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Theorem itgmulc2lem2 23349
Description: Lemma for itgmulc2 23350: real case. (Contributed by Mario Carneiro, 25-Aug-2014.)
Hypotheses
Ref Expression
itgmulc2.1 (𝜑𝐶 ∈ ℂ)
itgmulc2.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
itgmulc2.3 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
itgmulc2.4 (𝜑𝐶 ∈ ℝ)
itgmulc2.5 ((𝜑𝑥𝐴) → 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
StepHypRef Expression
1 itgmulc2.4 . . . . . . 7 (𝜑𝐶 ∈ ℝ)
21adantr 479 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
3 max0sub 11862 . . . . . 6 (𝐶 ∈ ℝ → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
42, 3syl 17 . . . . 5 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
54oveq1d 6541 . . . 4 ((𝜑𝑥𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
6 0re 9896 . . . . . . . 8 0 ∈ ℝ
7 ifcl 4079 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
81, 6, 7sylancl 692 . . . . . . 7 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
98recnd 9924 . . . . . 6 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
109adantr 479 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
111renegcld 10308 . . . . . . . 8 (𝜑 → -𝐶 ∈ ℝ)
12 ifcl 4079 . . . . . . . 8 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1311, 6, 12sylancl 692 . . . . . . 7 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1413recnd 9924 . . . . . 6 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
1514adantr 479 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
16 itgmulc2.5 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
1716recnd 9924 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
1810, 15, 17subdird 10338 . . . 4 ((𝜑𝑥𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
195, 18eqtr3d 2645 . . 3 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
2019itgeq2dv 23298 . 2 (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥)
218adantr 479 . . . 4 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
2221, 16remulcld 9926 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) ∈ ℝ)
23 itgmulc2.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
24 itgmulc2.3 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
259, 23, 24iblmulc2 23347 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ 𝐿1)
2613adantr 479 . . . 4 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
2726, 16remulcld 9926 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) ∈ ℝ)
2814, 23, 24iblmulc2 23347 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ 𝐿1)
2922, 25, 27, 28itgsub 23342 . 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 4079 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
3116, 6, 30sylancl 692 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
3221, 31remulcld 9926 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ ℝ)
3316iblre 23310 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
3424, 33mpbid 220 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
3534simpld 473 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
369, 31, 35iblmulc2 23347 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
3716renegcld 10308 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
38 ifcl 4079 . . . . . . . 8 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
3937, 6, 38sylancl 692 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
4021, 39remulcld 9926 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ ℝ)
4134simprd 477 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
429, 39, 41iblmulc2 23347 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
4332, 36, 40, 42itgsub 23342 . . . . 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 11862 . . . . . . . . 9 (𝐵 ∈ ℝ → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
4516, 44syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
4645oveq2d 6542 . . . . . . 7 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵))
4731recnd 9924 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
4839recnd 9924 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
4910, 47, 48subdid 10337 . . . . . . 7 ((𝜑𝑥𝐴) → (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))))
5046, 49eqtr3d 2645 . . . . . 6 ((𝜑𝑥𝐴) → (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))))
5150itgeq2dv 23298 . . . . 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)
5216, 24itgreval 23313 . . . . . . 7 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
5352oveq2d 6542 . . . . . 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)))
5431, 35itgcl 23300 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
5539, 41itgcl 23300 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
569, 54, 55subdid 10337 . . . . . 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 11851 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
586, 1, 57sylancr 693 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
59 max1 11851 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
