fourierdlem40 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem40 41863

Description: 𝐻 is a continuous function on any partition interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem40.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem40.a	⊢ (𝜑 → 𝐴 ∈ (-π[,]π))
fourierdlem40.b	⊢ (𝜑 → 𝐵 ∈ (-π[,]π))
fourierdlem40.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem40.nxelab	⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
fourierdlem40.fcn	⊢ (𝜑 → (𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋))) ∈ (((𝐴 + 𝑋)(,)(𝐵 + 𝑋))–cn→ℂ))
fourierdlem40.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
fourierdlem40.w	⊢ (𝜑 → 𝑊 ∈ ℝ)
fourierdlem40.h	⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))

**Assertion**
Ref	Expression
fourierdlem40	⊢ (𝜑 → (𝐻 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))

Distinct variable groups: 𝐴,𝑠 𝐵,𝑠 𝐹,𝑠 𝑊,𝑠 𝑋,𝑠 𝑌,𝑠 𝜑,𝑠 Allowed substitution hint: 𝐻(𝑠)

**Proof of Theorem fourierdlem40**
Step	Hyp	Ref	Expression
1		fourierdlem40.h	. . . . 5 ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
2	1	reseq1i 5685	. . . 4 ⊢ (𝐻 ↾ (𝐴(,)𝐵)) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ (𝐴(,)𝐵))
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐻 ↾ (𝐴(,)𝐵)) = ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ (𝐴(,)𝐵)))
4		pire 24741	. . . . . . . . 9 ⊢ π ∈ ℝ
5	4	renegcli 10742	. . . . . . . 8 ⊢ -π ∈ ℝ
6	5	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -π ∈ ℝ)
7	4	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → π ∈ ℝ)
8		elioore 12578	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
9	8	adantl 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
10	5	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -π ∈ ℝ)
11	4	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → π ∈ ℝ)
12	10, 11	iccssred 41211	. . . . . . . . . . 11 ⊢ (𝜑 → (-π[,]π) ⊆ ℝ)
13		fourierdlem40.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (-π[,]π))
14	12, 13	sseldd 3853	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
15	14	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
16	5, 4	elicc2i 12612	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (-π[,]π) ↔ (𝐴 ∈ ℝ ∧ -π ≤ 𝐴 ∧ 𝐴 ≤ π))
17	16	simp2bi 1126	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-π[,]π) → -π ≤ 𝐴)
18	13, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → -π ≤ 𝐴)
19	18	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -π ≤ 𝐴)
20	15	rexrd 10484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
21		fourierdlem40.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ (-π[,]π))
22	12, 21	sseldd 3853	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ)
23	22	rexrd 10484	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
24	23	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
25		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (𝐴(,)𝐵))
26		ioogtlb 41201	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
27	20, 24, 25, 26	syl3anc 1351	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
28	6, 15, 9, 19, 27	lelttrd 10592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -π < 𝑠)
29	6, 9, 28	ltled 10582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -π ≤ 𝑠)
30	22	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
31		iooltub 41217	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
32	20, 24, 25, 31	syl3anc 1351	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
33	5, 4	elicc2i 12612	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (-π[,]π) ↔ (𝐵 ∈ ℝ ∧ -π ≤ 𝐵 ∧ 𝐵 ≤ π))
34	33	simp3bi 1127	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (-π[,]π) → 𝐵 ≤ π)
35	21, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≤ π)
36	35	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ≤ π)
37	9, 30, 7, 32, 36	ltletrd 10594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < π)
38	9, 7, 37	ltled 10582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≤ π)
39	6, 7, 9, 29, 38	eliccd 41210	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
40	39	ex 405	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ (-π[,]π)))
41	40	ssrdv 3858	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
42	41	resmptd 5747	. . 3 ⊢ (𝜑 → ((𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) ↾ (𝐴(,)𝐵)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))))
43		eleq1 2847	. . . . . . . . 9 ⊢ (𝑠 = 0 → (𝑠 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵)))
44	43	biimpac 471	. . . . . . . 8 ⊢ ((𝑠 ∈ (𝐴(,)𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
45	44	adantll 701	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
46		fourierdlem40.nxelab	. . . . . . . 8 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
47	46	ad2antrr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
48	45, 47	pm2.65da 804	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
49	48	iffalsed 4355	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
50		fourierdlem40.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
51	50	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
52		fourierdlem40.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ)
53	52	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
54	53, 9	readdcld 10463	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
55	51, 54	ffvelrnd 6671	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
56		fourierdlem40.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ)
57		fourierdlem40.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ ℝ)
58	56, 57	ifcld 4389	. . . . . . . . 9 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
59	58	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
60	55, 59	resubcld 10863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
61	60	recnd 10462	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
62	9	recnd 10462	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
63	48	neqned 2968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
64	61, 62, 63	divrecd 11214	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (1 / 𝑠)))
65	49, 64	eqtrd 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (1 / 𝑠)))
66	65	mpteq2dva 5016	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (1 / 𝑠))))
67	3, 42, 66	3eqtrd 2812	. 2 ⊢ (𝜑 → (𝐻 ↾ (𝐴(,)𝐵)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (1 / 𝑠))))
68	55	recnd 10462	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
69	59	recnd 10462	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
70	68, 69	negsubd 10798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) + -if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)))
71	70	eqcomd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) + -if(0 < 𝑠, 𝑌, 𝑊)))
72	71	mpteq2dva 5016	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) + -if(0 < 𝑠, 𝑌, 𝑊))))
73	14, 52	readdcld 10463	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝑋) ∈ ℝ)
74	73	rexrd 10484	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝑋) ∈ ℝ^*)
75	74	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐴 + 𝑋) ∈ ℝ^*)
76	22, 52	readdcld 10463	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝑋) ∈ ℝ)
77	76	rexrd 10484	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝑋) ∈ ℝ^*)
78	77	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐵 + 𝑋) ∈ ℝ^*)
79	14	recnd 10462	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
80	52	recnd 10462	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ℂ)
81	79, 80	addcomd 10636	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝑋) = (𝑋 + 𝐴))
82	81	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐴 + 𝑋) = (𝑋 + 𝐴))
83	15, 9, 53, 27	ltadd2dd 10593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) < (𝑋 + 𝑠))
84	82, 83	eqbrtrd 4945	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐴 + 𝑋) < (𝑋 + 𝑠))
