itgperiod - Mathbox for Glauco Siliprandi

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Theorem itgperiod 47258

Description: The integral of a periodic function, with period 𝑇 stays the same if the domain of integration is shifted. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
itgperiod.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
itgperiod.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
itgperiod.aleb	⊢ (𝜑 → 𝐴 ≤ 𝐵)
itgperiod.t	⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
itgperiod.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
itgperiod.fper	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
itgperiod.fcn	⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))

**Assertion**
Ref	Expression
itgperiod	⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝑥,𝑇 𝜑,𝑥

Proof of Theorem itgperiod Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itgperiod.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		itgperiod.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		itgperiod.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
4	3	rpred 13202	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		itgperiod.aleb	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 4, 5	leadd1dd 11968	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ≤ (𝐵 + 𝑇))
7	6	ditgpos 26220	. . 3 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐹‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐹‘𝑥) d𝑥)
8	1, 4	readdcld 11375	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
9	2, 4	readdcld 11375	. . . 4 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
10		itgperiod.f	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
11	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐹:ℝ⟶ℂ)
12	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
13	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ)
14		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
15		eliccre 46784	. . . . . 6 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
16	12, 13, 14, 15	syl3anc 1399	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
17	11, 16	ffvelcdmd 7115	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘𝑥) ∈ ℂ)
18	8, 9, 17	itgioo 26180	. . 3 ⊢ (𝜑 → ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥)
19	7, 18	eqtr2d 2799	. 2 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐹‘𝑥) d𝑥)
20		eqid 2763	. . . 4 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) = (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇))
21	4	recnd 11374	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℂ)
22	20	addccncf 25282	. . . . 5 ⊢ (𝑇 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) ∈ (ℂ–cn→ℂ))
23	21, 22	syl 18	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) ∈ (ℂ–cn→ℂ))
24	1, 2	iccssred 13603	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
25		ax-resscn 11294	. . . . 5 ⊢ ℝ ⊆ ℂ
26	24, 25	sstrdi 3925	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
27	8, 9	iccssred 13603	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℝ)
28	27, 25	sstrdi 3925	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℂ)
29	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ∈ ℝ)
30	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐵 + 𝑇) ∈ ℝ)
31	24	sselda 3913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ)
32	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
33	31, 32	readdcld 11375	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ℝ)
34	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
35		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
36	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
37		elicc2 13580	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
38	34, 36, 37	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
39	35, 38	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))
40	39	simp2d 1161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑦)
41	34, 31, 32, 40	leadd1dd 11968	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑦 + 𝑇))
42	39	simp3d 1162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ≤ 𝐵)
43	31, 36, 32, 42	leadd1dd 11968	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ≤ (𝐵 + 𝑇))
44	29, 30, 33, 41, 43	eliccd 46783	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
45	20, 23, 26, 28, 44	cncfmptssg 47148	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
46		eqeq1 2767	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
47	46	rexbidv 3190	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
48		oveq1 7456	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
49	48	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
50	49	cbvrexvw 3245	. . . . . . 7 ⊢ (∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇))
51	47, 50	bitrdi 290	. . . . . 6 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)))
52	51	cbvrabv 3426	. . . . 5 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}
53	10	ffdmd 6768	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
54		simp3 1156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 = (𝑧 + 𝑇))
55	24	sselda 3913	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ∈ ℝ)
56	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
57	55, 56	readdcld 11375	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 + 𝑇) ∈ ℝ)
58	57	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑤 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ℝ)
59	54, 58	eqeltrd 2864	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 ∈ ℝ)
60	59	rexlimdv3a 3171	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ ℝ))
61	60	ralrimivw 3162	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ ℂ (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ ℝ))
62		rabss 3998	. . . . . . 7 ⊢ ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ⊆ ℝ ↔ ∀𝑤 ∈ ℂ (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ ℝ))
63	61, 62	sylibr 237	. . . . . 6 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ⊆ ℝ)
64	10	fdmd 6748	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = ℝ)
65	63, 64	sseqtrrd 3950	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ⊆ dom 𝐹)
66		itgperiod.fper	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
67		itgperiod.fcn	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
68	26, 4, 52, 53, 65, 66, 67	cncfperiod 47156	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) ∈ ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ))
69	47	elrab 3648	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ↔ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
