Mathbox for Brendan Leahy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itgmulc2nclem2 Structured version   Visualization version   GIF version

Theorem itgmulc2nclem2 35397
 Description: Lemma for itgmulc2nc 35398; cf. itgmulc2lem2 24525. (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 12623 . . . . . . 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 7166 . . . . 5 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
54adantr 485 . . . 4 ((𝜑𝑥𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
6 0re 10674 . . . . . . . 8 0 ∈ ℝ
7 ifcl 4466 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
81, 6, 7sylancl 590 . . . . . . 7 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
98recnd 10700 . . . . . 6 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
109adantr 485 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
111renegcld 11098 . . . . . . . 8 (𝜑 → -𝐶 ∈ ℝ)
12 ifcl 4466 . . . . . . . 8 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1311, 6, 12sylancl 590 . . . . . . 7 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1413recnd 10700 . . . . . 6 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
1514adantr 485 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
16 itgmulc2nc.5 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
1716recnd 10700 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
1810, 15, 17subdird 11128 . . . 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 2796 . . 3 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
2019itgeq2dv 24474 . 2 (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥)
21 ovexd 7186 . . 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 7186 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) ∈ V)
2523, 24mbfdm2 24330 . . . . . 6 (𝜑𝐴 ∈ dom vol)
268adantr 485 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
27 fconstmpt 5584 . . . . . . 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 2760 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3025, 26, 16, 28, 29offval2 7425 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)))
31 iblmbf 24460 . . . . . . 7 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
3222, 31syl 17 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
3317fmpttd 6871 . . . . . 6 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℂ)
3432, 8, 33mbfmulc2re 24341 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) ∈ MblFn)
3530, 34eqeltrrd 2854 . . . 4 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ MblFn)
369, 16, 22, 35iblmulc2nc 35395 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ 𝐿1)
37 ovexd 7186 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) ∈ V)
3813adantr 485 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
39 fconstmpt 5584 . . . . . . 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 7425 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
4232, 13, 33mbfmulc2re 24341 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴𝐵)) ∈ MblFn)
4341, 42eqeltrrd 2854 . . . 4 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ MblFn)
4414, 16, 22, 43iblmulc2nc 35395 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ 𝐿1)
4519mpteq2dva 5128 . . . 4 (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) = (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))))
4645, 23eqeltrrd 2854 . . 3 (𝜑 → (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))) ∈ MblFn)
4721, 36, 37, 44, 46itgsubnc 35392 . 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 7186 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
49 ifcl 4466 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5016, 6, 49sylancl 590 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5116iblre 24486 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
5222, 51mpbid 235 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
5352simpld 499 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
54 eqidd 2760 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
5525, 26, 50, 28, 54offval2 7425 . . . . . . . 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 24344 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
5750recnd 10700 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
5857fmpttd 6871 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℂ)
5956, 8, 58mbfmulc2re 24341 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
6055, 59eqeltrrd 2854 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
619, 50, 53, 60iblmulc2nc 35395 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
62 ovexd 7186 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
6316renegcld 11098 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
64 ifcl 4466 . . . . . . . 8 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
6563, 6, 64sylancl 590 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
6652simprd 500 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
67 eqidd 2760 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
6825, 26, 65, 28, 67offval2 7425 . . . . . . . 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 24343 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ MblFn)
7063, 69mbfpos 24344 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
7165recnd 10700 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
7271fmpttd 6871 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℂ)
7370, 8, 72mbfmulc2re 24341 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
7468, 73eqeltrrd 2854 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
759, 65, 66, 74iblmulc2nc 35395 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
