itgperiod - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > itgperiod		Structured version Visualization version GIF version

Theorem itgperiod 46431

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 12984	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		itgperiod.aleb	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 4, 5	leadd1dd 11762	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ≤ (𝐵 + 𝑇))
7	6	ditgpos 25848	. . 3 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐹‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐹‘𝑥) d𝑥)
8	1, 4	readdcld 11172	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
9	2, 4	readdcld 11172	. . . 4 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
10		itgperiod.f	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
11	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐹:ℝ⟶ℂ)
12	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
13	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ)
14		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
15		eliccre 45957	. . . . . 6 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
16	12, 13, 14, 15	syl3anc 1379	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
17	11, 16	ffvelcdmd 7033	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘𝑥) ∈ ℂ)
18	8, 9, 17	itgioo 25808	. . 3 ⊢ (𝜑 → ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥)
19	7, 18	eqtr2d 2776	. 2 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐹‘𝑥) d𝑥)
20		eqid 2740	. . . 4 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) = (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇))
21	4	recnd 11171	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℂ)
22	20	addccncf 24909	. . . . 5 ⊢ (𝑇 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) ∈ (ℂ–cn→ℂ))
23	21, 22	syl 17	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) ∈ (ℂ–cn→ℂ))
24	1, 2	iccssred 13385	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
25		ax-resscn 11093	. . . . 5 ⊢ ℝ ⊆ ℂ
26	24, 25	sstrdi 3934	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
27	8, 9	iccssred 13385	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℝ)
28	27, 25	sstrdi 3934	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℂ)
29	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ∈ ℝ)
30	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐵 + 𝑇) ∈ ℝ)
31	24	sselda 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ)
32	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
33	31, 32	readdcld 11172	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ℝ)
34	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
35		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
36	2	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
37		elicc2 13362	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
38	34, 36, 37	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
39	35, 38	mpbid 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))
40	39	simp2d 1149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑦)
41	34, 31, 32, 40	leadd1dd 11762	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑦 + 𝑇))
42	39	simp3d 1150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ≤ 𝐵)
43	31, 36, 32, 42	leadd1dd 11762	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ≤ (𝐵 + 𝑇))
44	29, 30, 33, 41, 43	eliccd 45956	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
45	20, 23, 26, 28, 44	cncfmptssg 46321	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
46		eqeq1 2744	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
47	46	rexbidv 3164	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
48		oveq1 7370	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
49	48	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
50	49	cbvrexvw 3219	. . . . . . 7 ⊢ (∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇))
51	47, 50	bitrdi 288	. . . . . 6 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)))
52	51	cbvrabv 3402	. . . . 5 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}
53	10	ffdmd 6692	. . . . 5 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)
54		simp3 1144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 = (𝑧 + 𝑇))
55	24	sselda 3922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ∈ ℝ)
56	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
57	55, 56	readdcld 11172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 + 𝑇) ∈ ℝ)
58	57	3adant3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑤 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ℝ)
59	54, 58	eqeltrd 2840	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑤 = (𝑧 + 𝑇)) → 𝑤 ∈ ℝ)
60	59	rexlimdv3a 3145	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ ℝ))
61	60	ralrimivw 3136	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ ℂ (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ ℝ))
62		rabss 4008	. . . . . . 7 ⊢ ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ⊆ ℝ ↔ ∀𝑤 ∈ ℂ (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) → 𝑤 ∈ ℝ))
63	61, 62	sylibr 235	. . . . . 6 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ⊆ ℝ)
64	10	fdmd 6672	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = ℝ)
65	63, 64	sseqtrrd 3959	. . . . 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 46329	. . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) ∈ ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ))
69	47	elrab 3636	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ↔ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
70		simprr 778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
71		nfv 1921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧𝜑
72		nfv 1921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧 𝑥 ∈ ℂ
73		nfre1 3265	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)
74	72, 73	nfan 1906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
75	71, 74	nfan 1906	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
76		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))
77		simp3 1144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
78	1	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
