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Theorem itgparts 26014

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 24860	. . . . . . 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 7065	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐵 ∈ ℂ)
5		ioossicc 13386	. . . . . . 7 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
6	5	sseli 3917	. . . . . 6 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
7		itgparts.c	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ))
8		cncff 24860	. . . . . . . 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 7065	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐶 ∈ ℂ)
11	6, 10	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ)
12	4, 11	mulcld 11165	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐵 · 𝐶) ∈ ℂ)
13		itgparts.bc	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿¹)
14	12, 13	itgcl 25751	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 ∈ ℂ)
15		itgparts.a	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ))
16		cncff 24860	. . . . . . . 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 7065	. . . . . 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 24860	. . . . . . 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 7065	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐷 ∈ ℂ)
24	19, 23	mulcld 11165	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐴 · 𝐷) ∈ ℂ)
25		itgparts.ad	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿¹)
26	24, 25	itgcl 25751	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 ∈ ℂ)
27	14, 26	pncan2d 11507	. 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 25792	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥))
29		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡))
30		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑡((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥)
31		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑥ℝ
32		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑥 D
33		nfmpt1 5184	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
34	31, 32, 33	nfov 7397	. . . . . . . 8 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))
35		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
36	34, 35	nffv 6850	. . . . . . 7 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡)
37	29, 30, 36	cbvitg 25743	. . . . . 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 13382	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
44	38, 39, 43	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
45	18, 10	mulcld 11165	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → (𝐴 · 𝐶) ∈ ℂ)
46		tgioo4 24770	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
47		eqid 2736	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
48		iccntr 24787	. . . . . . . . . . 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 25938	. . . . . . . . 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 25938	. . . . . . . . . . 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 2773	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
56	42, 44, 10, 46, 47, 49	dvmptntr 25938	. . . . . . . . . . 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 2773	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷))
59	52, 19, 4, 55, 11, 23, 58	dvmptmul 25928	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
60	23, 19	mulcomd 11166	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐷 · 𝐴) = (𝐴 · 𝐷))
61	60	oveq2d 7383	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐵 · 𝐶) + (𝐷 · 𝐴)) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
62	61	mpteq2dva 5178	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
63	50, 59, 62	3eqtrd 2775	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
64	47	addcn 24831	. . . . . . . . . 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 6002	. . . . . . . . . . . 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 24864	. . . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ ((𝑋(,)𝑌)–cn→ℂ))
71	1, 70	mulcncf 25413	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
72		resmpt 6002	. . . . . . . . . . . 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 24864	. . . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴) ∈ ((𝑋(,)𝑌)–cn→ℂ))
77	76, 20	mulcncf 25413	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
78	47, 65, 71, 77	cncfmpt2f 24882	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
79	63, 78	eqeltrd 2836	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
80	12, 13, 24, 25	ibladd 25788	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ 𝐿¹)
81	63, 80	eqeltrd 2836	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ 𝐿¹)
82	15, 7	mulcncf 25413	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) ∈ ((𝑋[,]𝑌)–cn→ℂ))
83	38, 39, 40, 79, 81, 82	ftc2 26011	. . . . . 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 2783	. . . . 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 6842	. . . . . . . 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 7400	. . . . . . . 8 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) ∈ V
89		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
90	89	fvmpt2 6959	. . . . . . . 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 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
93	92	itgeq2dv 25749	. . . . 5 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) d𝑥 = ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥)
94	38	rexrd 11195	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
95	39	rexrd 11195	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
96		ubicc2 13418	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ (𝑋[,]𝑌))
97	94, 95, 40, 96	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋[,]𝑌))
98		ovex 7400	. . . . . . . . 9 ⊢ (𝐴 · 𝐶) ∈ V
99	98	csbex 5246	. . . . . . . 8 ⊢ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V
100		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) = (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
101	100	fvmpts 6951	. . . . . . . 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 3873	. . . . . . 7 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) = 𝐹)
105	102, 104	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = 𝐹)
106		lbicc2 13417	. . . . . . . . 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 5246	. . . . . . . 8 ⊢ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V
109	100	fvmpts 6951	. . . . . . . 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 3873	. . . . . . 7 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) = 𝐸)
113	110, 112	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = 𝐸)
114	105, 113	oveq12d 7385	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) − ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋)) = (𝐹 − 𝐸))
115	84, 93, 114	3eqtr3d 2779	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (𝐹 − 𝐸))
116	28, 115	eqtr3d 2773	. . 3 ⊢ (𝜑 → (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥) = (𝐹 − 𝐸))
117	116	oveq1d 7382	. 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 2773	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 3429 ⦋csb 3837 ⊆ wss 3889 {cpr 4569 class class class wbr 5085 ↦ cmpt 5166 ran crn 5632 ↾ cres 5633 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 + caddc 11041 · cmul 11043 ℝ^*cxr 11178 ≤ cle 11180 − cmin 11377 (,)cioo 13298 [,]cicc 13301 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21352 intcnt 22982 Cn ccn 23189 ×_t ctx 23525 –cn→ccncf 24843 𝐿¹cibl 25584 ∫citg 25585 D cdv 25830

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-symdif 4193 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-ovol 25431 df-vol 25432 df-mbf 25586 df-itg1 25587 df-itg2 25588 df-ibl 25589 df-itg 25590 df-0p 25637 df-limc 25833 df-dv 25834

This theorem is referenced by: lcmineqlem10 42477 itgsinexplem1 46382 fourierdlem39 46574

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