itgsbtaddcnst - Mathbox for Glauco Siliprandi

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

Theorem itgsbtaddcnst 38874

Description: Integral substitution, adding a constant to the function's argument. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
itgsbtaddcnst.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
itgsbtaddcnst.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
itgsbtaddcnst.aleb	⊢ (𝜑 → 𝐴 ≤ 𝐵)
itgsbtaddcnst.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
itgsbtaddcnst.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))

**Assertion**
Ref	Expression
itgsbtaddcnst	⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)

Distinct variable groups: 𝐴,𝑠,𝑡 𝐵,𝑠,𝑡 𝐹,𝑠,𝑡 𝑋,𝑠,𝑡 𝜑,𝑠,𝑡

**Proof of Theorem itgsbtaddcnst**
Step	Hyp	Ref	Expression
1		itgsbtaddcnst.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		itgsbtaddcnst.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		itgsbtaddcnst.aleb	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
4	1, 2	iccssred 38574	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5	4	sselda 3568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℝ)
6	5	recnd 9947	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℂ)
7		itgsbtaddcnst.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 9947	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℂ)
10	6, 9	negsubd 10277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 + -𝑋) = (𝑡 − 𝑋))
11	10	eqcomd 2616	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) = (𝑡 + -𝑋))
12	11	mpteq2dva 4672	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)))
13	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
14	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
15	13, 14	resubcld 10337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ∈ ℝ)
16	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
17	16, 14	resubcld 10337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ)
18	5, 14	resubcld 10337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℝ)
19		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
20	1, 2	jca 553	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
22		elicc2 12109	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ (𝐴[,]𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 ≤ 𝑡 ∧ 𝑡 ≤ 𝐵)))
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴[,]𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 ≤ 𝑡 ∧ 𝑡 ≤ 𝐵)))
24	19, 23	mpbid 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ℝ ∧ 𝐴 ≤ 𝑡 ∧ 𝑡 ≤ 𝐵))
25	24	simp2d 1067	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑡)
26	13, 5, 14, 25	lesub1dd 10522	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ≤ (𝑡 − 𝑋))
27	24	simp3d 1068	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ≤ 𝐵)
28	5, 16, 14, 27	lesub1dd 10522	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ≤ (𝐵 − 𝑋))
29	15, 17, 18, 26, 28	eliccd 38573	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
30		eqid 2610	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))
31	29, 30	fmptd 6292	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
32	12, 31	feq1dd 38341	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
33	1, 7	resubcld 10337	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
34	2, 7	resubcld 10337	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
35	33, 34	iccssred 38574	. . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℝ)
36		ax-resscn 9872	. . . . . . 7 ⊢ ℝ ⊆ ℂ
37	35, 36	syl6ss 3580	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℂ)
38	4, 36	syl6ss 3580	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
39	38	resmptd 5371	. . . . . . . 8 ⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)))
40		ssid 3587	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
41		cncfmptid 22523	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
42	40, 40, 41	mp2an 704	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ)
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
44	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ℂ ⊆ ℂ)
45		id 22	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
46	44, 45, 44	constcncfg 38756	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
47	43, 46	subcncf 38754	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
48	8, 47	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
49		rescncf 22508	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ) → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
50	38, 48, 49	sylc 63	. . . . . . . 8 ⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
51	39, 50	eqeltrrd 2689	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
52	12, 51	eqeltrrd 2689	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
53		cncffvrn 22509	. . . . . 6 ⊢ ((((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℂ ∧ (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ((𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))) ↔ (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
54	37, 52, 53	syl2anc 691	. . . . 5 ⊢ (𝜑 → ((𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))) ↔ (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
55	32, 54	mpbird 246	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
56	12, 55	eqeltrd 2688	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
57		eqid 2610	. . . . 5 ⊢ (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠))
58	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
59		simpr 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
60	58, 59	addcomd 10117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
61	60	mpteq2dva 4672	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
62		eqid 2610	. . . . . . . 8 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
63	62	addccncf 22527	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
64	8, 63	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
65	61, 64	eqeltrd 2688	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
66	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
67	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐵 ∈ ℝ)
68	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑋 ∈ ℝ)
69	35	sselda 3568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ℝ)
70	68, 69	readdcld 9948	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ ℝ)
71		simpr 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
72	33	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
73	34	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
74		elicc2 12109	. . . . . . . . . 10 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ) → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (𝐵 − 𝑋))))
75	72, 73, 74	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (𝐵 − 𝑋))))
76	71, 75	mpbid 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑠 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (𝐵 − 𝑋)))
77	76	simp2d 1067	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ≤ 𝑠)