85	9, 30, 53, 32	ltadd2dd 10593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝑋 + 𝐵))
86	22	recnd 10462	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
87	80, 86	addcomd 10636	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 + 𝐵) = (𝐵 + 𝑋))
88	87	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐵) = (𝐵 + 𝑋))
89	85, 88	breqtrd 4949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝐵 + 𝑋))
90	75, 78, 54, 84, 89	eliood 41204	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))
91		fvres 6512	. . . . . . . . 9 ⊢ ((𝑋 + 𝑠) ∈ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)) → ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
92	90, 91	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
93	92	eqcomd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠)))
94	93	mpteq2dva 5016	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠))))
95		ioosscn 41200	. . . . . . . 8 ⊢ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)) ⊆ ℂ
96	95	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)) ⊆ ℂ)
97		fourierdlem40.fcn	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋))) ∈ (((𝐴 + 𝑋)(,)(𝐵 + 𝑋))–cn→ℂ))
98		ioosscn 41200	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℂ
99	98	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
100	96, 97, 99, 80, 90	fourierdlem23 41846	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
101	94, 100	eqeltrd 2860	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
102		0red 10437	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
103	14	ad2antrr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
104	8	adantl 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
105		simplr 756	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 ≤ 𝐴)
106	27	adantlr 702	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
107	102, 103, 104, 105, 106	lelttrd 10592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 < 𝑠)
108	107	iftrued 4352	. . . . . . . . 9 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌)
109	108	negeqd 10674	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -if(0 < 𝑠, 𝑌, 𝑊) = -𝑌)
110	109	mpteq2dva 5016	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑌))
111	56	renegcld 10862	. . . . . . . . . 10 ⊢ (𝜑 → -𝑌 ∈ ℝ)
112	111	recnd 10462	. . . . . . . . 9 ⊢ (𝜑 → -𝑌 ∈ ℂ)
113		ssid 3873	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
114	113	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
115	99, 112, 114	constcncfg 41584	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑌) ∈ ((𝐴(,)𝐵)–cn→ℂ))
116	115	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑌) ∈ ((𝐴(,)𝐵)–cn→ℂ))
117	110, 116	eqeltrd 2860	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
118	14	rexrd 10484	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
119	118	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 𝐴 ∈ ℝ^*)
120	23	ad2antrr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 𝐵 ∈ ℝ^*)
121		0red 10437	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 0 ∈ ℝ)
122		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → ¬ 0 ≤ 𝐴)
123	14	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
124		0red 10437	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → 0 ∈ ℝ)
125	123, 124	ltnled 10581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → (𝐴 < 0 ↔ ¬ 0 ≤ 𝐴))
126	122, 125	mpbird 249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → 𝐴 < 0)
127	126	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 𝐴 < 0)
128		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → ¬ 𝐵 ≤ 0)
129		0red 10437	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → 0 ∈ ℝ)
130	22	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → 𝐵 ∈ ℝ)
131	129, 130	ltnled 10581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → (0 < 𝐵 ↔ ¬ 𝐵 ≤ 0))
132	128, 131	mpbird 249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → 0 < 𝐵)
133	132	adantlr 702	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 0 < 𝐵)
134	119, 120, 121, 127, 133	eliood 41204	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 0 ∈ (𝐴(,)𝐵))
135	46	ad2antrr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → ¬ 0 ∈ (𝐴(,)𝐵))
136	134, 135	condan 805	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → 𝐵 ≤ 0)
137	8	adantl 474	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
138		0red 10437	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
139	22	ad2antrr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
140	32	adantlr 702	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
141		simplr 756	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ≤ 0)
142	137, 139, 138, 140, 141	ltletrd 10594	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 0)
143	137, 138, 142	ltnsymd 10583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 0 < 𝑠)
144	143	iffalsed 4355	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑊)
145	144	negeqd 10674	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -if(0 < 𝑠, 𝑌, 𝑊) = -𝑊)
146	145	mpteq2dva 5016	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑊))
147	57	recnd 10462	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ℂ)
148	147	negcld 10779	. . . . . . . . . 10 ⊢ (𝜑 → -𝑊 ∈ ℂ)
149	99, 148, 114	constcncfg 41584	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑊) ∈ ((𝐴(,)𝐵)–cn→ℂ))
150	149	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑊) ∈ ((𝐴(,)𝐵)–cn→ℂ))
151	146, 150	eqeltrd 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
152	136, 151	syldan 582	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
153	117, 152	pm2.61dan 800	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
154	101, 153	addcncf 41586	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) + -if(0 < 𝑠, 𝑌, 𝑊))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
155	72, 154	eqeltrd 2860	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
156		eqid 2772	. . . 4 ⊢ (𝑠 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑠)) = (𝑠 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑠))
157		1cnd 10428	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
158	156	cdivcncf 23222	. . . . 5 ⊢ (1 ∈ ℂ → (𝑠 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑠)) ∈ ((ℂ ∖ {0})–cn→ℂ))
159	157, 158	syl 17	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑠)) ∈ ((ℂ ∖ {0})–cn→ℂ))
160		velsn 4451	. . . . . . . 8 ⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0)
161	48, 160	sylnibr 321	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 ∈ {0})
162	62, 161	eldifd 3834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (ℂ ∖ {0}))
163	162	ralrimiva 3126	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (𝐴(,)𝐵)𝑠 ∈ (ℂ ∖ {0}))
164		dfss3 3841	. . . . 5 ⊢ ((𝐴(,)𝐵) ⊆ (ℂ ∖ {0}) ↔ ∀𝑠 ∈ (𝐴(,)𝐵)𝑠 ∈ (ℂ ∖ {0}))
165	163, 164	sylibr 226	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (ℂ ∖ {0}))
166	9, 63	rereccld 11262	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 𝑠) ∈ ℝ)
167	166	recnd 10462	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 𝑠) ∈ ℂ)
168	156, 159, 165, 114, 167	cncfmptssg 41583	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑠)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
169	155, 168	mulcncf 23744	. 2 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (1 / 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
170	67, 169	eqeltrd 2860	1 ⊢ (𝜑 → (𝐻 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3082 ∖ cdif 3820 ⊆ wss 3823 ifcif 4344 {csn 4435 class class class wbr 4923 ↦ cmpt 5002 ↾ cres 5403 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ℂcc 10327 ℝcr 10328 0cc0 10329 1c1 10330 + caddc 10332 · cmul 10334 ℝ^*cxr 10467 < clt 10468 ≤ cle 10469 − cmin 10664 -cneg 10665 / cdiv 11092 (,)cioo 12548 [,]cicc 12551 πcpi 15274 –cn→ccncf 23181

