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Theorem itgparts 26025

Description: Integration by parts. If 𝐵(𝑥) is the derivative of 𝐴(𝑥) and 𝐷(𝑥) is the derivative of 𝐶(𝑥), and 𝐸 = (𝐴 · 𝐵)(𝑋) and 𝐹 = (𝐴 · 𝐵)(𝑌), then under suitable integrability and differentiability assumptions, the integral of 𝐴 · 𝐷 from 𝑋 to 𝑌 is equal to 𝐹 − 𝐸 minus the integral of 𝐵 · 𝐶. (Contributed by Mario Carneiro, 3-Sep-2014.)

**Hypotheses**
Ref	Expression
itgparts.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
itgparts.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
itgparts.le	⊢ (𝜑 → 𝑋 ≤ 𝑌)
itgparts.a	⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ))
itgparts.c	⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ))
itgparts.b	⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ))
itgparts.d	⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ))
itgparts.ad	⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿¹)
itgparts.bc	⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿¹)
itgparts.da	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
itgparts.dc	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷))
itgparts.e	⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴 · 𝐶) = 𝐸)
itgparts.f	⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝐴 · 𝐶) = 𝐹)

**Assertion**
Ref	Expression
itgparts	⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 = ((𝐹 − 𝐸) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥))

Distinct variable groups: 𝜑,𝑥 𝑥,𝑋 𝑥,𝑌 𝑥,𝐸 𝑥,𝐹 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝐶(𝑥) 𝐷(𝑥)

