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Theorem itgmulc2nclem2 33957
Description: Lemma for itgmulc2nc 33958; cf. itgmulc2lem2 23937. (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 12272 . . . . . . 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 6891 . . . . 5 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
54adantr 473 . . . 4 ((𝜑𝑥𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
6 0re 10328 . . . . . . . 8 0 ∈ ℝ
7 ifcl 4319 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
81, 6, 7sylancl 581 . . . . . . 7 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
98recnd 10355 . . . . . 6 (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
109adantr 473 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
111renegcld 10747 . . . . . . . 8 (𝜑 → -𝐶 ∈ ℝ)
12 ifcl 4319 . . . . . . . 8 ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1311, 6, 12sylancl 581 . . . . . . 7 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
1413recnd 10355 . . . . . 6 (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
1514adantr 473 . . . . 5 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
16 itgmulc2nc.5 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
1716recnd 10355 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
1810, 15, 17subdird 10777 . . . 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 2833 . . 3 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
2019itgeq2dv 23886 . 2 (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥)
21 ovexd 6910 . . 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 6910 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 · 𝐵) ∈ V)
2523, 24mbfdm2 23742 . . . . . 6 (𝜑𝐴 ∈ dom vol)
268adantr 473 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
27 fconstmpt 5366 . . . . . . 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 2798 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3025, 26, 16, 28, 29offval2 7146 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘𝑓 · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)))
31 iblmbf 23872 . . . . . . 7 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
3222, 31syl 17 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
3317fmpttd 6609 . . . . . 6 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℂ)
3432, 8, 33mbfmulc2re 23753 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘𝑓 · (𝑥𝐴𝐵)) ∈ MblFn)
3530, 34eqeltrrd 2877 . . . 4 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ MblFn)
369, 16, 22, 35iblmulc2nc 33955 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ 𝐿1)
37 ovexd 6910 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) ∈ V)
3813adantr 473 . . . . . 6 ((𝜑𝑥𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
39 fconstmpt 5366 . . . . . . 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 7146 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘𝑓 · (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
4232, 13, 33mbfmulc2re 23753 . . . . 5 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘𝑓 · (𝑥𝐴𝐵)) ∈ MblFn)
4341, 42eqeltrrd 2877 . . . 4 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ MblFn)
4414, 16, 22, 43iblmulc2nc 33955 . . 3 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ 𝐿1)
4519mpteq2dva 4935 . . . 4 (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) = (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))))
4645, 23eqeltrrd 2877 . . 3 (𝜑 → (𝑥𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))) ∈ MblFn)
4721, 36, 37, 44, 46itgsubnc 33952 . 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 6910 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
49 ifcl 4319 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5016, 6, 49sylancl 581 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5116iblre 23898 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
5222, 51mpbid 224 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
5352simpld 489 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
54 eqidd 2798 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
5525, 26, 50, 28, 54offval2 7146 . . . . . . . 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 23756 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
5750recnd 10355 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
5857fmpttd 6609 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℂ)
5956, 8, 58mbfmulc2re 23753 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘𝑓 · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
6055, 59eqeltrrd 2877 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
619, 50, 53, 60iblmulc2nc 33955 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
62 ovexd 6910 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
6316renegcld 10747 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
64 ifcl 4319 . . . . . . . 8 ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
6563, 6, 64sylancl 581 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
6652simprd 490 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
67 eqidd 2798 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) = (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
6825, 26, 65, 28, 67offval2 7146 . . . . . . . 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 23755 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ MblFn)
7063, 69mbfpos 23756 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
7165recnd 10355 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
7271fmpttd 6609 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℂ)
7370, 8, 72mbfmulc2re 23753 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘𝑓 · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
7468, 73eqeltrrd 2877 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
759, 65, 66, 74iblmulc2nc 33955 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