79		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ∈ (𝐴[,]𝐵))
80	2	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
81		elicc2 13362	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑧 ∈ (𝐴[,]𝐵) ↔ (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵)))
82	78, 80, 81	syl2anc 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 ∈ (𝐴[,]𝐵) ↔ (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵)))
83	79, 82	mpbid 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵))
84	83	simp2d 1149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑧)
85	78, 55, 56, 84	leadd1dd 11762	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑧 + 𝑇))
86	83	simp3d 1150	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ≤ 𝐵)
87	55, 80, 56, 86	leadd1dd 11762	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 + 𝑇) ≤ (𝐵 + 𝑇))
88	57, 85, 87	3jca 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇)))
89	88	3adant3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇)))
90	8	3ad2ant1 1139	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝐴 + 𝑇) ∈ ℝ)
91	9	3ad2ant1 1139	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝐵 + 𝑇) ∈ ℝ)
92		elicc2 13362	. . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
95	77, 94	eqeltrd 2840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
96	95	3exp 1125	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ (𝐴[,]𝐵) → (𝑥 = (𝑧 + 𝑇) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))))
97	96	adantr 481	. . . . . . . . . . 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 3247	. . . . . . . . . 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 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
100	69, 99	sylan2b 600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
101	16	recnd 11171	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℂ)
102	1	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ∈ ℝ)
103	2	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐵 ∈ ℝ)
104	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℝ)
105	16, 104	resubcld 11576	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
106	1	recnd 11171	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℂ)
107	106, 21	pncand 11504	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
108	107	eqcomd 2746	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
109	108	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
110		elicc2 13362	. . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇))))
112	14, 111	mpbid 233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇)))
113	112	simp2d 1149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
114	12, 16, 104, 113	lesub1dd 11764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
115	109, 114	eqbrtrd 5101	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ≤ (𝑥 − 𝑇))
116	112	simp3d 1150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
117	16, 13, 104, 116	lesub1dd 11764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ ((𝐵 + 𝑇) − 𝑇))
118	2	recnd 11171	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℂ)
119	118, 21	pncand 11504	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
120	119	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
121	117, 120	breqtrd 5105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
122	102, 103, 105, 115, 121	eliccd 45956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
123	21	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℂ)
124	101, 123	npcand 11507	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
125	124	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
126		oveq1 7370	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑧 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
127	126	rspceeqv 3590	. . . . . . . . . 10 ⊢ (((𝑥 − 𝑇) ∈ (𝐴[,]𝐵) ∧ 𝑥 = ((𝑥 − 𝑇) + 𝑇)) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
128	122, 125, 127	syl2anc 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
129	101, 128, 69	sylanbrc 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})
130	100, 129	impbida 806	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ↔ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
131	130	eqrdv 2738	. . . . . 6 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} = ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
132	131	reseq2d 5938	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) = (𝐹 ↾ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
133	131, 65	eqsstrrd 3957	. . . . . 6 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ dom 𝐹)
134	53, 133	feqresmpt 6903	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘𝑥)))
135	132, 134	eqtr2d 2776	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘𝑥)) = (𝐹 ↾ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}))
136	1, 2, 4	iccshift 45970	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})
137	136	oveq1d 7378	. . . 4 ⊢ (𝜑 → (((𝐴 + 𝑇)[,](𝐵 + 𝑇))–cn→ℂ) = ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ))
138	68, 135, 137	3eltr4d 2855	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘𝑥)) ∈ (((𝐴 + 𝑇)[,](𝐵 + 𝑇))–cn→ℂ))
139		ioosscn 13359	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
140	139	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
141		1cnd 11137	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
142		ssid 3944	. . . . . 6 ⊢ ℂ ⊆ ℂ
143	142	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
144	140, 141, 143	constcncfg 46322	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
145		fconstmpt 5687	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1)
146		ioombl 25557	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
147	146	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
148		ioovolcl 25562	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
149	1, 2, 148	syl2anc 590	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
150		iblconst 25810	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ (vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ 1 ∈ ℂ) → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
151	147, 149, 141, 150	syl3anc 1379	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
