itgparts - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > itgparts		Structured version Visualization version GIF version

Theorem itgparts 26110

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 24956	. . . . . . 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 7095	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐵 ∈ ℂ)
5		ioossicc 13438	. . . . . . 7 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
6	5	sseli 3933	. . . . . 6 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
7		itgparts.c	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ))
8		cncff 24956	. . . . . . . 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 7095	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐶 ∈ ℂ)
11	6, 10	sylan2 602	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℂ)
12	4, 11	mulcld 11203	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐵 · 𝐶) ∈ ℂ)
13		itgparts.bc	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿¹)
14	12, 13	itgcl 25847	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 ∈ ℂ)
15		itgparts.a	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ))
16		cncff 24956	. . . . . . . 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 7095	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ ℂ)
19	6, 18	sylan2 602	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ ℂ)
20		itgparts.d	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ))
21		cncff 24956	. . . . . . 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 7095	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐷 ∈ ℂ)
24	19, 23	mulcld 11203	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐴 · 𝐷) ∈ ℂ)
25		itgparts.ad	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿¹)
26	24, 25	itgcl 25847	. . 3 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 ∈ ℂ)
27	14, 26	pncan2d 11545	. 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 25888	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥))
29		fveq2 6868	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡))
30		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑡((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥)
31		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑥ℝ
32		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑥 D
33		nfmpt1 5200	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
34	31, 32, 33	nfov 7427	. . . . . . . 8 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))
35		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥𝑡
36	34, 35	nffv 6878	. . . . . . 7 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑡)
37	29, 30, 36	cbvitg 25839	. . . . . 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 11131	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
42	41	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
43		iccssre 13434	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
44	38, 39, 43	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
45	18, 10	mulcld 11203	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → (𝐴 · 𝐶) ∈ ℂ)
46		tgioo4 24866	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
47		eqid 2763	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
48		iccntr 24883	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌))
49	38, 39, 48	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝑋[,]𝑌)) = (𝑋(,)𝑌))
50	42, 44, 45, 46, 47, 49	dvmptntr 26034	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))))
51		reelprrecn 11166	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
52	51	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
53	42, 44, 18, 46, 47, 49	dvmptntr 26034	. . . . . . . . . . 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 2800	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
56	42, 44, 10, 46, 47, 49	dvmptntr 26034	. . . . . . . . . . 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 2800	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷))
59	52, 19, 4, 55, 11, 23, 58	dvmptmul 26024	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
60	23, 19	mulcomd 11204	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (𝐷 · 𝐴) = (𝐴 · 𝐷))
61	60	oveq2d 7413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝐵 · 𝐶) + (𝐷 · 𝐴)) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
62	61	mpteq2dva 5194	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
63	50, 59, 62	3eqtrd 2802	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))))
64	47	addcn 24927	. . . . . . . . . 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 6027	. . . . . . . . . . . 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 24960	. . . . . . . . . . . 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 2868	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ ((𝑋(,)𝑌)–cn→ℂ))
71	1, 70	mulcncf 25509	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
72		resmpt 6027	. . . . . . . . . . . 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 24960	. . . . . . . . . . . 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 2868	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐴) ∈ ((𝑋(,)𝑌)–cn→ℂ))
77	76, 20	mulcncf 25509	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
78	47, 65, 71, 77	cncfmpt2f 24978	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
79	63, 78	eqeltrd 2863	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
80	12, 13, 24, 25	ibladd 25884	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) ∈ 𝐿¹)
81	63, 80	eqeltrd 2863	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))) ∈ 𝐿¹)
82	15, 7	mulcncf 25509	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) ∈ ((𝑋[,]𝑌)–cn→ℂ))
83	38, 39, 40, 79, 81, 82	ftc2 26107	. . . . . 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 2810	. . . . 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 6870	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥))
