Theorem itgmulc2lem2 26060

Description: Lemma for itgmulc2 26061: real case. (Contributed by Mario Carneiro, 25-Aug-2014.)

Hypotheses

Ref	Expression
itgmulc2.1	⊢ (𝜑 → 𝐶 ∈ ℂ)
itgmulc2.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
itgmulc2.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)
itgmulc2.4	⊢ (𝜑 → 𝐶 ∈ ℝ)
itgmulc2.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
itgmulc2lem2	⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥   𝑥,𝑉

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem itgmulc2lem2

Step	Hyp	Ref	Expression
1		itgmulc2.4	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
2	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
3		max0sub 13248	. . . . . 6 ⊢ (𝐶 ∈ ℝ → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
4	2, 3	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
5	4	oveq1d 7428	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
6		0re 11234	. . . . . . . 8 ⊢ 0 ∈ ℝ
7		ifcl 4528	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
8	1, 6, 7	sylancl 598	. . . . . . 7 ⊢ (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
9	8	recnd 11261	. . . . . 6 ⊢ (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
10	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
11	1	renegcld 11665	. . . . . . . 8 ⊢ (𝜑 → -𝐶 ∈ ℝ)
12		ifcl 4528	. . . . . . . 8 ⊢ ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
13	11, 6, 12	sylancl 598	. . . . . . 7 ⊢ (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
14	13	recnd 11261	. . . . . 6 ⊢ (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
15	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
16		itgmulc2.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
17	16	recnd 11261	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
18	10, 15, 17	subdird 11695	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
19	5, 18	eqtr3d 2797	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
20	19	itgeq2dv 26009	. 2 ⊢ (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥)
21	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
22	21, 16	remulcld 11263	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) ∈ ℝ)
23		itgmulc2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
24		itgmulc2.3	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)
25	9, 23, 24	iblmulc2 26058	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ 𝐿1)
26	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
27	26, 16	remulcld 11263	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) ∈ ℝ)
28	14, 23, 24	iblmulc2 26058	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ 𝐿1)
29	22, 25, 27, 28	itgsub 26053	. 2 ⊢ (𝜑 → ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥 = (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥))
30		ifcl 4528	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
31	16, 6, 30	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
32	21, 31	remulcld 11263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ ℝ)
33	16	iblre 26021	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
34	24, 33	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
35	34	simpld 500	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
36	9, 31, 35	iblmulc2 26058	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
37	16	renegcld 11665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
38		ifcl 4528	. . . . . . . 8 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
39	37, 6, 38	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
40	21, 39	remulcld 11263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ ℝ)
41	34	simprd 501	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
42	9, 39, 41	iblmulc2 26058	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
43	32, 36, 40, 42	itgsub 26053	. . . . 5 ⊢ (𝜑 → ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) d𝑥 = (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
44		max0sub 13248	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
45	16, 44	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
46	45	oveq2d 7429	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵))
47	31	recnd 11261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
48	39	recnd 11261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
49	10, 47, 48	subdid 11694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
50	46, 49	eqtr3d 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
51	50	itgeq2dv 26009	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) d𝑥)
52	16, 24	itgreval 26024	. . . . . . 7 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
53	52	oveq2d 7429	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥) = (if(0 ≤ 𝐶, 𝐶, 0) · (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)))
54	31, 35	itgcl 26011	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
55	39, 41	itgcl 26011	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
56	9, 54, 55	subdid 11694	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) = ((if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) − (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)))
57		max1 13237	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
58	6, 1, 57	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
59		max1 13237	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
60	6, 16, 59	sylancr 599	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
61	9, 31, 35, 8, 31, 58, 60	itgmulc2lem1 26059	. . . . . . 7 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) = ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥)
62		max1 13237	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
63	6, 37, 62	sylancr 599	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
64	9, 39, 41, 8, 39, 58, 63	itgmulc2lem1 26059	. . . . . . 7 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) = ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥)
65	61, 64	oveq12d 7431	. . . . . 6 ⊢ (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) − (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) = (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
66	53, 56, 65	3eqtrd 2799	. . . . 5 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥) = (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
67	43, 51, 66	3eqtr4d 2805	. . . 4 ⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥))
68	26, 31	remulcld 11263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ ℝ)
69	14, 31, 35	iblmulc2 26058	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
70	26, 39	remulcld 11263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ ℝ)
71	14, 39, 41	iblmulc2 26058	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
72	68, 69, 70, 71	itgsub 26053	. . . . 5 ⊢ (𝜑 → ∫𝐴((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) d𝑥 = (∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
73	45	oveq2d 7429	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))
74	15, 47, 48	subdid 11694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
75	73, 74	eqtr3d 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) = ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
76	75	itgeq2dv 26009	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) d𝑥)
77	52	oveq2d 7429	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥) = (if(0 ≤ -𝐶, -𝐶, 0) · (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)))
78	14, 54, 55	subdid 11694	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) = ((if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) − (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)))
79		max1 13237	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
80	6, 11, 79	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
81	14, 31, 35, 13, 31, 80, 60	itgmulc2lem1 26059	. . . . . . 7 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) = ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥)
82	14, 39, 41, 13, 39, 80, 63	itgmulc2lem1 26059	. . . . . . 7 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) = ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥)
83	81, 82	oveq12d 7431	. . . . . 6 ⊢ (𝜑 → ((if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) − (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) = (∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
84	77, 78, 83	3eqtrd 2799	. . . . 5 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥) = (∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
85	72, 76, 84	3eqtr4d 2805	. . . 4 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥))
86	67, 85	oveq12d 7431	. . 3 ⊢ (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = ((if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥) − (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥)))
87	23, 24	itgcl 26011	. . . 4 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
88	9, 14, 87	subdird 11695	. . 3 ⊢ (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = ((if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥) − (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥)))
89	1, 3	syl 18	. . . 4 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
90	89	oveq1d 7428	. . 3 ⊢ (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
91	86, 88, 90	3eqtr2d 2801	. 2 ⊢ (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
92	20, 29, 91	3eqtrrd 2800	1 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186  (class class class)co 7413  ℂcc 11122  ℝcr 11123  0cc0 11124   · cmul 11129   ≤ cle 11268   − cmin 11465  -cneg 11466  𝐿¹cibl 25845  ∫citg 25846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cc 10437  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202  ax-addf 11203

