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Theorem itgmulc2nclem2 34946
Description: Lemma for itgmulc2nc 34947; cf. itgmulc2lem2 24425. (Contributed by Brendan Leahy, 19-Nov-2017.)
Hypotheses
Ref Expression
itgmulc2nc.1 (𝜑𝐶 ∈ ℂ)
itgmulc2nc.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
itgmulc2nc.3 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
itgmulc2nc.m (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
itgmulc2nc.4 (𝜑𝐶 ∈ ℝ)
itgmulc2nc.5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
itgmulc2nclem2 (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 itgmulc2nclem2
StepHypRef Expression
1 itgmulc2nc.4 . . . . . . 7 (𝜑𝐶 ∈ ℝ)
2 max0sub 12581 . . . . . . 7 (𝐶 ∈ ℝ → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
31, 2syl 17 . . . . . 6 (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
43oveq1d 7163 . . . . 5 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
54adantr 483 . . . 4 ((𝜑𝑥𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
6 0re 10635 . . . . . . . 8 0 ∈ ℝ
7 ifcl 4509 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
81, 6, 7sylancl 588 . . . . . . 7 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
98recnd 10661 . . . . . 6 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
109adantr 483 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
111renegcld 11059 . . . . . . . 8 (𝜑 → -𝐶 ∈ ℝ)
12 ifcl 4509 . . . . . . . 8 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1311, 6, 12sylancl 588 . . . . . . 7 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1413recnd 10661 . . . . . 6 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
1514adantr 483 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
16 itgmulc2nc.5 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
1716recnd 10661 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
1810, 15, 17subdird 11089 . . . 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 2856 . . 3 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
2019itgeq2dv 24374 . 2 (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥)
21 ovexd 7183 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) ∈ V)
22 itgmulc2nc.3 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
23 itgmulc2nc.m . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
24 ovexd 7183 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) ∈ V)
2523, 24mbfdm2 24230 . . . . . 6 (𝜑𝐴 ∈ dom vol)
268adantr 483 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
27 fconstmpt 5607 . . . . . . 7 (𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) = (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0))
2827a1i 11 . . . . . 6 (𝜑 → (𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) = (𝑥𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)))
29 eqidd 2820 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3025, 26, 16, 28, 29offval2 7418 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)))
31 iblmbf 24360 . . . . . . 7 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
3222, 31syl 17 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
3317fmpttd 6872 . . . . . 6 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℂ)
3432, 8, 33mbfmulc2re 24241 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) ∈ MblFn)
3530, 34eqeltrrd 2912 . . . 4 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ MblFn)
369, 16, 22, 35iblmulc2nc 34944 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ 𝐿1)
37 ovexd 7183 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) ∈ V)
3813adantr 483 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
39 fconstmpt 5607 . . . . . . 7 (𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) = (𝑥𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0))
4039a1i 11 . . . . . 6 (𝜑 → (𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) = (𝑥𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)))
4125, 38, 16, 40, 29offval2 7418 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
4232, 13, 33mbfmulc2re 24241 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) ∈ MblFn)
4341, 42eqeltrrd 2912 . . . 4 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ MblFn)
4414, 16, 22, 43iblmulc2nc 34944 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ 𝐿1)
4519mpteq2dva 5152 . . . 4 (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) = (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))))
4645, 23eqeltrrd 2912 . . 3 (𝜑 → (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))) ∈ MblFn)
4721, 36, 37, 44, 46itgsubnc 34941 . 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))
48 ovexd 7183 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
49 ifcl 4509 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5016, 6, 49sylancl 588 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5116iblre 24386 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
5222, 51mpbid 234 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
5352simpld 497 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
54 eqidd 2820 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
5525, 26, 50, 28, 54offval2 7418 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) = (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))))
5616, 32mbfpos 24244 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
5750recnd 10661 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
5857fmpttd 6872 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℂ)
5956, 8, 58mbfmulc2re 24241 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
6055, 59eqeltrrd 2912 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
619, 50, 53, 60iblmulc2nc 34944 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
62 ovexd 7183 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
6316renegcld 11059 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
64 ifcl 4509 . . . . . . . 8 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
6563, 6, 64sylancl 588 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
6652simprd 498 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
67 eqidd 2820 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
6825, 26, 65, 28, 67offval2 7418 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) = (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
6916, 32mbfneg 24243 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ MblFn)
7063, 69mbfpos 24244 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
7165recnd 10661 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
