itgsbtaddcnst - Mathbox for Glauco Siliprandi

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Theorem itgsbtaddcnst 46676

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 13462	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5	4	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℝ)
6	5	recnd 11238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℂ)
7		itgsbtaddcnst.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 11238	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9	8	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℂ)
10	6, 9	negsubd 11576	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 + -𝑋) = (𝑡 − 𝑋))
11	10	eqcomd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) = (𝑡 + -𝑋))
12	11	mpteq2dva 5205	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)))
13	1	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
14	7	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
15	13, 14	resubcld 11643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ∈ ℝ)
16	2	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
17	16, 14	resubcld 11643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ)
18	5, 14	resubcld 11643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℝ)
19		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
20	1, 2	jca 520	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
21	20	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
22		elicc2 13439	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ (𝐴[,]𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 ≤ 𝑡 ∧ 𝑡 ≤ 𝐵)))
23	21, 22	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴[,]𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 ≤ 𝑡 ∧ 𝑡 ≤ 𝐵)))
24	19, 23	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ℝ ∧ 𝐴 ≤ 𝑡 ∧ 𝑡 ≤ 𝐵))
25	24	simp2d 1161	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑡)
26	13, 5, 14, 25	lesub1dd 11831	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ≤ (𝑡 − 𝑋))
27	24	simp3d 1162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ≤ 𝐵)
28	5, 16, 14, 27	lesub1dd 11831	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ≤ (𝐵 − 𝑋))
29	15, 17, 18, 26, 28	eliccd 46200	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
30	29	fmpttd 7112	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
31	12, 30	feq1dd 6690	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
32	1, 7	resubcld 11643	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
33	2, 7	resubcld 11643	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
34	32, 33	iccssred 13462	. . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℝ)
35		ax-resscn 11158	. . . . . . 7 ⊢ ℝ ⊆ ℂ
36	34, 35	sstrdi 3950	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℂ)
37	4, 35	sstrdi 3950	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
38	37	resmptd 6044	. . . . . . . 8 ⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)))
39		ssid 3960	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
40		cncfmptid 25053	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
41	39, 39, 40	mp2an 704	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ)
42	41	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
43	39	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ℂ ⊆ ℂ)
44		id 23	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
45	43, 44, 43	constcncfg 46566	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
46	42, 45	subcncf 25585	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
47	8, 46	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
48		rescncf 25037	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ ℂ → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ) → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
49	37, 47, 48	sylc 66	. . . . . . . 8 ⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
50	38, 49	eqeltrrd 2864	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
51	12, 50	eqeltrrd 2864	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
52		cncfcdm 25038	. . . . . 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))))
53	36, 51, 52	syl2anc 595	. . . . 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))))
54	31, 53	mpbird 260	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
55	12, 54	eqeltrd 2863	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
56		eqid 2763	. . . . 5 ⊢ (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠))
57	8	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
58		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
59	57, 58	addcomd 11413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
60	59	mpteq2dva 5205	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
61		eqid 2763	. . . . . . . 8 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
62	61	addccncf 25057	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
63	8, 62	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
64	60, 63	eqeltrd 2863	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
65	1	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
66	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐵 ∈ ℝ)
67	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑋 ∈ ℝ)
68	34	sselda 3938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ℝ)
69	67, 68	readdcld 11239	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ ℝ)
70		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
71	32	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
72	33	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
73		elicc2 13439	. . . . . . . . . 10 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ) → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (𝐵 − 𝑋))))
74	71, 72, 73	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (𝐵 − 𝑋))))
75	70, 74	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑠 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (𝐵 − 𝑋)))
76	75	simp2d 1161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ≤ 𝑠)
77	65, 67, 68	lesubadd2d 11814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝐴 − 𝑋) ≤ 𝑠 ↔ 𝐴 ≤ (𝑋 + 𝑠)))
78	76, 77	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑠))
79	75	simp3d 1162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ≤ (𝐵 − 𝑋))
80	67, 68, 66	leaddsub2d 11817	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝑋 + 𝑠) ≤ 𝐵 ↔ 𝑠 ≤ (𝐵 − 𝑋)))
81	79, 80	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ≤ 𝐵)
82	65, 66, 69, 78, 81	eliccd 46200	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ (𝐴[,]𝐵))
83	56, 64, 36, 37, 82	cncfmptssg 46565	. . . 4 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝑋 + 𝑠)) ∈ (((𝐴 − 𝑋)[,](𝐵 − 𝑋))–cn→(𝐴[,]𝐵)))
84		itgsbtaddcnst.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
85	83, 84	cncfcompt 46577	. . 3 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝐴 − 𝑋)[,](𝐵 − 𝑋))–cn→ℂ))