78	66, 68, 69	lesubadd2d 10505	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝐴 − 𝑋) ≤ 𝑠 ↔ 𝐴 ≤ (𝑋 + 𝑠)))
79	77, 78	mpbid 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑠))
80	76	simp3d 1068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ≤ (𝐵 − 𝑋))
81	68, 69, 67	leaddsub2d 10508	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝑋 + 𝑠) ≤ 𝐵 ↔ 𝑠 ≤ (𝐵 − 𝑋)))
82	80, 81	mpbird 246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ≤ 𝐵)
83	66, 67, 70, 79, 82	eliccd 38573	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ (𝐴[,]𝐵))
84	57, 65, 37, 38, 83	cncfmptssg 38755	. . . 4 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝑋 + 𝑠)) ∈ (((𝐴 − 𝑋)[,](𝐵 − 𝑋))–cn→(𝐴[,]𝐵)))
85		itgsbtaddcnst.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
86	84, 85	cncfcompt 38768	. . 3 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝐴 − 𝑋)[,](𝐵 − 𝑋))–cn→ℂ))
87		ax-1cn 9873	. . . . . 6 ⊢ 1 ∈ ℂ
88		ioosscn 38563	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
89		cncfmptc 22522	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
90	87, 88, 40, 89	mp3an 1416	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ)
91	90	a1i 11	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
92		fconstmpt 5085	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1)
93		ioombl 23140	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
94	93	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
95		volioo 38840	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
96	1, 2, 3, 95	syl3anc 1318	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
97	2, 1	resubcld 10337	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
98	96, 97	eqeltrd 2688	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
99		1cnd 9935	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
100		iblconst 23390	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ (vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ 1 ∈ ℂ) → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
101	94, 98, 99, 100	syl3anc 1318	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
102	92, 101	syl5eqelr 2693	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
103	91, 102	elind 3760	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
104	36	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
105	18	recnd 9947	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℂ)
106		eqid 2610	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107	106	tgioo2 22414	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
108		iccntr 22432	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
109	20, 108	syl 17	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
110	104, 4, 105, 107, 106, 109	dvmptntr 23540	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))))
111		reelprrecn 9907	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
112	111	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
113		ioossre 12106	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
114	113	sseli 3564	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ ℝ)
115	114	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℝ)
116	115	recnd 9947	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℂ)
117		1cnd 9935	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
118	104	sselda 3568	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℂ)
119		1cnd 9935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 1 ∈ ℂ)
120	112	dvmptid 23526	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑡)) = (𝑡 ∈ ℝ ↦ 1))
121	113	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
122		iooretop 22379	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
123	122	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
124	112, 118, 119, 120, 121, 107, 106, 123	dvmptres 23532	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑡)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
125	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℂ)
126		0cnd 9912	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
127	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ ℂ)
128		0cnd 9912	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 0 ∈ ℂ)
129	112, 8	dvmptc 23527	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑋)) = (𝑡 ∈ ℝ ↦ 0))
130	112, 127, 128, 129, 121, 107, 106, 123	dvmptres 23532	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑋)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 0))
131	112, 116, 117, 124, 125, 126, 130	dvmptsub 23536	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)))
132	117	subid1d 10260	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (1 − 0) = 1)
133	132	mpteq2dva 4672	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
134	110, 131, 133	3eqtrd 2648	. . 3 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
135		oveq2 6557	. . . 4 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝑋 + 𝑠) = (𝑋 + (𝑡 − 𝑋)))
136	135	fveq2d 6107	. . 3 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + (𝑡 − 𝑋))))
137		oveq1 6556	. . 3 ⊢ (𝑡 = 𝐴 → (𝑡 − 𝑋) = (𝐴 − 𝑋))
138		oveq1 6556	. . 3 ⊢ (𝑡 = 𝐵 → (𝑡 − 𝑋) = (𝐵 − 𝑋))
139	1, 2, 3, 56, 86, 103, 134, 136, 137, 138, 33, 34	itgsubsticc 38868	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡)
140	125, 116	pncan3d 10274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝑋 + (𝑡 − 𝑋)) = 𝑡)
141	140	fveq2d 6107	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + (𝑡 − 𝑋))) = (𝐹‘𝑡))
142	141	oveq1d 6564	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = ((𝐹‘𝑡) · 1))
143		cncff 22504	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
144	85, 143	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
145	144	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
146		ioossicc 12130	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
147	146	sseli 3564	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ (𝐴[,]𝐵))
148	147	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
149	145, 148	ffvelrnd 6268	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ℂ)
150	149	mulid1d 9936	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑡) · 1) = (𝐹‘𝑡))
151	142, 150	eqtrd 2644	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = (𝐹‘𝑡))
152	3, 151	ditgeq3d 38856	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)
153	139, 152	eqtrd 2644	1 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ran crn 5039 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 − cmin 10145 -cneg 10146 (,)cioo 12046 [,]cicc 12049 TopOpenctopn 15905 topGenctg 15921 ℂ_fldccnfld 19567 intcnt 20631 –cn→ccncf 22487 volcvol 23039 𝐿¹cibl 23192 ⨜cdit 23416 D cdv 23433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-ovol 23040 df-vol 23041 df-mbf 23194 df-itg1 23195 df-itg2 23196 df-ibl 23197 df-itg 23198 df-0p 23243 df-ditg 23417 df-limc 23436 df-dv 23437

This theorem is referenced by: fourierdlem82 39081

W3C validator