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 45964

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 13355	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5	4	sselda 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℝ)
6	5	recnd 11162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℂ)
7		itgsbtaddcnst.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ)
8	7	recnd 11162	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑋 ∈ ℂ)
10	6, 9	negsubd 11499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 + -𝑋) = (𝑡 − 𝑋))
11	10	eqcomd 2735	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) = (𝑡 + -𝑋))
12	11	mpteq2dva 5188	. . . 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 11566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ∈ ℝ)
16	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
17	16, 14	resubcld 11566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ)
18	5, 14	resubcld 11566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℝ)
19		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
20	1, 2	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
22		elicc2 13332	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ℝ ∧ 𝐴 ≤ 𝑡 ∧ 𝑡 ≤ 𝐵))
25	24	simp2d 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑡)
26	13, 5, 14, 25	lesub1dd 11754	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑋) ≤ (𝑡 − 𝑋))
27	24	simp3d 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ≤ 𝐵)
28	5, 16, 14, 27	lesub1dd 11754	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ≤ (𝐵 − 𝑋))
29	15, 17, 18, 26, 28	eliccd 45486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
30	29	fmpttd 7053	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
31	12, 30	feq1dd 6639	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)):(𝐴[,]𝐵)⟶((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
32	1, 7	resubcld 11566	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
33	2, 7	resubcld 11566	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
34	32, 33	iccssred 13355	. . . . . . 7 ⊢ (𝜑 → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ⊆ ℝ)
35		ax-resscn 11085	. . . . . . 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 5995	. . . . . . . 8 ⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)))
39		ssid 3960	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
40		cncfmptid 24822	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
41	39, 39, 40	mp2an 692	. . . . . . . . . . . 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 22	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
45	43, 44, 43	constcncfg 45854	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ 𝑋) ∈ (ℂ–cn→ℂ))
46	42, 45	subcncf 25361	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
47	8, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ∈ (ℂ–cn→ℂ))
48		rescncf 24806	. . . . . . . . 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 65	. . . . . . . 8 ⊢ (𝜑 → ((𝑡 ∈ ℂ ↦ (𝑡 − 𝑋)) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
50	38, 49	eqeltrrd 2829	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
51	12, 50	eqeltrrd 2829	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
52		cncfcdm 24807	. . . . . 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 584	. . . . 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 257	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 + -𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
55	12, 54	eqeltrd 2828	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 − 𝑋)[,](𝐵 − 𝑋))))
56		eqid 2729	. . . . 5 ⊢ (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠))
57	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑋 ∈ ℂ)
58		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
59	57, 58	addcomd 11336	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℂ) → (𝑋 + 𝑠) = (𝑠 + 𝑋))
60	59	mpteq2dva 5188	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)))
61		eqid 2729	. . . . . . . 8 ⊢ (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) = (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋))
62	61	addccncf 24826	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
63	8, 62	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑠 + 𝑋)) ∈ (ℂ–cn→ℂ))
64	60, 63	eqeltrd 2828	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ (𝑋 + 𝑠)) ∈ (ℂ–cn→ℂ))
65	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
66	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐵 ∈ ℝ)
67	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑋 ∈ ℝ)
68	34	sselda 3937	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ℝ)
69	67, 68	readdcld 11163	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ ℝ)
70		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
71	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
72	33	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
73		elicc2 13332	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑠 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (𝐵 − 𝑋))))
75	70, 74	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑠 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑠 ∧ 𝑠 ≤ (𝐵 − 𝑋)))
76	75	simp2d 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ≤ 𝑠)
77	65, 67, 68	lesubadd2d 11737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝐴 − 𝑋) ≤ 𝑠 ↔ 𝐴 ≤ (𝑋 + 𝑠)))
78	76, 77	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑠))
79	75	simp3d 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑠 ≤ (𝐵 − 𝑋))
80	67, 68, 66	leaddsub2d 11740	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝑋 + 𝑠) ≤ 𝐵 ↔ 𝑠 ≤ (𝐵 − 𝑋)))
81	79, 80	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ≤ 𝐵)
82	65, 66, 69, 78, 81	eliccd 45486	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑠) ∈ (𝐴[,]𝐵))
83	56, 64, 36, 37, 82	cncfmptssg 45853	. . . 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 45865	. . 3 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (((𝐴 − 𝑋)[,](𝐵 − 𝑋))–cn→ℂ))
86		ax-1cn 11086	. . . . . 6 ⊢ 1 ∈ ℂ
87		ioosscn 13329	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
88		cncfmptc 24821	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
89	86, 87, 39, 88	mp3an 1463	. . . . 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 5685	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1)