76 max0sub 12623 . . . . . . . . . . . 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 7167 . . . . . . . . . 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 11127 . . . . . . . . . 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 2796 . . . . . . . . 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 5128 . . . . . . . 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 2794 . . . . . . 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 2854 . . . . . 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 35392 . . . . 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 24474 . . . . 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 24489 . . . . . . 7 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
8786oveq2d 7167 . . . . . 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 24476 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
8965, 66itgcl 24476 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
909, 88, 89subdid 11127 . . . . . 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 12612 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
926, 1, 91sylancr 591 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
93 max1 12612 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
946, 16, 93sylancr 591 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
959, 50, 53, 60, 8, 50, 92, 94itgmulc2nclem1 35396 . . . . . . 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 12612 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
976, 63, 96sylancr 591 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
989, 65, 66, 74, 8, 65, 92, 97itgmulc2nclem1 35396 . . . . . . 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 7169 . . . . . 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 2798 . . . . 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 2804 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥))
102 ovexd 7186 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
10325, 38, 50, 40, 54offval2 7425 . . . . . . . 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 24341 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
105103, 104eqeltrrd 2854 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
10614, 50, 53, 105iblmulc2nc 35395 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
107 ovexd 7186 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
10825, 38, 65, 40, 67offval2 7425 . . . . . . . 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 24341 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘f · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
110108, 109eqeltrrd 2854 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
11114, 65, 66, 110iblmulc2nc 35395 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
11277oveq2d 7167 . . . . . . . . . 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 11127 . . . . . . . . . 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 2796 . . . . . . . . 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 5128 . . . . . . . 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 2794 . . . . . . 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 2854 . . . . . 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 35392 . . . . 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 24474 . . . . 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 7167 . . . . . 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 11127 . . . . . 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 12612 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
1236, 11, 122sylancr 591 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
12414, 50, 53, 105, 13, 50, 123, 94itgmulc2nclem1 35396 . . . . . . 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 35396 . . . . . . 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 7169 . . . . . 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 2798 . . . . 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 2804 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥))
129101, 128oveq12d 7169 . . 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 24476 . . . 4 (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
1319, 14, 130subdird 11128 . . 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 7166 . . 3 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
133129, 131, 1323eqtr2d 2800 . 2 (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
13420, 47, 1333eqtrrd 2799 1 (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  Vcvv 3410  ifcif 4421  {csn 4523   class class class wbr 5033   ↦ cmpt 5113   × cxp 5523  dom cdm 5525  (class class class)co 7151   ∘f cof 7404  ℂcc 10566  ℝcr 10567  0cc0 10568   · cmul 10573   ≤ cle 10707   − cmin 10901  -cneg 10902  volcvol 24156  MblFncmbf 24307  𝐿1cibl 24310  ∫citg 24311 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-inf2 9130  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645  ax-pre-sup 10646  ax-addf 10647 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7406  df-ofr 7407  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-2o 8114  df-oadd 8117  df-er 8300  df-map 8419  df-pm 8420  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-fi 8901  df-sup 8932  df-inf 8933  df-oi 9000  df-dju 9356  df-card 9394  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-div 11329  df-nn 11668  df-2 11730  df-3 11731  df-4 11732  df-n0 11928  df-z 12014  df-uz 12276  df-q 12382  df-rp 12424  df-xneg 12541  df-xadd 12542  df-xmul 12543  df-ioo 12776  df-ico 12778  df-icc 12779  df-fz 12933  df-fzo 13076  df-fl 13204  df-mod 13280  df-seq 13412  df-exp 13473  df-hash 13734  df-cj 14499  df-re 14500  df-im 14501  df-sqrt 14635  df-abs 14636  df-clim 14886  df-sum 15084  df-rest 16747  df-topgen 16768  df-psmet 20151  df-xmet 20152  df-met 20153  df-bl 20154  df-mopn 20155  df-top 21587  df-topon 21604  df-bases 21639  df-cmp 22080  df-ovol 24157  df-vol 24158  df-mbf 24312  df-itg1 24313  df-itg2 24314  df-ibl 24315  df-itg 24316  df-0p 24363 This theorem is referenced by:  itgmulc2nc  35398
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