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Theorem itgparts 26002

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 24833	. . . . . . 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 7128	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐵 ∈ ℂ)
5		ioossicc 13450	. . . . . . 7 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
6	5	sseli 3978	. . . . . 6 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
7		itgparts.c	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ))
8		cncff 24833	. . . . . . . 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 7128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐶 ∈ ℂ)
11	6, 10	sylan2 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ)
12	4, 11	mulcld 11272	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐵 · 𝐶) ∈ ℂ)
13		itgparts.bc	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿¹)
14	12, 13	itgcl 25733	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 ∈ ℂ)
15		itgparts.a	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ))
16		cncff 24833	. . . . . . . 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 7128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ ℂ)
19	6, 18	sylan2 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℂ)
20		itgparts.d	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ))
21		cncff 24833	. . . . . . 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 7128	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐷 ∈ ℂ)
24	19, 23	mulcld 11272	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐴 · 𝐷) ∈ ℂ)
25		itgparts.ad	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿¹)
26	24, 25	itgcl 25733	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 ∈ ℂ)
27	14, 26	pncan2d 11611	. 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 25774	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥))
29		fveq2 6902	. . . . . . 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 5260	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
34	31, 32, 33	nfov 7456	. . . . . . . 8 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))
35		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
36	34, 35	nffv 6912	. . . . . . 7 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡)
37	29, 30, 36	cbvitg 25725	. . . . . 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 11203	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
42	41	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
43		iccssre 13446	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
44	38, 39, 43	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
45	18, 10	mulcld 11272	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → (𝐴 · 𝐶) ∈ ℂ)
46		eqid 2728	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
47	46	tgioo2 24739	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
48		iccntr 24757	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌))
49	38, 39, 48	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌))
50	42, 44, 45, 47, 46, 49	dvmptntr 25923	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))))
51		reelprrecn 11238	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
52	51	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
53	42, 44, 18, 47, 46, 49	dvmptntr 25923	. . . . . . . . . . 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 2770	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
56	42, 44, 10, 47, 46, 49	dvmptntr 25923	. . . . . . . . . . 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 2770	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷))
59	52, 19, 4, 55, 11, 23, 58	dvmptmul 25913	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
60	23, 19	mulcomd 11273	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐷 · 𝐴) = (𝐴 · 𝐷))
61	60	oveq2d 7442	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐵 · 𝐶) + (𝐷 · 𝐴)) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
62	61	mpteq2dva 5252	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
63	50, 59, 62	3eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
64	46	addcn 24801	. . . . . . . . . 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 6046	. . . . . . . . . . . 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 24837	. . . . . . . . . . . 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 2834	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ ((𝑋(,)𝑌)–cn→ℂ))
71	1, 70	mulcncf 25394	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
72		resmpt 6046	. . . . . . . . . . . 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 24837	. . . . . . . . . . . 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 2834	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴) ∈ ((𝑋(,)𝑌)–cn→ℂ))
77	76, 20	mulcncf 25394	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
78	46, 65, 71, 77	cncfmpt2f 24855	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
79	63, 78	eqeltrd 2829	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
80	12, 13, 24, 25	ibladd 25770	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ 𝐿¹)
81	63, 80	eqeltrd 2829	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ 𝐿¹)
82	15, 7	mulcncf 25394	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) ∈ ((𝑋[,]𝑌)–cn→ℂ))
83	38, 39, 40, 79, 81, 82	ftc2 25999	. . . . . 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 2780	. . . . 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 6904	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥))
86	85	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥))
87		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋(,)𝑌))
88		ovex 7459	. . . . . . . 8 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) ∈ V
89		eqid 2728	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
90	89	fvmpt2 7021	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
92	86, 91	eqtrd 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
93	92	itgeq2dv 25731	. . . . 5 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) d𝑥 = ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥)
94	38	rexrd 11302	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
95	39	rexrd 11302	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
96		ubicc2 13482	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ (𝑋[,]𝑌))
97	94, 95, 40, 96	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋[,]𝑌))
98		ovex 7459	. . . . . . . . 9 ⊢ (𝐴 · 𝐶) ∈ V
99	98	csbex 5315	. . . . . . . 8 ⊢ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V
100		eqid 2728	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) = (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
101	100	fvmpts 7013	. . . . . . . 8 ⊢ ((𝑌 ∈ (𝑋[,]𝑌) ∧ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶))
102	97, 99, 101	sylancl 584	. . . . . . 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 3932	. . . . . . 7 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) = 𝐹)
105	102, 104	eqtrd 2768	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = 𝐹)
106		lbicc2 13481	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ (𝑋[,]𝑌))
107	94, 95, 40, 106	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑋[,]𝑌))
108	98	csbex 5315	. . . . . . . 8 ⊢ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V
109	100	fvmpts 7013	. . . . . . . 8 ⊢ ((𝑋 ∈ (𝑋[,]𝑌) ∧ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶))
110	107, 108, 109	sylancl 584	. . . . . . 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 3932	. . . . . . 7 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) = 𝐸)
113	110, 112	eqtrd 2768	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = 𝐸)
114	105, 113	oveq12d 7444	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) − ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋)) = (𝐹 − 𝐸))
115	84, 93, 114	3eqtr3d 2776	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (𝐹 − 𝐸))
116	28, 115	eqtr3d 2770	. . 3 ⊢ (𝜑 → (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥) = (𝐹 − 𝐸))
117	116	oveq1d 7441	. 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 2770	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 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⦋csb 3894 ⊆ wss 3949 {cpr 4634 class class class wbr 5152 ↦ cmpt 5235 ran crn 5683 ↾ cres 5684 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 + caddc 11149 · cmul 11151 ℝ^*cxr 11285 ≤ cle 11287 − cmin 11482 (,)cioo 13364 [,]cicc 13367 TopOpenctopn 17410 topGenctg 17426 ℂ_fldccnfld 21286 intcnt 22941 Cn ccn 23148 ×_t ctx 23484 –cn→ccncf 24816 𝐿¹cibl 25566 ∫citg 25567 D cdv 25812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cc 10466 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-symdif 4245 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-cmp 23311 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-ovol 25413 df-vol 25414 df-mbf 25568 df-itg1 25569 df-itg2 25570 df-ibl 25571 df-itg 25572 df-0p 25619 df-limc 25815 df-dv 25816

This theorem is referenced by: lcmineqlem10 41541 itgsinexplem1 45371 fourierdlem39 45563

W3C validator