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-ofr 7679  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-omul 8460  df-er 8696  df-map 8828  df-pm 8829  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-fi 9381  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9906  df-card 9944  df-acn 9947  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-q 12998  df-rp 13043  df-xneg 13163  df-xadd 13164  df-xmul 13165  df-ioo 13402  df-ioc 13403  df-ico 13404  df-icc 13405  df-fz 13562  df-fzo 13710  df-fl 13853  df-mod 13931  df-seq 14066  df-exp 14126  df-hash 14395  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-rlim 15576  df-sum 15774  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-starv 17357  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ds 17364  df-unif 17365  df-hom 17366  df-cco 17367  df-rest 17507  df-topn 17508  df-0g 17526  df-gsum 17527  df-topgen 17528  df-pt 17529  df-prds 17532  df-xrs 17588  df-qtop 17593  df-imas 17594  df-xps 17596  df-mre 17670  df-mrc 17671  df-acs 17673  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-submnd 18892  df-mulg 19191  df-cntz 19444  df-cmn 19909  df-psmet 21577  df-xmet 21578  df-met 21579  df-bl 21580  df-mopn 21581  df-cnfld 21586  df-top 23119  df-topon 23136  df-topsp 23158  df-bases 23171  df-cn 23452  df-cnp 23453  df-cmp 23612  df-tx 23788  df-hmeo 23981  df-xms 24546  df-ms 24547  df-tms 24548  df-cncf 25106  df-ovol 25692  df-vol 25693  df-mbf 25847  df-itg1 25848  df-itg2 25849  df-ibl 25850  df-itg 25851  df-0p 25898

This theorem is used by:  itgmulc2  26061