Theorem itgmulc2lem2 26197

Description: Lemma for itgmulc2 26198: 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 13364	. . . . . 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 7464	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
6		0re 11347	. . . . . . . 8 ⊢ 0 ∈ ℝ
7		ifcl 4531	. . . . . . . 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 11374	. . . . . 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 11780	. . . . . . . 8 ⊢ (𝜑 → -𝐶 ∈ ℝ)
12		ifcl 4531	. . . . . . . 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 11374	. . . . . 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 11374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
18	10, 15, 17	subdird 11810	. . . 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 2800	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
20	19	itgeq2dv 26146	. 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 11376	. . 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 26195	. . 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 11376	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) ∈ ℝ)
28	14, 23, 24	iblmulc2 26195	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ 𝐿1)
29	22, 25, 27, 28	itgsub 26190	. 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 4531	. . . . . . . 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 11376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ ℝ)
33	16	iblre 26158	. . . . . . . . 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 26195	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
37	16	renegcld 11780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
38		ifcl 4531	. . . . . . . 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 11376	. . . . . 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 26195	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
43	32, 36, 40, 42	itgsub 26190	. . . . 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 13364	. . . . . . . . 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 7465	. . . . . . 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 11374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
48	39	recnd 11374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
49	10, 47, 48	subdid 11809	. . . . . . 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 2800	. . . . . 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 26146	. . . . 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 26161	. . . . . . 7 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
53	52	oveq2d 7465	. . . . . 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 26148	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
55	39, 41	itgcl 26148	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
56	9, 54, 55	subdid 11809	. . . . . 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 13353	. . . . . . . . 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 13353	. . . . . . . . 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 26196	. . . . . . 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 13353	. . . . . . . . 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 26196	. . . . . . 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 7467	. . . . . 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 2802	. . . . 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 2808	. . . 4 ⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥))
68	26, 31	remulcld 11376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ ℝ)
69	14, 31, 35	iblmulc2 26195	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
70	26, 39	remulcld 11376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ ℝ)
71	14, 39, 41	iblmulc2 26195	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
72	68, 69, 70, 71	itgsub 26190	. . . . 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 7465	. . . . . . 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 11809	. . . . . . 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 2800	. . . . . 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 26146	. . . . 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 7465	. . . . . 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 11809	. . . . . 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 13353	. . . . . . . . 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 26196	. . . . . . 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 26196	. . . . . . 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 7467	. . . . . 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 2802	. . . . 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 2808	. . . 4 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥))
86	67, 85	oveq12d 7467	. . 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 26148	. . . 4 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
88	9, 14, 87	subdird 11810	. . 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 7464	. . 3 ⊢ (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
91	86, 88, 90	3eqtr2d 2804	. 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 2803	1 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ifcif 4485   class class class wbr 5106   ↦ cmpt 5190  (class class class)co 7449  ℂcc 11235  ℝcr 11236  0cc0 11237   · cmul 11242   ≤ cle 11381   − cmin 11578   -cneg 11579  𝐿¹cibl 25982  ∫citg 25983

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-inf2 9677  ax-cc 10550  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314  ax-pre-sup 11315  ax-addf 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-disj 5074  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-isom 6573  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-of 7723  df-ofr 7724  df-om 7910  df-1st 8033  df-2nd 8034  df-supp 8213  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-2o 8513  df-oadd 8516  df-omul 8517  df-er 8753  df-map 8885  df-pm 8886  df-ixp 8962  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-fsupp 9390  df-fi 9438  df-sup 9469  df-inf 9470  df-oi 9539  df-dju 10017  df-card 10055  df-acn 10058  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-div 12012  df-nn 12374  df-2 12443  df-3 12444  df-4 12445  df-5 12446  df-6 12447  df-7 12448  df-8 12449  df-9 12450  df-n0 12645  df-z 12732  df-dec 12853  df-uz 13004  df-q 13114  df-rp 13159  df-xneg 13279  df-xadd 13280  df-xmul 13281  df-ioo 13518  df-ioc 13519  df-ico 13520  df-icc 13521  df-fz 13678  df-fzo 13827  df-fl 13970  df-mod 14048  df-seq 14183  df-exp 14243  df-hash 14514  df-cj 15307  df-re 15308  df-im 15309  df-sqrt 15443  df-abs 15444  df-clim 15696  df-rlim 15697  df-sum 15895  df-struct 17366  df-sets 17383  df-slot 17401  df-ndx 17413  df-base 17429  df-ress 17450  df-plusg 17482  df-mulr 17483  df-starv 17484  df-sca 17485  df-vsca 17486  df-ip 17487  df-tset 17488  df-ple 17489  df-ds 17491  df-unif 17492  df-hom 17493  df-cco 17494  df-rest 17634  df-topn 17635  df-0g 17653  df-gsum 17654  df-topgen 17655  df-pt 17656  df-prds 17659  df-xrs 17715  df-qtop 17720  df-imas 17721  df-xps 17723  df-mre 17797  df-mrc 17798  df-acs 17800  df-mgm 18857  df-sgrp 18949  df-mnd 18965  df-submnd 19020  df-mulg 19319  df-cntz 19573  df-cmn 20038  df-psmet 21712  df-xmet 21713  df-met 21714  df-bl 21715  df-mopn 21716  df-cnfld 21721  df-top 23256  df-topon 23273  df-topsp 23295  df-bases 23308  df-cn 23589  df-cnp 23590  df-cmp 23749  df-tx 23925  df-hmeo 24118  df-xms 24683  df-ms 24684  df-tms 24685  df-cncf 25243  df-ovol 25829  df-vol 25830  df-mbf 25984  df-itg1 25985  df-itg2 25986  df-ibl 25987  df-itg 25988  df-0p 26035

This theorem is used by:  itgmulc2  26198