fourierdlem40 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem40 46975

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 5968	. . . 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 26692	. . . . . . . . 9 ⊢ π ∈ ℝ
5	4	renegcli 11543	. . . . . . . 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 13428	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
9	8	adantl 487	. . . . . . 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 13487	. . . . . . . . . . 11 ⊢ (𝜑 → (-π[,]π) ⊆ ℝ)
13		fourierdlem40.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (-π[,]π))
14	12, 13	sseldd 3932	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
15	14	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
16	5, 4	elicc2i 13465	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (-π[,]π) ↔ (𝐴 ∈ ℝ ∧ -π ≤ 𝐴 ∧ 𝐴 ≤ π))
17	16	simp2bi 1164	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-π[,]π) → -π ≤ 𝐴)
18	13, 17	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → -π ≤ 𝐴)
19	18	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -π ≤ 𝐴)
20	15	rexrd 11283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
21		fourierdlem40.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ (-π[,]π))
22	12, 21	sseldd 3932	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ)
23	22	rexrd 11283	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
24	23	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
25		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (𝐴(,)𝐵))
26		ioogtlb 46325	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
27	20, 24, 25, 26	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
28	6, 15, 9, 19, 27	lelttrd 11392	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -π < 𝑠)
29	6, 9, 28	ltled 11382	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -π ≤ 𝑠)
30	22	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
31		iooltub 46340	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
32	20, 24, 25, 31	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
33	5, 4	elicc2i 13465	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (-π[,]π) ↔ (𝐵 ∈ ℝ ∧ -π ≤ 𝐵 ∧ 𝐵 ≤ π))
34	33	simp3bi 1165	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (-π[,]π) → 𝐵 ≤ π)
35	21, 34	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≤ π)
36	35	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ≤ π)
37	9, 30, 7, 32, 36	ltletrd 11394	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < π)
38	9, 7, 37	ltled 11382	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≤ π)
39	6, 7, 9, 29, 38	eliccd 46334	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
40	39	ex 418	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ (-π[,]π)))
41	40	ssrdv 3937	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
42	41	resmptd 6036	. . 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 2848	. . . . . . . . 9 ⊢ (𝑠 = 0 → (𝑠 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵)))
44	43	biimpac 484	. . . . . . . 8 ⊢ ((𝑠 ∈ (𝐴(,)𝐵) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
45	44	adantll 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → 0 ∈ (𝐴(,)𝐵))
46		fourierdlem40.nxelab	. . . . . . . 8 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
47	46	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) ∧ 𝑠 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
48	45, 47	pm2.65da 829	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
49	48	iffalsed 4493	. . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
52		fourierdlem40.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℝ)
53	52	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
54	53, 9	readdcld 11262	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
55	51, 54	ffvelcdmd 7078	. . . . . . . 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 4529	. . . . . . . . 9 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
59	58	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
60	55, 59	resubcld 11666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
61	60	recnd 11261	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
62	9	recnd 11261	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
63	48	neqned 2962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
64	61, 62, 63	divrecd 12018	. . . . 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 2795	. . . 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 5198	. . 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 2799	. 2 ⊢ (𝜑 → (𝐻 ↾ (𝐴(,)𝐵)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (1 / 𝑠))))
68	55	recnd 11261	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
69	59	recnd 11261	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
70	68, 69	negsubd 11599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) + -if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)))
71	70	eqcomd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) = ((𝐹‘(𝑋 + 𝑠)) + -if(0 < 𝑠, 𝑌, 𝑊)))
72	71	mpteq2dva 5198	. . . 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 11262	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝑋) ∈ ℝ)
74	73	rexrd 11283	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 + 𝑋) ∈ ℝ^*)
75	74	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐴 + 𝑋) ∈ ℝ^*)
76	22, 52	readdcld 11262	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 𝑋) ∈ ℝ)
77	76	rexrd 11283	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 𝑋) ∈ ℝ^*)
78	77	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐵 + 𝑋) ∈ ℝ^*)
79	14	recnd 11261	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
80	52	recnd 11261	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ℂ)
81	79, 80	addcomd 11436	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 + 𝑋) = (𝑋 + 𝐴))
82	81	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐴 + 𝑋) = (𝑋 + 𝐴))
83	15, 9, 53, 27	ltadd2dd 11393	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) < (𝑋 + 𝑠))
84	82, 83	eqbrtrd 5127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐴 + 𝑋) < (𝑋 + 𝑠))
85	9, 30, 53, 32	ltadd2dd 11393	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝑋 + 𝐵))
86	22	recnd 11261	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
87	80, 86	addcomd 11436	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 + 𝐵) = (𝐵 + 𝑋))
88	87	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐵) = (𝐵 + 𝑋))
89	85, 88	breqtrd 5131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝐵 + 𝑋))
90	75, 78, 54, 84, 89	eliood 46328	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))
91		fvres 6897	. . . . . . . . 9 ⊢ ((𝑋 + 𝑠) ∈ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)) → ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
92	90, 91	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + 𝑠)))
