itgsbtaddcnst - Mathbox for Glauco Siliprandi

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Theorem itgsbtaddcnst 47085

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 13568	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5	4	sselda 3908	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℝ)
6	5	recnd 11340	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℂ)
7		itgsbtaddcnst.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 11340	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9	8	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℂ)
10	6, 9	negsubd 11678	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 + -𝑋) = (𝑡 − 𝑋))
11	10	eqcomd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) = (𝑡 + -𝑋))
12	11	mpteq2dva 5197	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)))
13	1	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
14	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℝ)
15	13, 14	resubcld 11747	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ∈ ℝ)
16	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
17	16, 14	resubcld 11747	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ)
18	5, 14	resubcld 11747	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℝ)
19		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
20	1, 2	jca 521	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
21	20	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
22		elicc2 13545	. . . . . . . . . . . 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 11935	. . . . . . . 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 11935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ≤ (𝐵 − 𝑋))
29	15, 17, 18, 26, 28	eliccd 46609	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
30	29	fmpttd 7115	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
31	12, 30	feq1dd 6692	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
32	1, 7	resubcld 11747	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
33	2, 7	resubcld 11747	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
34	32, 33	iccssred 13568	. . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℝ)
35		ax-resscn 11260	. . . . . . 7 ⊢ ℝ ⊆ ℂ
36	34, 35	sstrdi 3920	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℂ)
37	4, 35	sstrdi 3920	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
38	37	resmptd 6031	. . . . . . . 8 ⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)))
39		ssid 3930	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
40		cncfmptid 25239	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
41	39, 39, 40	mp2an 705	. . . . . . . . . . . 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 46975	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
46	42, 45	subcncf 25771	. . . . . . . . . 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 25223	. . . . . . . . 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 2861	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
51	12, 50	eqeltrrd 2861	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
52		cncfcdm 25224	. . . . . 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 596	. . . . 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 2860	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
56		eqid 2760	. . . . 5 ⊢ (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠))
57	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
58		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
59	57, 58	addcomd 11515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
60	59	mpteq2dva 5197	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
61		eqid 2760	. . . . . . . 8 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
62	61	addccncf 25243	. . . . . . 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 2860	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
65	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
66	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐵 ∈ ℝ)
67	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑋 ∈ ℝ)
68	34	sselda 3908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ℝ)
69	67, 68	readdcld 11341	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ ℝ)
70		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
71	32	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
72	33	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
73		elicc2 13545	. . . . . . . . . 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 596	. . . . . . . . 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 11918	. . . . . . 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 11921	. . . . . . 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 46609	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ (𝐴[,]𝐵))
83	56, 64, 36, 37, 82	cncfmptssg 46974	. . . 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 46986	. . 3 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝐴 − 𝑋)[,](𝐵 − 𝑋))–cn→ℂ))
86		ax-1cn 11261	. . . . . 6 ⊢ 1 ∈ ℂ
87		ioosscn 13542	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
88		cncfmptc 25238	. . . . . 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 5712	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1)
92		ioombl 25891	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
93	92	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
94		volioo 25895	. . . . . . . 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 11747	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
97	95, 96	eqeltrd 2860	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
98		1cnd 11305	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
99		iblconst 26143	. . . . . 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 2865	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
102	90, 101	elind 4097	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
103	35	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
104	18	recnd 11340	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℂ)
105		tgioo4 25129	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106		eqid 2760	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107		iccntr 25146	. . . . . 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 26296	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))))
110		reelprrecn 11295	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
111	110	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
112		ioossre 13541	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
113	112	sseli 3904	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ ℝ)
114	113	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℝ)
115	114	recnd 11340	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℂ)
116		1cnd 11305	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
117	103	sselda 3908	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℂ)
118		1cnd 11305	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 1 ∈ ℂ)
119	111	dvmptid 26282	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑡)) = (𝑡 ∈ ℝ ↦ 1))
120	112	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
121		iooretop 25089	. . . . . . 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 26288	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑡)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
124	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℂ)
125		0cnd 11302	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
126	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ ℂ)
127		0cnd 11302	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 0 ∈ ℂ)
128	111, 8	dvmptc 26283	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑋)) = (𝑡 ∈ ℝ ↦ 0))
129	111, 126, 127, 128, 120, 105, 106, 122	dvmptres 26288	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑋)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 0))
130	111, 115, 116, 123, 124, 125, 129	dvmptsub 26292	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)))
131	116	subid1d 11661	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (1 − 0) = 1)
132	131	mpteq2dva 5197	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
133	109, 130, 132	3eqtrd 2799	. . 3 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
134		oveq2 7428	. . . 4 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝑋 + 𝑠) = (𝑋 + (𝑡 − 𝑋)))
135	134	fveq2d 6889	. . 3 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + (𝑡 − 𝑋))))
136		oveq1 7427	. . 3 ⊢ (𝑡 = 𝐴 → (𝑡 − 𝑋) = (𝐴 − 𝑋))
137		oveq1 7427	. . 3 ⊢ (𝑡 = 𝐵 → (𝑡 − 𝑋) = (𝐵 − 𝑋))
138	1, 2, 3, 55, 85, 102, 133, 135, 136, 137, 32, 33	itgsubsticc 47079	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡)
139	124, 115	pncan3d 11675	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝑋 + (𝑡 − 𝑋)) = 𝑡)
140	139	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + (𝑡 − 𝑋))) = (𝐹‘𝑡))
141	140	oveq1d 7435	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = ((𝐹‘𝑡) · 1))
142		cncff 25219	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
143	84, 142	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
144	143	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
145		ioossicc 13567	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
146	145	sseli 3904	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ (𝐴[,]𝐵))
147	146	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
148	144, 147	ffvelcdmd 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ℂ)
149	148	mulridd 11329	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑡) · 1) = (𝐹‘𝑡))
150	141, 149	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = (𝐹‘𝑡))
151	3, 150	ditgeq3d 47067	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)
152	138, 151	eqtrd 2795	1 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 {csn 4584 {cpr 4586 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 dom cdm 5650 ran crn 5651 ↾ cres 5652 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 + caddc 11206 · cmul 11208 ≤ cle 11347 − cmin 11544 -cneg 11545 (,)cioo 13479 [,]cicc 13482 TopOpenctopn 17597 topGenctg 17613 ℂ_fldccnfld 21683 intcnt 23340 –cn→ccncf 25202 volcvol 25789 𝐿¹cibl 25943 ⨜cdit 26171 D cdv 26188