70		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
71		nfv 1949	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝜑
72		nfv 1949	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧 𝑥 ∈ ℂ
73		nfre1 3291	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)
74	72, 73	nfan 1934	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
75	71, 74	nfan 1934	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
76		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))
77		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
78	1	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
79		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ∈ (𝐴[,]𝐵))
80	2	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
81		elicc2 13580	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑧 ∈ (𝐴[,]𝐵) ↔ (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵)))
82	78, 80, 81	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 ∈ (𝐴[,]𝐵) ↔ (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵)))
83	79, 82	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵))
84	83	simp2d 1161	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑧)
85	78, 55, 56, 84	leadd1dd 11968	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑧 + 𝑇))
86	83	simp3d 1162	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ≤ 𝐵)
87	55, 80, 56, 86	leadd1dd 11968	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 + 𝑇) ≤ (𝐵 + 𝑇))
88	57, 85, 87	3jca 1146	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇)))
89	88	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇)))
90	8	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝐴 + 𝑇) ∈ ℝ)
91	9	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝐵 + 𝑇) ∈ ℝ)
92		elicc2 13580	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ) → ((𝑧 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇))))
93	90, 91, 92	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → ((𝑧 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇))))
94	89, 93	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
95	77, 94	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
96	95	3exp 1137	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ (𝐴[,]𝐵) → (𝑥 = (𝑧 + 𝑇) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))))
97	96	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → (𝑧 ∈ (𝐴[,]𝐵) → (𝑥 = (𝑧 + 𝑇) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))))
98	75, 76, 97	rexlimd 3273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → (∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
99	70, 98	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
100	69, 99	sylan2b 606	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
101	16	recnd 11374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℂ)
102	1	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ∈ ℝ)
103	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐵 ∈ ℝ)
104	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℝ)
105	16, 104	resubcld 11781	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
106	1	recnd 11374	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℂ)
107	106, 21	pncand 11707	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
108	107	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
109	108	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
110		elicc2 13580	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ) → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇))))
111	12, 13, 110	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇))))
112	14, 111	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇)))
113	112	simp2d 1161	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
114	12, 16, 104, 113	lesub1dd 11970	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
115	109, 114	eqbrtrd 5131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ≤ (𝑥 − 𝑇))
116	112	simp3d 1162	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
117	16, 13, 104, 116	lesub1dd 11970	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ ((𝐵 + 𝑇) − 𝑇))
118	2	recnd 11374	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℂ)
119	118, 21	pncand 11707	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
120	119	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
121	117, 120	breqtrd 5135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
122	102, 103, 105, 115, 121	eliccd 46783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
123	21	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℂ)
124	101, 123	npcand 11710	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
125	124	eqcomd 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
126		oveq1 7456	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑧 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
127	126	rspceeqv 3602	. . . . . . . . . 10 ⊢ (((𝑥 − 𝑇) ∈ (𝐴[,]𝐵) ∧ 𝑥 = ((𝑥 − 𝑇) + 𝑇)) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
128	122, 125, 127	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
129	101, 128, 69	sylanbrc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})
130	100, 129	impbida 813	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ↔ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
131	130	eqrdv 2761	. . . . . 6 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} = ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
132	131	reseq2d 5984	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) = (𝐹 ↾ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
133	131, 65	eqsstrrd 3948	. . . . . 6 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ dom 𝐹)
134	53, 133	feqresmpt 6983	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘𝑥)))
135	132, 134	eqtr2d 2799	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘𝑥)) = (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}))
136	1, 2, 4	iccshift 46797	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})
137	136	oveq1d 7464	. . . 4 ⊢ (𝜑 → (((𝐴 + 𝑇)[,](𝐵 + 𝑇))–cn→ℂ) = ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ))
138	68, 135, 137	3eltr4d 2879	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘𝑥)) ∈ (((𝐴 + 𝑇)[,](𝐵 + 𝑇))–cn→ℂ))
139		ioosscn 13577	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
140	139	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
141		1cnd 11339	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
142		ssid 3935	. . . . . 6 ⊢ ℂ ⊆ ℂ
143	142	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
