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Theorem itgmulc2nclem2 36145
Description: Lemma for itgmulc2nc 36146; cf. itgmulc2lem2 25197. (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 13115 . . . . . . 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 7372 . . . . 5 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
54adantr 481 . . . 4 ((𝜑𝑥𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
6 0re 11157 . . . . . . . 8 0 ∈ ℝ
7 ifcl 4531 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
81, 6, 7sylancl 586 . . . . . . 7 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
98recnd 11183 . . . . . 6 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
109adantr 481 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
111renegcld 11582 . . . . . . . 8 (𝜑 → -𝐶 ∈ ℝ)
12 ifcl 4531 . . . . . . . 8 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1311, 6, 12sylancl 586 . . . . . . 7 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1413recnd 11183 . . . . . 6 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
1514adantr 481 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
16 itgmulc2nc.5 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
1716recnd 11183 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
1810, 15, 17subdird 11612 . . . 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 2778 . . 3 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
2019itgeq2dv 25146 . 2 (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥)
21 ovexd 7392 . . 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 7392 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) ∈ V)
2523, 24mbfdm2 25001 . . . . . 6 (𝜑𝐴 ∈ dom vol)
268adantr 481 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
27 fconstmpt 5694 . . . . . . 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 2737 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3025, 26, 16, 28, 29offval2 7637 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)))
31 iblmbf 25132 . . . . . . 7 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
3222, 31syl 17 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
3317fmpttd 7063 . . . . . 6 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℂ)
3432, 8, 33mbfmulc2re 25012 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) ∈ MblFn)
3530, 34eqeltrrd 2839 . . . 4 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ MblFn)
369, 16, 22, 35iblmulc2nc 36143 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ 𝐿1)
37 ovexd 7392 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) ∈ V)
3813adantr 481 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
39 fconstmpt 5694 . . . . . . 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 7637 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
4232, 13, 33mbfmulc2re 25012 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) ∈ MblFn)
4341, 42eqeltrrd 2839 . . . 4 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ MblFn)
4414, 16, 22, 43iblmulc2nc 36143 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ 𝐿1)
4519mpteq2dva 5205 . . . 4 (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) = (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))))
4645, 23eqeltrrd 2839 . . 3 (𝜑 → (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))) ∈ MblFn)
4721, 36, 37, 44, 46itgsubnc 36140 . 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 7392 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
49 ifcl 4531 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5016, 6, 49sylancl 586 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5116iblre 25158 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
5222, 51mpbid 231 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
5352simpld 495 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
54 eqidd 2737 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
5525, 26, 50, 28, 54offval2 7637 . . . . . . . 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 25015 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
5750recnd 11183 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
5857fmpttd 7063 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℂ)
5956, 8, 58mbfmulc2re 25012 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
6055, 59eqeltrrd 2839 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
619, 50, 53, 60iblmulc2nc 36143 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
62 ovexd 7392 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
6316renegcld 11582 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
64 ifcl 4531 . . . . . . . 8 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
6563, 6, 64sylancl 586 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
6652simprd 496 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
67 eqidd 2737 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
6825, 26, 65, 28, 67offval2 7637 . . . . . . . 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 25014 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ MblFn)
7063, 69mbfpos 25015 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
7165recnd 11183 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
7271fmpttd 7063 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℂ)
7370, 8, 72mbfmulc2re 25012 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
7468, 73eqeltrrd 2839 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
759, 65, 66, 74iblmulc2nc 36143 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
76 max0sub 13115 . . . . . . . . . . . 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 7373 . . . . . . . . . 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 11611 . . . . . . . . . 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 2778 . . . . . . . . 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 5205 . . . . . . . 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 2776 . . . . . . 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 2839 . . . . . 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 36140 . . . . 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 25146 . . . . 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 25161 . . . . . . 7 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
8786oveq2d 7373 . . . . . 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 25148 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
8965, 66itgcl 25148 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
909, 88, 89subdid 11611 . . . . . 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 13104 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
926, 1, 91sylancr 587 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
93 max1 13104 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
946, 16, 93sylancr 587 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
959, 50, 53, 60, 8, 50, 92, 94itgmulc2nclem1 36144 . . . . . . 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 13104 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
976, 63, 96sylancr 587 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
989, 65, 66, 74, 8, 65, 92, 97itgmulc2nclem1 36144 . . . . . . 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 7375 . . . . . 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 2780 . . . . 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 2786 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥))
102 ovexd 7392 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
10325, 38, 50, 40, 54offval2 7637 . . . . . . . 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 25012 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
105103, 104eqeltrrd 2839 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
10614, 50, 53, 105iblmulc2nc 36143 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
107 ovexd 7392 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
10825, 38, 65, 40, 67offval2 7637 . . . . . . . 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 25012 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
110108, 109eqeltrrd 2839 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
11114, 65, 66, 110iblmulc2nc 36143 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
11277oveq2d 7373 . . . . . . . . . 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 11611 . . . . . . . . . 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 2778 . . . . . . . . 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 5205 . . . . . . . 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 2776 . . . . . . 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 2839 . . . . . 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 36140 . . . . 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 25146 . . . . 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 7373 . . . . . 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 11611 . . . . . 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 13104 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
1236, 11, 122sylancr 587 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
12414, 50, 53, 105, 13, 50, 123, 94itgmulc2nclem1 36144 . . . . . . 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 36144 . . . . . . 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 7375 . . . . . 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 2780 . . . . 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 2786 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥))
129101, 128oveq12d 7375 . . 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 25148 . . . 4 (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
1319, 14, 130subdird 11612 . . 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 7372 . . 3 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
133129, 131, 1323eqtr2d 2782 . 2 (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
13420, 47, 1333eqtrrd 2781 1 (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  (class class class)co 7357  f cof 7615  cc 11049  cr 11050  0cc0 11051   · cmul 11056  cle 11190  cmin 11385  -cneg 11386  volcvol 24827  MblFncmbf 24978  𝐿1cibl 24981  citg 24982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984  df-itg2 24985  df-ibl 24986  df-itg 24987  df-0p 25034
This theorem is referenced by:  itgmulc2nc  36146
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