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cc 10516 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-symdif 4198 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 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 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-oadd 8483 df-omul 8484 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9507 df-dju 9985 df-card 10023 df-acn 10026 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ioc 13484 df-ico 13485 df-icc 13486 df-fz 13643 df-fzo 13792 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-hash 14478 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-limsup 15643 df-clim 15660 df-rlim 15661 df-sum 15859 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-rest 17598 df-topn 17599 df-0g 17617 df-gsum 17618 df-topgen 17619 df-pt 17620 df-prds 17623 df-xrs 17679 df-qtop 17684 df-imas 17685 df-xps 17687 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-submnd 18984 df-mulg 19283 df-cntz 19536 df-cmn 20001 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-fbas 21680 df-fg 21681 df-cnfld 21684 df-top 23217 df-topon 23234 df-topsp 23256 df-bases 23269 df-cld 23342 df-ntr 23343 df-cls 23344 df-nei 23421 df-lp 23459 df-perf 23460 df-cn 23550 df-cnp 23551 df-haus 23638 df-cmp 23710 df-tx 23886 df-hmeo 24079 df-fil 24170 df-fm 24262 df-flim 24263 df-flf 24264 df-xms 24644 df-ms 24645 df-tms 24646 df-cncf 25204 df-ovol 25790 df-vol 25791 df-mbf 25945 df-itg1 25946 df-itg2 25947 df-ibl 25948 df-itg 25949 df-0p 25996 df-ditg 26172 df-limc 26191 df-dv 26192

This theorem is used by: fourierdlem82 47291

W3C validator