Proof of Theorem itgparts Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		itgparts.b	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ))
2		cncff 24857	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ) → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵):(𝑋(,)𝑌)⟶ℂ)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵):(𝑋(,)𝑌)⟶ℂ)
4	3	fvmptelcdm 7067	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐵 ∈ ℂ)
5		ioossicc 13361	. . . . . . 7 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
6	5	sseli 3931	. . . . . 6 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
7		itgparts.c	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ))
8		cncff 24857	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶):(𝑋[,]𝑌)⟶ℂ)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶):(𝑋[,]𝑌)⟶ℂ)
10	9	fvmptelcdm 7067	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐶 ∈ ℂ)
11	6, 10	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ)
12	4, 11	mulcld 11164	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐵 · 𝐶) ∈ ℂ)
13		itgparts.bc	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿¹)
14	12, 13	itgcl 25756	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 ∈ ℂ)
15		itgparts.a	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ))
16		cncff 24857	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶ℂ)
17	15, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶ℂ)
18	17	fvmptelcdm 7067	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ ℂ)
19	6, 18	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℂ)
20		itgparts.d	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ))
21		cncff 24857	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ) → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷):(𝑋(,)𝑌)⟶ℂ)
22	20, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷):(𝑋(,)𝑌)⟶ℂ)
23	22	fvmptelcdm 7067	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐷 ∈ ℂ)
24	19, 23	mulcld 11164	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐴 · 𝐷) ∈ ℂ)
25		itgparts.ad	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿¹)
26	24, 25	itgcl 25756	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 ∈ ℂ)
27	14, 26	pncan2d 11506	. 2 ⊢ (𝜑 → ((∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥) = ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥)
28	12, 13, 24, 25	itgadd 25797	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥))
29		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡))
30		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑡((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥)
31		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥ℝ
32		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥 D
33		nfmpt1 5199	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
34	31, 32, 33	nfov 7398	. . . . . . . 8 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))
35		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
36	34, 35	nffv 6852	. . . . . . 7 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡)
37	29, 30, 36	cbvitg 25748	. . . . . 6 ⊢ ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) d𝑥 = ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡) d𝑡
38		itgparts.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
39		itgparts.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ℝ)
40		itgparts.le	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
41		ax-resscn 11095	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
42	41	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
43		iccssre 13357	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
44	38, 39, 43	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
45	18, 10	mulcld 11164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → (𝐴 · 𝐶) ∈ ℂ)
46		tgioo4 24764	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
47		eqid 2737	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
48		iccntr 24781	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌))
49	38, 39, 48	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌))
50	42, 44, 45, 46, 47, 49	dvmptntr 25946	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))))
51		reelprrecn 11130	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
52	51	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
53	42, 44, 18, 46, 47, 49	dvmptntr 25946	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴)))
54		itgparts.da	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
55	53, 54	eqtr3d 2774	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
56	42, 44, 10, 46, 47, 49	dvmptntr 25946	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶)) = (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶)))
57		itgparts.dc	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷))
58	56, 57	eqtr3d 2774	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷))
59	52, 19, 4, 55, 11, 23, 58	dvmptmul 25936	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
60	23, 19	mulcomd 11165	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐷 · 𝐴) = (𝐴 · 𝐷))
61	60	oveq2d 7384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐵 · 𝐶) + (𝐷 · 𝐴)) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
62	61	mpteq2dva 5193	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
63	50, 59, 62	3eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
64	47	addcn 24825	. . . . . . . . . 10 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
65	64	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
66		resmpt 6004	. . . . . . . . . . . 12 ⊢ ((𝑋(,)𝑌) ⊆ (𝑋[,]𝑌) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ↾ (𝑋(,)𝑌)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶))
67	5, 66	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ↾ (𝑋(,)𝑌)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶)
68		rescncf 24861	. . . . . . . . . . . 12 ⊢ ((𝑋(,)𝑌) ⊆ (𝑋[,]𝑌) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ)))
69	5, 7, 68	mpsyl 68	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
70	67, 69	eqeltrrid 2842	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ ((𝑋(,)𝑌)–cn→ℂ))
71	1, 70	mulcncf 25417	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
72		resmpt 6004	. . . . . . . . . . . 12 ⊢ ((𝑋(,)𝑌) ⊆ (𝑋[,]𝑌) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ↾ (𝑋(,)𝑌)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴))
73	5, 72	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ↾ (𝑋(,)𝑌)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴)
74		rescncf 24861	. . . . . . . . . . . 12 ⊢ ((𝑋(,)𝑌) ⊆ (𝑋[,]𝑌) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ)))
75	5, 15, 74	mpsyl 68	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
76	73, 75	eqeltrrid 2842	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴) ∈ ((𝑋(,)𝑌)–cn→ℂ))
77	76, 20	mulcncf 25417	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
78	47, 65, 71, 77	cncfmpt2f 24879	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
79	63, 78	eqeltrd 2837	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
80	12, 13, 24, 25	ibladd 25793	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ 𝐿¹)
81	63, 80	eqeltrd 2837	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ 𝐿¹)
82	15, 7	mulcncf 25417	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) ∈ ((𝑋[,]𝑌)–cn→ℂ))
83	38, 39, 40, 79, 81, 82	ftc2 26022	. . . . . 6 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡) d𝑡 = (((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) − ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋)))
84	37, 83	eqtrid 2784	. . . . 5 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) d𝑥 = (((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) − ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋)))
85	63	fveq1d 6844	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥))
86	85	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥))
87		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋(,)𝑌))
88		ovex 7401	. . . . . . . 8 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) ∈ V
89		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
90	89	fvmpt2 6961	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑋(,)𝑌) ∧ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) ∈ V) → ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
91	87, 88, 90	sylancl 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
92	86, 91	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
93	92	itgeq2dv 25754	. . . . 5 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) d𝑥 = ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥)
94	38	rexrd 11194	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
95	39	rexrd 11194	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
96		ubicc2 13393	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ (𝑋[,]𝑌))
97	94, 95, 40, 96	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋[,]𝑌))
98		ovex 7401	. . . . . . . . 9 ⊢ (𝐴 · 𝐶) ∈ V
99	98	csbex 5258	. . . . . . . 8 ⊢ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V
100		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) = (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
101	100	fvmpts 6953	. . . . . . . 8 ⊢ ((𝑌 ∈ (𝑋[,]𝑌) ∧ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶))
102	97, 99, 101	sylancl 587	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶))
103		itgparts.f	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝐴 · 𝐶) = 𝐹)
104	39, 103	csbied 3887	. . . . . . 7 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) = 𝐹)
105	102, 104	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = 𝐹)
106		lbicc2 13392	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ (𝑋[,]𝑌))
107	94, 95, 40, 106	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑋[,]𝑌))
108	98	csbex 5258	. . . . . . . 8 ⊢ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V
109	100	fvmpts 6953	. . . . . . . 8 ⊢ ((𝑋 ∈ (𝑋[,]𝑌) ∧ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶))
110	107, 108, 109	sylancl 587	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶))
111		itgparts.e	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴 · 𝐶) = 𝐸)
112	38, 111	csbied 3887	. . . . . . 7 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) = 𝐸)
113	110, 112	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = 𝐸)
114	105, 113	oveq12d 7386	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) − ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋)) = (𝐹 − 𝐸))
115	84, 93, 114	3eqtr3d 2780	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (𝐹 − 𝐸))
116	28, 115	eqtr3d 2774	. . 3 ⊢ (𝜑 → (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥) = (𝐹 − 𝐸))
117	116	oveq1d 7383	. 2 ⊢ (𝜑 → ((∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥) = ((𝐹 − 𝐸) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥))
118	27, 117	eqtr3d 2774	1 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 = ((𝐹 − 𝐸) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ⦋csb 3851 ⊆ wss 3903 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 ran crn 5633 ↾ cres 5634 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 + caddc 11041 · cmul 11043 ℝ^*cxr 11177 ≤ cle 11179 − cmin 11376 (,)cioo 13273 [,]cicc 13276 TopOpenctopn 17353 topGenctg 17369 ℂ_fldccnfld 21324 intcnt 22976 Cn ccn 23183 ×_t ctx 23519 –cn→ccncf 24840 𝐿¹cibl 25589 ∫citg 25590 D cdv 25835

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4207 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19013 df-cntz 19261 df-cmn 19726 df-psmet 21316 df-xmet 21317 df-met 21318 df-bl 21319 df-mopn 21320 df-fbas 21321 df-fg 21322 df-cnfld 21325 df-top 22853 df-topon 22870 df-topsp 22892 df-bases 22905 df-cld 22978 df-ntr 22979 df-cls 22980 df-nei 23057 df-lp 23095 df-perf 23096 df-cn 23186 df-cnp 23187 df-haus 23274 df-cmp 23346 df-tx 23521 df-hmeo 23714 df-fil 23805 df-fm 23897 df-flim 23898 df-flf 23899 df-xms 24279 df-ms 24280 df-tms 24281 df-cncf 24842 df-ovol 25436 df-vol 25437 df-mbf 25591 df-itg1 25592 df-itg2 25593 df-ibl 25594 df-itg 25595 df-0p 25642 df-limc 25838 df-dv 25839

This theorem is referenced by: lcmineqlem10 42412 itgsinexplem1 46316 fourierdlem39 46508

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