itgsbtaddcnst - Mathbox for Glauco Siliprandi

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Theorem itgsbtaddcnst 46554

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 13452	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5	4	sselda 3939	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℝ)
6	5	recnd 11225	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℂ)
7		itgsbtaddcnst.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 11225	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9	8	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℂ)
10	6, 9	negsubd 11563	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 + -𝑋) = (𝑡 − 𝑋))
11	10	eqcomd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) = (𝑡 + -𝑋))
12	11	mpteq2dva 5198	. . . 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 11630	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ∈ ℝ)
16	2	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
17	16, 14	resubcld 11630	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ)
18	5, 14	resubcld 11630	. . . . . . . 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 13429	. . . . . . . . . . . 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 1159	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑡)
26	13, 5, 14, 25	lesub1dd 11818	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ≤ (𝑡 − 𝑋))
27	24	simp3d 1160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ≤ 𝐵)
28	5, 16, 14, 27	lesub1dd 11818	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ≤ (𝐵 − 𝑋))
29	15, 17, 18, 26, 28	eliccd 46078	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
30	29	fmpttd 7100	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
31	12, 30	feq1dd 6678	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
32	1, 7	resubcld 11630	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
33	2, 7	resubcld 11630	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
34	32, 33	iccssred 13452	. . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℝ)
35		ax-resscn 11145	. . . . . . 7 ⊢ ℝ ⊆ ℂ
36	34, 35	sstrdi 3951	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℂ)
37	4, 35	sstrdi 3951	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
38	37	resmptd 6033	. . . . . . . 8 ⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)))
39		ssid 3961	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
40		cncfmptid 25033	. . . . . . . . . . . . 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 46444	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
46	42, 45	subcncf 25565	. . . . . . . . . 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 25017	. . . . . . . . 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 2866	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
51	12, 50	eqeltrrd 2866	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
52		cncfcdm 25018	. . . . . 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 2865	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
56		eqid 2765	. . . . 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 11400	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
60	59	mpteq2dva 5198	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
61		eqid 2765	. . . . . . . 8 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
62	61	addccncf 25037	. . . . . . 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 2865	. . . . 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 3939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ℝ)
69	67, 68	readdcld 11226	. . . . . 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 13429	. . . . . . . . . 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 1159	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ≤ 𝑠)
77	65, 67, 68	lesubadd2d 11801	. . . . . . 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 1160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ≤ (𝐵 − 𝑋))
80	67, 68, 66	leaddsub2d 11804	. . . . . . 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 46078	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ (𝐴[,]𝐵))
83	56, 64, 36, 37, 82	cncfmptssg 46443	. . . 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 46455	. . 3 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝐴 − 𝑋)[,](𝐵 − 𝑋))–cn→ℂ))
86		ax-1cn 11146	. . . . . 6 ⊢ 1 ∈ ℂ
87		ioosscn 13426	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
88		cncfmptc 25032	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
89	86, 87, 39, 88	mp3an 1485	. . . . 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 5714	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1)
92		ioombl 25685	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
93	92	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
94		volioo 25689	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
95	1, 2, 3, 94	syl3anc 1394	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
96	2, 1	resubcld 11630	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
97	95, 96	eqeltrd 2865	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
98		1cnd 11190	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
99		iblconst 25938	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ (vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ 1 ∈ ℂ) → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
100	93, 97, 98, 99	syl3anc 1394	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
101	91, 100	eqeltrrid 2870	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
102	90, 101	elind 4155	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
103	35	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
104	18	recnd 11225	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℂ)
105		tgioo4 24923	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106		eqid 2765	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107		iccntr 24940	. . . . . 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 26091	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))))
110		reelprrecn 11180	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
111	110	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
112		ioossre 13425	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
113	112	sseli 3935	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ ℝ)
114	113	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℝ)
115	114	recnd 11225	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℂ)
116		1cnd 11190	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
117	103	sselda 3939	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℂ)
118		1cnd 11190	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 1 ∈ ℂ)
119	111	dvmptid 26077	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑡)) = (𝑡 ∈ ℝ ↦ 1))
120	112	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
121		iooretop 24883	. . . . . . 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 26083	. . . . 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 11187	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
126	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ ℂ)
127		0cnd 11187	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 0 ∈ ℂ)
128	111, 8	dvmptc 26078	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑋)) = (𝑡 ∈ ℝ ↦ 0))
129	111, 126, 127, 128, 120, 105, 106, 122	dvmptres 26083	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑋)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 0))
130	111, 115, 116, 123, 124, 125, 129	dvmptsub 26087	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)))
131	116	subid1d 11546	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (1 − 0) = 1)
132	131	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
133	109, 130, 132	3eqtrd 2804	. . 3 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
134		oveq2 7408	. . . 4 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝑋 + 𝑠) = (𝑋 + (𝑡 − 𝑋)))
135	134	fveq2d 6875	. . 3 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + (𝑡 − 𝑋))))
136		oveq1 7407	. . 3 ⊢ (𝑡 = 𝐴 → (𝑡 − 𝑋) = (𝐴 − 𝑋))
137		oveq1 7407	. . 3 ⊢ (𝑡 = 𝐵 → (𝑡 − 𝑋) = (𝐵 − 𝑋))
138	1, 2, 3, 55, 85, 102, 133, 135, 136, 137, 32, 33	itgsubsticc 46548	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡)
139	124, 115	pncan3d 11560	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝑋 + (𝑡 − 𝑋)) = 𝑡)
140	139	fveq2d 6875	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + (𝑡 − 𝑋))) = (𝐹‘𝑡))
141	140	oveq1d 7415	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = ((𝐹‘𝑡) · 1))
142		cncff 25013	. . . . . . . 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 13451	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
146	145	sseli 3935	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ (𝐴[,]𝐵))
147	146	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
148	144, 147	ffvelcdmd 7070	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ℂ)
149	148	mulridd 11214	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑡) · 1) = (𝐹‘𝑡))
150	141, 149	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = (𝐹‘𝑡))
151	3, 150	ditgeq3d 46536	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)
152	138, 151	eqtrd 2800	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 1101 = wceq 1563 ∈ wcel 2145 ⊆ wss 3907 {csn 4585 {cpr 4587 class class class wbr 5105 ↦ cmpt 5186 × cxp 5650 dom cdm 5652 ran crn 5653 ↾ cres 5654 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 ≤ cle 11232 − cmin 11429 -cneg 11430 (,)cioo 13363 [,]cicc 13366 TopOpenctopn 17464 topGenctg 17480 ℂ_fldccnfld 21482 intcnt 23135 –cn→ccncf 24996 volcvol 25583 𝐿¹cibl 25737 ⨜cdit 25966 D cdv 25983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cc 10407 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-symdif 4208 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-disj 5073 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-ofr 7665 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-omul 8446 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9875 df-card 9913 df-acn 9916 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-hash 14358 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-haus 23433 df-cmp 23505 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-ovol 25584 df-vol 25585 df-mbf 25739 df-itg1 25740 df-itg2 25741 df-ibl 25742 df-itg 25743 df-0p 25790 df-ditg 25967 df-limc 25986 df-dv 25987

This theorem is referenced by: fourierdlem82 46760

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