144	140, 141, 143	constcncfg 47149	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
145		fconstmpt 5718	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1)
146		ioombl 25930	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
147	146	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
148		ioovolcl 25935	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
149	1, 2, 148	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
150		iblconst 26182	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ (vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ 1 ∈ ℂ) → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
151	147, 149, 141, 150	syl3anc 1399	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
152	145, 151	eqeltrrid 2869	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
153	144, 152	elind 4100	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
154	24	resmptd 6048	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵)) = (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)))
155	154	eqcomd 2769	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) = ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵)))
156	155	oveq2d 7465	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = (ℝ D ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵))))
157	25	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
158	157	sselda 3913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
159	21	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℂ)
160	158, 159	addcld 11365	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑇) ∈ ℂ)
161	160	fmpttd 7145	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)):ℝ⟶ℂ)
162		ssid 3935	. . . . . . 7 ⊢ ℝ ⊆ ℝ
163	162	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℝ)
164		eqid 2763	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
165		tgioo4 25168	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
166	164, 165	dvres 26275	. . . . . 6 ⊢ (((ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)):ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ)) → (ℝ D ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
167	157, 161, 163, 24, 166	syl22anc 852	. . . . 5 ⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
168	156, 167	eqtrd 2798	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
169		iccntr 25185	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
170	1, 2, 169	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
171	170	reseq2d 5984	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)))
172		reelprrecn 11329	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
173	172	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
174		1cnd 11339	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
175	173	dvmptid 26321	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
176		0cnd 11336	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℂ)
177	173, 21	dvmptc 26322	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑇)) = (𝑦 ∈ ℝ ↦ 0))
178	173, 158, 174, 175, 159, 176, 177	dvmptadd 26324	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) = (𝑦 ∈ ℝ ↦ (1 + 0)))
179	178	reseq1d 5983	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)))
180		ioossre 13576	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
181	180	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
182	181	resmptd 6048	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)))
183		1p0e1 12503	. . . . . . 7 ⊢ (1 + 0) = 1
184	183	mpteq2i 5205	. . . . . 6 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1)
185	184	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
186	179, 182, 185	3eqtrd 2802	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
187	168, 171, 186	3eqtrd 2802	. . 3 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
188		fveq2 6914	. . 3 ⊢ (𝑥 = (𝑦 + 𝑇) → (𝐹‘𝑥) = (𝐹‘(𝑦 + 𝑇)))
189		oveq1 7456	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑇) = (𝐴 + 𝑇))
190		oveq1 7456	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 + 𝑇) = (𝐵 + 𝑇))
191	1, 2, 5, 45, 138, 153, 187, 188, 189, 190, 8, 9	itgsubsticc 47253	. 2 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐹‘𝑥) d𝑥 = ⨜[𝐴 → 𝐵]((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦)
192	5	ditgpos 26220	. . 3 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴(,)𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦)
193	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℂ)
194	193, 33	ffvelcdmd 7115	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) ∈ ℂ)
195		1cnd 11339	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 1 ∈ ℂ)
196	194, 195	mulcld 11366	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑦 + 𝑇)) · 1) ∈ ℂ)
197	1, 2, 196	itgioo 26180	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦)
198		fvoveq1 7472	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘(𝑥 + 𝑇)))
199	198	oveq1d 7464	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘(𝑦 + 𝑇)) · 1) = ((𝐹‘(𝑥 + 𝑇)) · 1))
200	199	cbvitgv 26141	. . . 4 ⊢ ∫(𝐴[,]𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐹‘(𝑥 + 𝑇)) · 1) d𝑥
201	10	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℂ)
202	24	sselda 3913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
203	4	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
204	202, 203	readdcld 11375	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ∈ ℝ)
205	201, 204	ffvelcdmd 7115	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) ∈ ℂ)
206	205	mulridd 11363	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑥 + 𝑇)) · 1) = (𝐹‘(𝑥 + 𝑇)))
207	206, 66	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑥 + 𝑇)) · 1) = (𝐹‘𝑥))
208	207	itgeq2dv 26146	. . . 4 ⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝐹‘(𝑥 + 𝑇)) · 1) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
209	200, 208	eqtrid 2810	. . 3 ⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
210	192, 197, 209	3eqtrd 2802	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
211	19, 191, 210	3eqtrd 2802	1 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 {crab 3416 ⊆ wss 3900 {csn 4588 {cpr 4590 class class class wbr 5106 ↦ cmpt 5190 × cxp 5654 dom cdm 5656 ran crn 5657 ↾ cres 5658 ⟶wf 6560 ‘cfv 6564 (class class class)co 7449 ℂcc 11235 ℝcr 11236 0cc0 11237 1c1 11238 + caddc 11240 · cmul 11242 ≤ cle 11381 − cmin 11578 ℝ⁺crp 13158 (,)cioo 13514 [,]cicc 13517 TopOpenctopn 17633 topGenctg 17649 ℂ_fldccnfld 21720 intcnt 23379 –cn→ccncf 25241 volcvol 25828 𝐿¹cibl 25982 ∫citg 25983 ⨜cdit 26210 D cdv 26227