86	85	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥))
87		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝑥 ∈ (𝑋(,)𝑌))
88		ovex 7430	. . . . . . . 8 ⊢ ((𝐵 · 𝐶) + (𝐴 · 𝐷)) ∈ V
89		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
90	89	fvmpt2 6988	. . . . . . . 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 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((𝑥 ∈ (𝑋(,)𝑌) ↦ ((𝐵 · 𝐶) + (𝐴 · 𝐷)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
92	86, 91	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) = ((𝐵 · 𝐶) + (𝐴 · 𝐷)))
93	92	itgeq2dv 25845	. . . . 5 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)))‘𝑥) d𝑥 = ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥)
94	38	rexrd 11233	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
95	39	rexrd 11233	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
96		ubicc2 13470	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ (𝑋[,]𝑌))
97	94, 95, 40, 96	syl3anc 1391	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑋[,]𝑌))
98		ovex 7430	. . . . . . . . 9 ⊢ (𝐴 · 𝐶) ∈ V
99	98	csbex 5262	. . . . . . . 8 ⊢ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V
100		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶)) = (𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))
101	100	fvmpts 6980	. . . . . . . 8 ⊢ ((𝑌 ∈ (𝑋[,]𝑌) ∧ ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) ∈ V) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶))
102	97, 99, 101	sylancl 595	. . . . . . 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 3889	. . . . . . 7 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌(𝐴 · 𝐶) = 𝐹)
105	102, 104	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) = 𝐹)
106		lbicc2 13469	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑌 ∈ ℝ^* ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ (𝑋[,]𝑌))
107	94, 95, 40, 106	syl3anc 1391	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑋[,]𝑌))
108	98	csbex 5262	. . . . . . . 8 ⊢ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V
109	100	fvmpts 6980	. . . . . . . 8 ⊢ ((𝑋 ∈ (𝑋[,]𝑌) ∧ ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) ∈ V) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶))
110	107, 108, 109	sylancl 595	. . . . . . 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 3889	. . . . . . 7 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌(𝐴 · 𝐶) = 𝐸)
113	110, 112	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋) = 𝐸)
114	105, 113	oveq12d 7415	. . . . 5 ⊢ (𝜑 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑌) − ((𝑥 ∈ (𝑋[,]𝑌) ↦ (𝐴 · 𝐶))‘𝑋)) = (𝐹 − 𝐸))
115	84, 93, 114	3eqtr3d 2806	. . . 4 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐵 · 𝐶) + (𝐴 · 𝐷)) d𝑥 = (𝐹 − 𝐸))
116	28, 115	eqtr3d 2800	. . 3 ⊢ (𝜑 → (∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥 + ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥) = (𝐹 − 𝐸))
117	116	oveq1d 7412	. 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 2800	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 399 = wceq 1561 ∈ wcel 2143 Vcvv 3455 ⦋csb 3853 ⊆ wss 3905 {cpr 4585 class class class wbr 5101 ↦ cmpt 5182 ran crn 5649 ↾ cres 5650 ⟶wf 6518 ‘cfv 6522 (class class class)co 7397 ℂcc 11072 ℝcr 11073 + caddc 11077 · cmul 11079 ℝ^*cxr 11216 ≤ cle 11218 − cmin 11415 (,)cioo 13350 [,]cicc 13353 TopOpenctopn 17451 topGenctg 17467 ℂ_fldccnfld 21425 intcnt 23078 Cn ccn 23285 ×_t ctx 23621 –cn→ccncf 24939 𝐿¹cibl 25680 ∫citg 25681 D cdv 25926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-inf2 9597 ax-cc 10393 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-symdif 4206 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5069 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-isom 6531 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-of 7661 df-ofr 7662 df-om 7848 df-1st 7971 df-2nd 7972 df-supp 8142 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-2o 8439 df-oadd 8442 df-omul 8443 df-er 8679 df-map 8811 df-pm 8812 df-ixp 8881 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-fsupp 9309 df-fi 9358 df-sup 9389 df-inf 9390 df-oi 9459 df-dju 9860 df-card 9898 df-acn 9901 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-div 11846 df-nn 12212 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12483 df-z 12570 df-dec 12690 df-uz 12841 df-q 12951 df-rp 12995 df-xneg 13115 df-xadd 13116 df-xmul 13117 df-ioo 13354 df-ioc 13355 df-ico 13356 df-icc 13357 df-fz 13514 df-fzo 13661 df-fl 13803 df-mod 13881 df-seq 14016 df-exp 14076 df-hash 14345 df-cj 15127 df-re 15128 df-im 15129 df-sqrt 15263 df-abs 15264 df-clim 15516 df-rlim 15517 df-sum 15715 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17247 df-ress 17268 df-plusg 17300 df-mulr 17301 df-starv 17302 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-unif 17310 df-hom 17311 df-cco 17312 df-rest 17452 df-topn 17453 df-0g 17471 df-gsum 17472 df-topgen 17473 df-pt 17474 df-prds 17477 df-xrs 17533 df-qtop 17538 df-imas 17539 df-xps 17541 df-mre 17615 df-mrc 17616 df-acs 17618 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-submnd 18819 df-mulg 19111 df-cntz 19358 df-cmn 19823 df-psmet 21417 df-xmet 21418 df-met 21419 df-bl 21420 df-mopn 21421 df-fbas 21422 df-fg 21423 df-cnfld 21426 df-top 22955 df-topon 22972 df-topsp 22994 df-bases 23007 df-cld 23080 df-ntr 23081 df-cls 23082 df-nei 23159 df-lp 23197 df-perf 23198 df-cn 23288 df-cnp 23289 df-haus 23376 df-cmp 23448 df-tx 23623 df-hmeo 23816 df-fil 23907 df-fm 23999 df-flim 24000 df-flf 24001 df-xms 24381 df-ms 24382 df-tms 24383 df-cncf 24941 df-ovol 25527 df-vol 25528 df-mbf 25682 df-itg1 25683 df-itg2 25684 df-ibl 25685 df-itg 25686 df-0p 25733 df-limc 25929 df-dv 25930

This theorem is referenced by: lcmineqlem10 42656 itgsinexplem1 46529 fourierdlem39 46721

W3C validator