86		ax-1cn 11159	. . . . . 6 ⊢ 1 ∈ ℂ
87		ioosscn 13436	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
88		cncfmptc 25052	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
89	86, 87, 39, 88	mp3an 1490	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ)
90	89	a1i 11	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
91		fconstmpt 5725	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1)
92		ioombl 25705	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
93	92	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
94		volioo 25709	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
95	1, 2, 3, 94	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
96	2, 1	resubcld 11643	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
97	95, 96	eqeltrd 2863	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
98		1cnd 11203	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
99		iblconst 25958	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ (vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ 1 ∈ ℂ) → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
100	93, 97, 98, 99	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
101	91, 100	eqeltrrid 2868	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
102	90, 101	elind 4154	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
103	35	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
104	18	recnd 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℂ)
105		tgioo4 24943	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106		eqid 2763	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107		iccntr 24960	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
108	20, 107	syl 18	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
109	103, 4, 104, 105, 106, 108	dvmptntr 26111	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))))
110		reelprrecn 11193	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
111	110	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
112		ioossre 13435	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
113	112	sseli 3934	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ ℝ)
114	113	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℝ)
115	114	recnd 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℂ)
116		1cnd 11203	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
117	103	sselda 3938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℂ)
118		1cnd 11203	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 1 ∈ ℂ)
119	111	dvmptid 26097	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑡)) = (𝑡 ∈ ℝ ↦ 1))
120	112	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
121		iooretop 24903	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
122	121	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
123	111, 117, 118, 119, 120, 105, 106, 122	dvmptres 26103	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑡)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
124	8	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℂ)
125		0cnd 11200	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
126	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ ℂ)
127		0cnd 11200	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 0 ∈ ℂ)
128	111, 8	dvmptc 26098	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑋)) = (𝑡 ∈ ℝ ↦ 0))
129	111, 126, 127, 128, 120, 105, 106, 122	dvmptres 26103	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑋)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 0))
130	111, 115, 116, 123, 124, 125, 129	dvmptsub 26107	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)))
131	116	subid1d 11559	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (1 − 0) = 1)
132	131	mpteq2dva 5205	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
133	109, 130, 132	3eqtrd 2802	. . 3 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
134		oveq2 7420	. . . 4 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝑋 + 𝑠) = (𝑋 + (𝑡 − 𝑋)))
135	134	fveq2d 6887	. . 3 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + (𝑡 − 𝑋))))
136		oveq1 7419	. . 3 ⊢ (𝑡 = 𝐴 → (𝑡 − 𝑋) = (𝐴 − 𝑋))
137		oveq1 7419	. . 3 ⊢ (𝑡 = 𝐵 → (𝑡 − 𝑋) = (𝐵 − 𝑋))
138	1, 2, 3, 55, 85, 102, 133, 135, 136, 137, 32, 33	itgsubsticc 46670	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡)
139	124, 115	pncan3d 11573	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝑋 + (𝑡 − 𝑋)) = 𝑡)
140	139	fveq2d 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + (𝑡 − 𝑋))) = (𝐹‘𝑡))
141	140	oveq1d 7427	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = ((𝐹‘𝑡) · 1))
142		cncff 25033	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
143	84, 142	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
144	143	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
145		ioossicc 13461	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
146	145	sseli 3934	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ (𝐴[,]𝐵))
147	146	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
148	144, 147	ffvelcdmd 7082	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ℂ)
149	148	mulridd 11227	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑡) · 1) = (𝐹‘𝑡))
150	141, 149	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = (𝐹‘𝑡))
151	3, 150	ditgeq3d 46658	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)
152	138, 151	eqtrd 2798	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ⊆ wss 3906 {csn 4590 {cpr 4592 class class class wbr 5110 ↦ cmpt 5193 × cxp 5661 dom cdm 5663 ran crn 5664 ↾ cres 5665 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 ≤ cle 11245 − cmin 11442 -cneg 11443 (,)cioo 13373 [,]cicc 13376 TopOpenctopn 17475 topGenctg 17491 ℂ_fldccnfld 21503 intcnt 23155 –cn→ccncf 25016 volcvol 25603 𝐿¹cibl 25757 ⨜cdit 25986 D cdv 26003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cc 10420 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-symdif 4207 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-disj 5078 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-oadd 8458 df-omul 8459 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-dju 9888 df-card 9926 df-acn 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-mulg 19135 df-cntz 19388 df-cmn 19853 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-fg 21501 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-lp 23274 df-perf 23275 df-cn 23365 df-cnp 23366 df-haus 23453 df-cmp 23525 df-tx 23700 df-hmeo 23893 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-xms 24458 df-ms 24459 df-tms 24460 df-cncf 25018 df-ovol 25604 df-vol 25605 df-mbf 25759 df-itg1 25760 df-itg2 25761 df-ibl 25762 df-itg 25763 df-0p 25810 df-ditg 25987 df-limc 26006 df-dv 26007

This theorem is referenced by: fourierdlem82 46882

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