606, 16, 59sylancr 693 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
619, 31, 35, 8, 31, 58, 60itgmulc2lem1 23348 . . . . . . 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 11851 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
636, 37, 62sylancr 693 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
649, 39, 41, 8, 39, 58, 63itgmulc2lem1 23348 . . . . . . 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)
6561, 64oveq12d 6544 . . . . . 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))
6653, 56, 653eqtrd 2647 . . . . 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))
6743, 51, 663eqtr4d 2653 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥))
6826, 31remulcld 9926 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ ℝ)
6914, 31, 35iblmulc2 23347 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
7026, 39remulcld 9926 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ ℝ)
7114, 39, 41iblmulc2 23347 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
7268, 69, 70, 71itgsub 23342 . . . . 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))
7345oveq2d 6542 . . . . . . 7 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))
7415, 47, 48subdid 10337 . . . . . . 7 ((𝜑𝑥𝐴) → (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))))
7573, 74eqtr3d 2645 . . . . . 6 ((𝜑𝑥𝐴) → (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))))
7675itgeq2dv 23298 . . . . 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)
7752oveq2d 6542 . . . . . 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)))
7814, 54, 55subdid 10337 . . . . . 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 11851 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
806, 11, 79sylancr 693 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
8114, 31, 35, 13, 31, 80, 60itgmulc2lem1 23348 . . . . . . 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)
8214, 39, 41, 13, 39, 80, 63itgmulc2lem1 23348 . . . . . . 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)
8381, 82oveq12d 6544 . . . . . 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))
8477, 78, 833eqtrd 2647 . . . . 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))
8572, 76, 843eqtr4d 2653 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥))
8667, 85oveq12d 6544 . . 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)))
8723, 24itgcl 23300 . . . 4 (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
889, 14, 87subdird 10338 . . 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)))
891, 3syl 17 . . . 4 (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
9089oveq1d 6541 . . 3 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
9186, 88, 903eqtr2d 2649 . 2 (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
9220, 29, 913eqtrrd 2648 1 (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  ifcif 4035   class class class wbr 4577  cmpt 4637  (class class class)co 6526  cc 9790  cr 9791  0cc0 9792   · cmul 9797  cle 9931  cmin 10117  -cneg 10118  𝐿1cibl 23136  citg 23137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-cc 9117  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-disj 4548  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-of 6772  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-omul 7429  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-acn 8628  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-4 10930  df-5 10931  df-6 10932  df-7 10933  df-8 10934  df-9 10935  df-n0 11142  df-z 11213  df-dec 11328  df-uz 11522  df-q 11623  df-rp 11667  df-xneg 11780  df-xadd 11781  df-xmul 11782  df-ioo 12008  df-ioc 12009  df-ico 12010  df-icc 12011  df-fz 12155  df-fzo 12292  df-fl 12412  df-mod 12488  df-seq 12621  df-exp 12680  df-hash 12937  df-cj 13635  df-re 13636  df-im 13637  df-sqrt 13771  df-abs 13772  df-clim 14015  df-rlim 14016  df-sum 14213  df-struct 15645  df-ndx 15646  df-slot 15647  df-base 15648  df-sets 15649  df-ress 15650  df-plusg 15729  df-mulr 15730  df-starv 15731  df-sca 15732  df-vsca 15733  df-ip 15734  df-tset 15735  df-ple 15736  df-ds 15739  df-unif 15740  df-hom 15741  df-cco 15742  df-rest 15854  df-topn 15855  df-0g 15873  df-gsum 15874  df-topgen 15875  df-pt 15876  df-prds 15879  df-xrs 15933  df-qtop 15938  df-imas 15939  df-xps 15941  df-mre 16017  df-mrc 16018  df-acs 16020  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-submnd 17107  df-mulg 17312  df-cntz 17521  df-cmn 17966  df-psmet 19507  df-xmet 19508  df-met 19509  df-bl 19510  df-mopn 19511  df-cnfld 19516  df-top 20468  df-bases 20469  df-topon 20470  df-topsp 20471  df-cn 20788  df-cnp 20789  df-cmp 20947  df-tx 21122  df-hmeo 21315  df-xms 21882  df-ms 21883  df-tms 21884  df-cncf 22436  df-ovol 22984  df-vol 22985  df-mbf 23138  df-itg1 23139  df-itg2 23140  df-ibl 23141  df-itg 23142  df-0p 23187
This theorem is referenced by:  itgmulc2  23350
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