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 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 ax-addf 10408 ax-mulf 10409

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 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-er 8083 df-map 8202 df-pm 8203 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-fi 8664 df-sup 8695 df-inf 8696 df-oi 8763 df-card 9156 df-cda 9382 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-q 12157 df-rp 12199 df-xneg 12318 df-xadd 12319 df-xmul 12320 df-ioo 12552 df-ioc 12553 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-fl 12971 df-seq 13179 df-exp 13239 df-fac 13443 df-bc 13472 df-hash 13500 df-shft 14281 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-limsup 14683 df-clim 14700 df-rlim 14701 df-sum 14898 df-ef 15275 df-sin 15277 df-cos 15278 df-pi 15280 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-rest 16546 df-topn 16547 df-0g 16565 df-gsum 16566 df-topgen 16567 df-pt 16568 df-prds 16571 df-xrs 16625 df-qtop 16630 df-imas 16631 df-xps 16633 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-cnfld 20242 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-cld 21325 df-ntr 21326 df-cls 21327 df-nei 21404 df-lp 21442 df-perf 21443 df-cn 21533 df-cnp 21534 df-haus 21621 df-tx 21868 df-hmeo 22061 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-xms 22627 df-ms 22628 df-tms 22629 df-cncf 23183 df-limc 24161 df-dv 24162

This theorem is referenced by: fourierdlem103 41925 fourierdlem104 41926

W3C validator