76 max0sub 12272 . . . . . . . . . . . 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 6892 . . . . . . . . . 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 10776 . . . . . . . . . 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 2833 . . . . . . . . 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 4935 . . . . . . . 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 2831 . . . . . . 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 2877 . . . . . 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 33952 . . . . 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 23886 . . . . 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 23901 . . . . . . 7 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
8786oveq2d 6892 . . . . . 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 23888 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
8965, 66itgcl 23888 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
909, 88, 89subdid 10776 . . . . . 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 12261 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
926, 1, 91sylancr 582 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
93 max1 12261 . . . . . . . . 9 ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
946, 16, 93sylancr 582 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
959, 50, 53, 60, 8, 50, 92, 94itgmulc2nclem1 33956 . . . . . . 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 12261 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
976, 63, 96sylancr 582 . . . . . . . 8 ((𝜑𝑥𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
989, 65, 66, 74, 8, 65, 92, 97itgmulc2nclem1 33956 . . . . . . 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 6894 . . . . . 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 2835 . . . . 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 2841 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥))
102 ovexd 6910 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
10325, 38, 50, 40, 54offval2 7146 . . . . . . . 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 23753 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘𝑓 · (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
105103, 104eqeltrrd 2877 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
10614, 50, 53, 105iblmulc2nc 33955 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿1)
107 ovexd 6910 . . . . . 6 ((𝜑𝑥𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
10825, 38, 65, 40, 67offval2 7146 . . . . . . . 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 23753 . . . . . . . 8 (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘𝑓 · (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
110108, 109eqeltrrd 2877 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
11114, 65, 66, 110iblmulc2nc 33955 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿1)
11277oveq2d 6892 . . . . . . . . . 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 10776 . . . . . . . . . 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 2833 . . . . . . . . 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 4935 . . . . . . . 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 2831 . . . . . . 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 2877 . . . . . 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 33952 . . . . 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 23886 . . . . 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 6892 . . . . . 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 10776 . . . . . 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 12261 . . . . . . . . 9 ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
1236, 11, 122sylancr 582 . . . . . . . 8 (𝜑 → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
12414, 50, 53, 105, 13, 50, 123, 94itgmulc2nclem1 33956 . . . . . . 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 33956 . . . . . . 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 6894 . . . . . 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 2835 . . . . 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 2841 . . . 4 (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥))
129101, 128oveq12d 6894 . . 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 23888 . . . 4 (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
1319, 14, 130subdird 10777 . . 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 6891 . . 3 (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
133129, 131, 1323eqtr2d 2837 . 2 (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
13420, 47, 1333eqtrrd 2836 1 (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3383  ifcif 4275  {csn 4366   class class class wbr 4841  cmpt 4920   × cxp 5308  dom cdm 5310  (class class class)co 6876  𝑓 cof 7127  cc 10220  cr 10221  0cc0 10222   · cmul 10227  cle 10362  cmin 10554  -cneg 10555  volcvol 23568  MblFncmbf 23719  𝐿1cibl 23722  citg 23723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-inf2 8786  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299  ax-pre-sup 10300  ax-addf 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-disj 4810  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-se 5270  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-of 7129  df-ofr 7130  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-2o 7798  df-oadd 7801  df-er 7980  df-map 8095  df-pm 8096  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-fi 8557  df-sup 8588  df-inf 8589  df-oi 8655  df-card 9049  df-cda 9276  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-div 10975  df-nn 11311  df-2 11372  df-3 11373  df-4 11374  df-n0 11577  df-z 11663  df-uz 11927  df-q 12030  df-rp 12071  df-xneg 12189  df-xadd 12190  df-xmul 12191  df-ioo 12424  df-ico 12426  df-icc 12427  df-fz 12577  df-fzo 12717  df-fl 12844  df-mod 12920  df-seq 13052  df-exp 13111  df-hash 13367  df-cj 14177  df-re 14178  df-im 14179  df-sqrt 14313  df-abs 14314  df-clim 14557  df-sum 14755  df-rest 16395  df-topgen 16416  df-psmet 20057  df-xmet 20058  df-met 20059  df-bl 20060  df-mopn 20061  df-top 21024  df-topon 21041  df-bases 21076  df-cmp 21516  df-ovol 23569  df-vol 23570  df-mbf 23724  df-itg1 23725  df-itg2 23726  df-ibl 23727  df-itg 23728  df-0p 23775
This theorem is referenced by:  itgmulc2nc  33958
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