7271fmpttd 6872 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℂ)
7370, 8, 72mbfmulc2re 24241 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
7468, 73eqeltrrd 2912 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
759, 65, 66, 74iblmulc2nc 34944 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
76 max0sub 12581 . . . . . . . . . . . 12 (𝐵 ∈ ℝ → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
7716, 76syl 17 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
7877oveq2d 7164 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵))
7910, 57, 71subdid 11088 . . . . . . . . . 10 ((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))))
8078, 79eqtr3d 2856 . . . . . . . . 9 ((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))))
8180mpteq2dva 5152 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) = (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))))
8230, 81eqtrd 2854 . . . . . . 7 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))))
8382, 34eqeltrrd 2912 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))) ∈ MblFn)
8448, 61, 62, 75, 83itgsubnc 34941 . . . . 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))
8580itgeq2dv 24374 . . . . 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)
8616, 22itgreval 24389 . . . . . . 7 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
8786oveq2d 7164 . . . . . 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)))
8850, 53itgcl 24376 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
8965, 66itgcl 24376 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
909, 88, 89subdid 11088 . . . . . 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)))
91 max1 12570 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
926, 1, 91sylancr 589 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
93 max1 12570 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
946, 16, 93sylancr 589 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
959, 50, 53, 60, 8, 50, 92, 94itgmulc2nclem1 34945 . . . . . . 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)
96 max1 12570 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
976, 63, 96sylancr 589 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
989, 65, 66, 74, 8, 65, 92, 97itgmulc2nclem1 34945 . . . . . . 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)
9995, 98oveq12d 7166 . . . . . 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))
10087, 90, 993eqtrd 2858 . . . . 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))
10184, 85, 1003eqtr4d 2864 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥))
102 ovexd 7183 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
10325, 38, 50, 40, 54offval2 7418 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) = (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))))
10456, 13, 58mbfmulc2re 24241 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
105103, 104eqeltrrd 2912 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
10614, 50, 53, 105iblmulc2nc 34944 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
107 ovexd 7183 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
10825, 38, 65, 40, 67offval2 7418 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) = (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
10970, 13, 72mbfmulc2re 24241 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
110108, 109eqeltrrd 2912 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
11114, 65, 66, 110iblmulc2nc 34944 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
11277oveq2d 7164 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))
11315, 57, 71subdid 11088 . . . . . . . . . 10 ((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))))
114112, 113eqtr3d 2856 . . . . . . . . 9 ((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))))
115114mpteq2dva 5152 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) = (𝑥𝐴 ↦ ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))))
11641, 115eqtrd 2854 . . . . . . 7 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))))
117116, 42eqeltrrd 2912 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))) ∈ MblFn)
118102, 106, 107, 111, 117itgsubnc 34941 . . . . 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))
119114itgeq2dv 24374 . . . . 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)
12086oveq2d 7164 . . . . . 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)))
12114, 88, 89subdid 11088 . . . . . 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)))
122 max1 12570 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
1236, 11, 122sylancr 589 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
12414, 50, 53, 105, 13, 50, 123, 94itgmulc2nclem1 34945 . . . . . . 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)
12514, 65, 66, 110, 13, 65, 123, 97itgmulc2nclem1 34945 . . . . . . 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)
126124, 125oveq12d 7166 . . . . . 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))
127120, 121, 1263eqtrd 2858 . . . . 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))
128118, 119, 1273eqtr4d 2864 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥))
129101, 128oveq12d 7166 . . 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)))
13016, 22itgcl 24376 . . . 4 (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
1319, 14, 130subdird 11089 . . 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)))
1323oveq1d 7163 . . 3 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
133129, 131, 1323eqtr2d 2860 . 2 (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
13420, 47, 1333eqtrrd 2859 1 (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  ifcif 4465  {csn 4559   class class class wbr 5057  cmpt 5137   × cxp 5546  dom cdm 5548  (class class class)co 7148  f cof 7399  cc 10527  cr 10528  0cc0 10529   · cmul 10534  cle 10668  cmin 10862  -cneg 10863  volcvol 24056  MblFncmbf 24207  𝐿1cibl 24210  citg 24211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-ofr 7402  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-mod 13230  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-rest 16688  df-topgen 16709  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-top 21494  df-topon 21511  df-bases 21546  df-cmp 21987  df-ovol 24057  df-vol 24058  df-mbf 24212  df-itg1 24213  df-itg2 24214  df-ibl 24215  df-itg 24216  df-0p 24263
This theorem is referenced by:  itgmulc2nc  34947
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