152	145, 151	eqeltrrid 2845	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
153	144, 152	elind 4136	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
154	24	resmptd 5999	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵)) = (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)))
155	154	eqcomd 2746	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) = ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵)))
156	155	oveq2d 7379	. . . . 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 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
159	21	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℂ)
160	158, 159	addcld 11162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑇) ∈ ℂ)
161	160	fmpttd 7063	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)):ℝ⟶ℂ)
162		ssid 3944	. . . . . . 7 ⊢ ℝ ⊆ ℝ
163	162	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℝ)
164		eqid 2740	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
165		tgioo4 24795	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
166	164, 165	dvres 25903	. . . . . 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 844	. . . . 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 2775	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
169		iccntr 24812	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
170	1, 2, 169	syl2anc 590	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
171	170	reseq2d 5938	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)))
172		reelprrecn 11128	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
173	172	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
174		1cnd 11137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
175	173	dvmptid 25949	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
176		0cnd 11135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℂ)
177	173, 21	dvmptc 25950	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑇)) = (𝑦 ∈ ℝ ↦ 0))
178	173, 158, 174, 175, 159, 176, 177	dvmptadd 25952	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) = (𝑦 ∈ ℝ ↦ (1 + 0)))
179	178	reseq1d 5937	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)))
180		ioossre 13358	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
181	180	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
182	181	resmptd 5999	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)))
183		1p0e1 12298	. . . . . . 7 ⊢ (1 + 0) = 1
184	183	mpteq2i 5175	. . . . . 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 2779	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
187	168, 171, 186	3eqtrd 2779	. . 3 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
188		fveq2 6834	. . 3 ⊢ (𝑥 = (𝑦 + 𝑇) → (𝐹‘𝑥) = (𝐹‘(𝑦 + 𝑇)))
189		oveq1 7370	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑇) = (𝐴 + 𝑇))
190		oveq1 7370	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 + 𝑇) = (𝐵 + 𝑇))
191	1, 2, 5, 45, 138, 153, 187, 188, 189, 190, 8, 9	itgsubsticc 46426	. 2 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐹‘𝑥) d𝑥 = ⨜[𝐴 → 𝐵]((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦)
192	5	ditgpos 25848	. . 3 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴(,)𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦)
193	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℂ)
194	193, 33	ffvelcdmd 7033	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑦 + 𝑇)) ∈ ℂ)
195		1cnd 11137	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 1 ∈ ℂ)
196	194, 195	mulcld 11163	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑦 + 𝑇)) · 1) ∈ ℂ)
197	1, 2, 196	itgioo 25808	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦)
198		fvoveq1 7386	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘(𝑥 + 𝑇)))
199	198	oveq1d 7378	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘(𝑦 + 𝑇)) · 1) = ((𝐹‘(𝑥 + 𝑇)) · 1))
200	199	cbvitgv 25769	. . . 4 ⊢ ∫(𝐴[,]𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐹‘(𝑥 + 𝑇)) · 1) d𝑥
201	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℂ)
202	24	sselda 3922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
203	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
204	202, 203	readdcld 11172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ∈ ℝ)
205	201, 204	ffvelcdmd 7033	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) ∈ ℂ)
206	205	mulridd 11160	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑥 + 𝑇)) · 1) = (𝐹‘(𝑥 + 𝑇)))
207	206, 66	eqtrd 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘(𝑥 + 𝑇)) · 1) = (𝐹‘𝑥))
208	207	itgeq2dv 25774	. . . 4 ⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝐹‘(𝑥 + 𝑇)) · 1) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
209	200, 208	eqtrid 2787	. . 3 ⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
210	192, 197, 209	3eqtrd 2779	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐹‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
211	19, 191, 210	3eqtrd 2779	1 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3054 ∃wrex 3064 {crab 3392 ⊆ wss 3890 {csn 4562 {cpr 4564 class class class wbr 5079 ↦ cmpt 5160 × cxp 5623 dom cdm 5625 ran crn 5626 ↾ cres 5627 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 ≤ cle 11178 − cmin 11375 ℝ⁺crp 12940 (,)cioo 13296 [,]cicc 13299 TopOpenctopn 17382 topGenctg 17398 ℂ_fldccnfld 21354 intcnt 23007 –cn→ccncf 24868 volcvol 25455 𝐿¹cibl 25609 ∫citg 25610 ⨜cdit 25838 D cdv 25855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cc 10355 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-symdif 4188 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-disj 5047 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-ofr 7628 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-oadd 8406 df-omul 8407 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-dju 9823 df-card 9861 df-acn 9864 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-cmp 23377 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-ovol 25456 df-vol 25457 df-mbf 25611 df-itg1 25612 df-itg2 25613 df-ibl 25614 df-itg 25615 df-0p 25662 df-ditg 25839 df-limc 25858 df-dv 25859

This theorem is referenced by: (None)

W3C validator