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-inf2 9677 ax-cc 10550 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314 ax-pre-sup 11315 ax-addf 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-symdif 4201 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5074 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-isom 6573 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-of 7723 df-ofr 7724 df-om 7910 df-1st 8033 df-2nd 8034 df-supp 8213 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-2o 8513 df-oadd 8516 df-omul 8517 df-er 8753 df-map 8885 df-pm 8886 df-ixp 8962 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-fsupp 9390 df-fi 9438 df-sup 9469 df-inf 9470 df-oi 9539 df-dju 10017 df-card 10055 df-acn 10058 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-3 12444 df-4 12445 df-5 12446 df-6 12447 df-7 12448 df-8 12449 df-9 12450 df-n0 12645 df-z 12732 df-dec 12853 df-uz 13004 df-q 13114 df-rp 13159 df-xneg 13279 df-xadd 13280 df-xmul 13281 df-ioo 13518 df-ioc 13519 df-ico 13520 df-icc 13521 df-fz 13678 df-fzo 13827 df-fl 13970 df-mod 14048 df-seq 14183 df-exp 14243 df-hash 14514 df-cj 15307 df-re 15308 df-im 15309 df-sqrt 15443 df-abs 15444 df-limsup 15679 df-clim 15696 df-rlim 15697 df-sum 15895 df-struct 17366 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-mulr 17483 df-starv 17484 df-sca 17485 df-vsca 17486 df-ip 17487 df-tset 17488 df-ple 17489 df-ds 17491 df-unif 17492 df-hom 17493 df-cco 17494 df-rest 17634 df-topn 17635 df-0g 17653 df-gsum 17654 df-topgen 17655 df-pt 17656 df-prds 17659 df-xrs 17715 df-qtop 17720 df-imas 17721 df-xps 17723 df-mre 17797 df-mrc 17798 df-acs 17800 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-submnd 19020 df-mulg 19319 df-cntz 19573 df-cmn 20038 df-psmet 21712 df-xmet 21713 df-met 21714 df-bl 21715 df-mopn 21716 df-fbas 21717 df-fg 21718 df-cnfld 21721 df-top 23256 df-topon 23273 df-topsp 23295 df-bases 23308 df-cld 23381 df-ntr 23382 df-cls 23383 df-nei 23460 df-lp 23498 df-perf 23499 df-cn 23589 df-cnp 23590 df-haus 23677 df-cmp 23749 df-tx 23925 df-hmeo 24118 df-fil 24209 df-fm 24301 df-flim 24302 df-flf 24303 df-xms 24683 df-ms 24684 df-tms 24685 df-cncf 25243 df-ovol 25829 df-vol 25830 df-mbf 25984 df-itg1 25985 df-itg2 25986 df-ibl 25987 df-itg 25988 df-0p 26035 df-ditg 26211 df-limc 26230 df-dv 26231

This theorem is used by: (None)

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