))))
391		eqeq1 2782	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑆‘𝑖) ↔ 𝑥 = (𝑆‘𝑖)))
392		eqeq1 2782	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 = (𝑆‘(𝑖 + 1)) ↔ 𝑥 = (𝑆‘(𝑖 + 1))))
393	392, 29	ifbieq2d 4376	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → if(𝑦 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑦)) = if(𝑥 = (𝑆‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))
394	391, 393	ifbieq2d 4376	. . . . . . 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))))
395	394	cbvmptv 5029	. . . . . 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))))
396		eqid 2778	. . . . . 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)))
397	77, 79, 80, 81, 84, 170, 228, 231, 232, 349, 385, 390, 395, 396	fourierdlem81 41904	. . . . 5 ⊢ ((𝜑 ∧ 0 < -𝑇) → ∫(((𝐴 + 𝑇) + -𝑇)[,]((𝐵 + 𝑇) + -𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥)
398	74, 397	eqtr2d 2815	. . . 4 ⊢ ((𝜑 ∧ 0 < -𝑇) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
399	49, 59, 398	syl2anc 576	. . 3 ⊢ (((𝜑 ∧ ¬ 0 < 𝑇) ∧ ¬ 𝑇 = 0) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
400	48, 399	pm2.61dan 800	. 2 ⊢ ((𝜑 ∧ ¬ 0 < 𝑇) → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
401	34, 400	pm2.61dan 800	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 198 ∧ wa 387 ∨ wo 833 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ∃wrex 3089 {crab 3092 Vcvv 3415 ⊆ wss 3831 ifcif 4351 class class class wbr 4930 ↦ cmpt 5009 dom cdm 5408 ↾ cres 5410 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ↑_𝑚 cmap 8208 ℂcc 10335 ℝcr 10336 0cc0 10337 1c1 10338 + caddc 10340 ℝ^*cxr 10475 < clt 10476 ≤ cle 10477 − cmin 10672 -cneg 10673 ℕcn 11441 ℕ₀cn0 11710 ℤcz 11796 ℤ_≥cuz 12061 (,)cioo 12557 [,]cicc 12560 ...cfz 12711 ..^cfzo 12852 –cn→ccncf 23190 ∫citg 23925 lim_ℂ climc 24166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cc 9657 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-symdif 4108 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-disj 4899 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-ofr 7230 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-omul 7912 df-er 8091 df-map 8210 df-pm 8211 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-dju 9126 df-card 9164 df-acn 9167 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ioc 12562 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-fl 12980 df-mod 13056 df-seq 13188 df-exp 13248 df-hash 13509 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-limsup 14692 df-clim 14709 df-rlim 14710 df-sum 14907 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-mulg 18015 df-cntz 18221 df-cmn 18671 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-lp 21451 df-perf 21452 df-cn 21542 df-cnp 21543 df-haus 21630 df-cmp 21702 df-tx 21877 df-hmeo 22070 df-fil 22161 df-fm 22253 df-flim 22254 df-flf 22255 df-xms 22636 df-ms 22637 df-tms 22638 df-cncf 23192 df-ovol 23771 df-vol 23772 df-mbf 23926 df-itg1 23927 df-itg2 23928 df-ibl 23929 df-itg 23930 df-0p 23977 df-ditg 24151 df-limc 24170 df-dv 24171

This theorem is referenced by: fourierdlem107 41930

W3C validator