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Theorem itgparts 26033

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 24879	. . . . . . 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 7055	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐵 ∈ ℂ)
5		ioossicc 13378	. . . . . . 7 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
6	5	sseli 3911	. . . . . 6 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
7		itgparts.c	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ))
8		cncff 24879	. . . . . . . 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 7055	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐶 ∈ ℂ)
11	6, 10	sylan2 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ)
12	4, 11	mulcld 11157	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐵 · 𝐶) ∈ ℂ)
13		itgparts.bc	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿¹)
14	12, 13	itgcl 25770	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 ∈ ℂ)
15		itgparts.a	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ))
16		cncff 24879	. . . . . . . 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 7055	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ ℂ)
19	6, 18	sylan2 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℂ)
20		itgparts.d	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ))
21		cncff 24879	. . . . . . 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 7055	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐷 ∈ ℂ)
24	19, 23	mulcld 11157	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐴 · 𝐷) ∈ ℂ)
25		itgparts.ad	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿¹)
26	24, 25	itgcl 25770	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 ∈ ℂ)
27	14, 26	pncan2d 11499	. 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 25811	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥))
29		fveq2 6828	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡))
30		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑡((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥)
31		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑥ℝ
32		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑥 D
33		nfmpt1 5172	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
34	31, 32, 33	nfov 7387	. . . . . . . 8 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))
35		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
36	34, 35	nffv 6838	. . . . . . 7 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡)
37	29, 30, 36	cbvitg 25762	. . . . . 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 11087	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
42	41	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
43		iccssre 13374	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
44	38, 39, 43	syl2anc 590	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
45	18, 10	mulcld 11157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → (𝐴 · 𝐶) ∈ ℂ)
46		tgioo4 24789	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
47		eqid 2739	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
48		iccntr 24806	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌))
49	38, 39, 48	syl2anc 590	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌))
50	42, 44, 45, 46, 47, 49	dvmptntr 25957	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))))
51		reelprrecn 11122	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
52	51	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
53	42, 44, 18, 46, 47, 49	dvmptntr 25957	. . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
56	42, 44, 10, 46, 47, 49	dvmptntr 25957	. . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷))
59	52, 19, 4, 55, 11, 23, 58	dvmptmul 25947	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
60	23, 19	mulcomd 11158	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐷 · 𝐴) = (𝐴 · 𝐷))
61	60	oveq2d 7373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐵 · 𝐶) + (𝐷 · 𝐴)) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
62	61	mpteq2dva 5166	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
63	50, 59, 62	3eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
64	47	addcn 24850	. . . . . . . . . 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 5990	. . . . . . . . . . . 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 24883	. . . . . . . . . . . 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 2844	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ ((𝑋(,)𝑌)–cn→ℂ))
71	1, 70	mulcncf 25432	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
72		resmpt 5990	. . . . . . . . . . . 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 24883	. . . . . . . . . . . 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 2844	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴) ∈ ((𝑋(,)𝑌)–cn→ℂ))
77	76, 20	mulcncf 25432	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
78	47, 65, 71, 77	cncfmpt2f 24901	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
79	63, 78	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
80	12, 13, 24, 25	ibladd 25807	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ 𝐿¹)
81	63, 80	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ 𝐿¹)
82	15, 7	mulcncf 25432	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) ∈ ((𝑋[,]𝑌)–cn→ℂ))
83	38, 39, 40, 79, 81, 82	ftc2 26030	. . . . . 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 2786	. . . . 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 6830	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥))
86	85	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥))
87		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋(,)𝑌))
88		ovex 7390	. . . . . . . 8 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) ∈ V
89		eqid 2739	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
90	89	fvmpt2 6948	. . . . . . . 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 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
92	86, 91	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
93	92	itgeq2dv 25768	. . . . 5 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) d𝑥 = ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥)
94	38	rexrd 11187	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
95	39	rexrd 11187	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
96		ubicc2 13410	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ (𝑋[,]𝑌))
97	94, 95, 40, 96	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋[,]𝑌))
98		ovex 7390	. . . . . . . . 9 ⊢ (𝐴 · 𝐶) ∈ V
99	98	csbex 5234	. . . . . . . 8 ⊢ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V
100		eqid 2739	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) = (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
101	100	fvmpts 6940	. . . . . . . 8 ⊢ ((𝑌 ∈ (𝑋[,]𝑌) ∧ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶))
102	97, 99, 101	sylancl 592	. . . . . . 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 3867	. . . . . . 7 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) = 𝐹)
105	102, 104	eqtrd 2774	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = 𝐹)
106		lbicc2 13409	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ (𝑋[,]𝑌))
107	94, 95, 40, 106	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑋[,]𝑌))
108	98	csbex 5234	. . . . . . . 8 ⊢ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V
109	100	fvmpts 6940	. . . . . . . 8 ⊢ ((𝑋 ∈ (𝑋[,]𝑌) ∧ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶))
110	107, 108, 109	sylancl 592	. . . . . . 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 3867	. . . . . . 7 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) = 𝐸)
113	110, 112	eqtrd 2774	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = 𝐸)
114	105, 113	oveq12d 7375	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) − ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋)) = (𝐹 − 𝐸))
115	84, 93, 114	3eqtr3d 2782	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (𝐹 − 𝐸))
116	28, 115	eqtr3d 2776	. . 3 ⊢ (𝜑 → (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥) = (𝐹 − 𝐸))
117	116	oveq1d 7372	. 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 2776	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 396 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ⦋csb 3831 ⊆ wss 3883 {cpr 4558 class class class wbr 5073 ↦ cmpt 5154 ran crn 5620 ↾ cres 5621 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ℂcc 11028 ℝcr 11029 + caddc 11033 · cmul 11035 ℝ^*cxr 11170 ≤ cle 11172 − cmin 11369 (,)cioo 13290 [,]cicc 13293 TopOpenctopn 17376 topGenctg 17392 ℂ_fldccnfld 21348 intcnt 23001 Cn ccn 23208 ×_t ctx 23544 –cn→ccncf 24862 𝐿¹cibl 25603 ∫citg 25604 D cdv 25849

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 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

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 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-symdif 4182 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-disj 5041 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-ofr 7622 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9817 df-card 9855 df-acn 9858 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-ioc 13295 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-fl 13743 df-mod 13821 df-seq 13956 df-exp 14016 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15442 df-rlim 15443 df-sum 15641 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-nei 23082 df-lp 23120 df-perf 23121 df-cn 23211 df-cnp 23212 df-haus 23299 df-cmp 23371 df-tx 23546 df-hmeo 23739 df-fil 23830 df-fm 23922 df-flim 23923 df-flf 23924 df-xms 24304 df-ms 24305 df-tms 24306 df-cncf 24864 df-ovol 25450 df-vol 25451 df-mbf 25605 df-itg1 25606 df-itg2 25607 df-ibl 25608 df-itg 25609 df-0p 25656 df-limc 25852 df-dv 25853

This theorem is referenced by: lcmineqlem10 42532 itgsinexplem1 46405 fourierdlem39 46597

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