92		ioombl 25482	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
93	92	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
94		volioo 25486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
95	1, 2, 3, 94	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
96	2, 1	resubcld 11566	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
97	95, 96	eqeltrd 2828	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
98		1cnd 11129	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
99		iblconst 25735	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ (vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ 1 ∈ ℂ) → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
100	93, 97, 98, 99	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
101	91, 100	eqeltrrid 2833	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
102	90, 101	elind 4153	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
103	35	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
104	18	recnd 11162	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 − 𝑋) ∈ ℂ)
105		tgioo4 24709	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106		eqid 2729	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107		iccntr 24726	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
108	20, 107	syl 17	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
109	103, 4, 104, 105, 106, 108	dvmptntr 25891	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))))
110		reelprrecn 11120	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
111	110	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
112		ioossre 13328	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
113	112	sseli 3933	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ ℝ)
114	113	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℝ)
115	114	recnd 11162	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ ℂ)
116		1cnd 11129	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
117	103	sselda 3937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℂ)
118		1cnd 11129	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 1 ∈ ℂ)
119	111	dvmptid 25877	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑡)) = (𝑡 ∈ ℝ ↦ 1))
120	112	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
121		iooretop 24669	. . . . . . 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 25883	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑡)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
124	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℂ)
125		0cnd 11127	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 0 ∈ ℂ)
126	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ ℂ)
127		0cnd 11127	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → 0 ∈ ℂ)
128	111, 8	dvmptc 25878	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑡 ∈ ℝ ↦ 𝑋)) = (𝑡 ∈ ℝ ↦ 0))
129	111, 126, 127, 128, 120, 105, 106, 122	dvmptres 25883	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ 𝑋)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 0))
130	111, 115, 116, 123, 124, 125, 129	dvmptsub 25887	. . . 4 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)))
131	116	subid1d 11482	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (1 − 0) = 1)
132	131	mpteq2dva 5188	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ (1 − 0)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
133	109, 130, 132	3eqtrd 2768	. . 3 ⊢ (𝜑 → (ℝ D (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 − 𝑋))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ 1))
134		oveq2 7361	. . . 4 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝑋 + 𝑠) = (𝑋 + (𝑡 − 𝑋)))
135	134	fveq2d 6830	. . 3 ⊢ (𝑠 = (𝑡 − 𝑋) → (𝐹‘(𝑋 + 𝑠)) = (𝐹‘(𝑋 + (𝑡 − 𝑋))))
136		oveq1 7360	. . 3 ⊢ (𝑡 = 𝐴 → (𝑡 − 𝑋) = (𝐴 − 𝑋))
137		oveq1 7360	. . 3 ⊢ (𝑡 = 𝐵 → (𝑡 − 𝑋) = (𝐵 − 𝑋))
138	1, 2, 3, 55, 85, 102, 133, 135, 136, 137, 32, 33	itgsubsticc 45958	. 2 ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡)
139	124, 115	pncan3d 11496	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝑋 + (𝑡 − 𝑋)) = 𝑡)
140	139	fveq2d 6830	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + (𝑡 − 𝑋))) = (𝐹‘𝑡))
141	140	oveq1d 7368	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = ((𝐹‘𝑡) · 1))
142		cncff 24802	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
143	84, 142	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
144	143	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
145		ioossicc 13354	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
146	145	sseli 3933	. . . . . . 7 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → 𝑡 ∈ (𝐴[,]𝐵))
147	146	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
148	144, 147	ffvelcdmd 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ℂ)
149	148	mulridd 11151	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑡) · 1) = (𝐹‘𝑡))
150	141, 149	eqtrd 2764	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) = (𝐹‘𝑡))
151	3, 150	ditgeq3d 45946	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑋 + (𝑡 − 𝑋))) · 1) d𝑡 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)
152	138, 151	eqtrd 2764	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⊆ wss 3905 {csn 4579 {cpr 4581 class class class wbr 5095 ↦ cmpt 5176 × cxp 5621 dom cdm 5623 ran crn 5624 ↾ cres 5625 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 ≤ cle 11169 − cmin 11365 -cneg 11366 (,)cioo 13266 [,]cicc 13269 TopOpenctopn 17343 topGenctg 17359 ℂ_fldccnfld 21279 intcnt 22920 –cn→ccncf 24785 volcvol 25380 𝐿¹cibl 25534 ⨜cdit 25763 D cdv 25780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cc 10348 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-symdif 4206 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-disj 5063 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-ofr 7618 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-cmp 23290 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-ovol 25381 df-vol 25382 df-mbf 25536 df-itg1 25537 df-itg2 25538 df-ibl 25539 df-itg 25540 df-0p 25587 df-ditg 25764 df-limc 25783 df-dv 25784

This theorem is referenced by: fourierdlem82 46170

W3C validator