93	92	eqcomd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) = ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠)))
94	93	mpteq2dva 5198	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋)))‘(𝑋 + 𝑠))))
95		ioosscn 13461	. . . . . . . 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 13461	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℂ
99	98	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
100	96, 97, 99, 80, 90	fourierdlem23 46958	. . . . . 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 11235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
103	14	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
104	8	adantl 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
105		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 ≤ 𝐴)
106	27	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
107	102, 103, 104, 105, 106	lelttrd 11392	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 < 𝑠)
108	107	iftrued 4490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑌)
109	108	negeqd 11475	. . . . . . . 8 ⊢ (((𝜑 ∧ 0 ≤ 𝐴) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -if(0 < 𝑠, 𝑌, 𝑊) = -𝑌)
110	109	mpteq2dva 5198	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐴) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑌))
111	56	renegcld 11665	. . . . . . . . . 10 ⊢ (𝜑 → -𝑌 ∈ ℝ)
112	111	recnd 11261	. . . . . . . . 9 ⊢ (𝜑 → -𝑌 ∈ ℂ)
113		ssid 3953	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
114	113	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ⊆ ℂ)
115	99, 112, 114	constcncfg 46700	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑌) ∈ ((𝐴(,)𝐵)–cn→ℂ))
116	115	adantr 486	. . . . . . 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 11283	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
119	118	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 𝐴 ∈ ℝ^*)
120	23	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 𝐵 ∈ ℝ^*)
121		0red 11235	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 0 ∈ ℝ)
122		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → ¬ 0 ≤ 𝐴)
123	14	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
124		0red 11235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → 0 ∈ ℝ)
125	123, 124	ltnled 11381	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → (𝐴 < 0 ↔ ¬ 0 ≤ 𝐴))
126	122, 125	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → 𝐴 < 0)
127	126	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 𝐴 < 0)
128		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → ¬ 𝐵 ≤ 0)
129		0red 11235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → 0 ∈ ℝ)
130	22	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → 𝐵 ∈ ℝ)
131	129, 130	ltnled 11381	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → (0 < 𝐵 ↔ ¬ 𝐵 ≤ 0))
132	128, 131	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 0) → 0 < 𝐵)
133	132	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 0 < 𝐵)
134	119, 120, 121, 127, 133	eliood 46328	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → 0 ∈ (𝐴(,)𝐵))
135	46	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 0 ≤ 𝐴) ∧ ¬ 𝐵 ≤ 0) → ¬ 0 ∈ (𝐴(,)𝐵))
136	134, 135	condan 830	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → 𝐵 ≤ 0)
137	8	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
138		0red 11235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
139	22	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
140	32	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
141		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ≤ 0)
142	137, 139, 138, 140, 141	ltletrd 11394	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 0)
143	137, 138, 142	ltnsymd 11383	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 0 < 𝑠)
144	143	iffalsed 4493	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑊)
145	144	negeqd 11475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ≤ 0) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → -if(0 < 𝑠, 𝑌, 𝑊) = -𝑊)
146	145	mpteq2dva 5198	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≤ 0) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑊))
147	57	recnd 11261	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ℂ)
148	147	negcld 11580	. . . . . . . . . 10 ⊢ (𝜑 → -𝑊 ∈ ℂ)
149	99, 148, 114	constcncfg 46700	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ -𝑊) ∈ ((𝐴(,)𝐵)–cn→ℂ))
150	149	adantr 486	. . . . . . . 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 603	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 ≤ 𝐴) → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
153	117, 152	pm2.61dan 825	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ -if(0 < 𝑠, 𝑌, 𝑊)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
154	101, 153	addcncf 25672	. . . 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 2760	. . . 4 ⊢ (𝑠 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑠)) = (𝑠 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑠))
157		1cnd 11226	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
158	156	cdivcncf 25149	. . . . 5 ⊢ (1 ∈ ℂ → (𝑠 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑠)) ∈ ((ℂ ∖ {0})–cn→ℂ))
159	157, 158	syl 18	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑠)) ∈ ((ℂ ∖ {0})–cn→ℂ))
160		velsn 4600	. . . . . . . 8 ⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0)
161	48, 160	sylnibr 332	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 ∈ {0})
162	62, 161	eldifd 3910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (ℂ ∖ {0}))
163	162	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (𝐴(,)𝐵)𝑠 ∈ (ℂ ∖ {0}))
164		dfss3 3920	. . . . 5 ⊢ ((𝐴(,)𝐵) ⊆ (ℂ ∖ {0}) ↔ ∀𝑠 ∈ (𝐴(,)𝐵)𝑠 ∈ (ℂ ∖ {0}))
165	163, 164	sylibr 237	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (ℂ ∖ {0}))
166	9, 63	rereccld 12066	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 𝑠) ∈ ℝ)
167	166	recnd 11261	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 𝑠) ∈ ℂ)
168	156, 159, 165, 114, 167	cncfmptssg 46699	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑠)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
169	155, 168	mulcncf 25674	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∖ cdif 3896 ⊆ wss 3899 ifcif 4482 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 − cmin 11465 -cneg 11466 / cdiv 11895 (,)cioo 13398 [,]cicc 13401 πcpi 16152 –cn→ccncf 25104

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 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 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16153 df-sin 16155 df-cos 16156 df-pi 16158 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-limc 26093 df-dv 26094

This theorem is used by: fourierdlem103 47037 fourierdlem104 47038

W3C validator