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Theorem mbfposadd 33812
Description: If the sum of two measurable functions is measurable, the sum of their nonnegative parts is measurable. (Contributed by Brendan Leahy, 2-Apr-2018.)
Hypotheses
Ref Expression
mbfposadd.1 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
mbfposadd.2 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
mbfposadd.3 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
mbfposadd.4 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
mbfposadd.5 (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)
Assertion
Ref Expression
mbfposadd (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem mbfposadd
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfposadd.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
2 0re 10295 . . . . 5 0 ∈ ℝ
3 ifcl 4287 . . . . 5 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
41, 2, 3sylancl 580 . . . 4 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5 mbfposadd.4 . . . . 5 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
6 ifcl 4287 . . . . 5 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
75, 2, 6sylancl 580 . . . 4 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
84, 7readdcld 10323 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
98fmpttd 6575 . 2 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))):𝐴⟶ℝ)
10 ssrab2 3847 . . . 4 {𝑥𝐴 ∣ 0 ≤ 𝐶} ⊆ 𝐴
11 fssres 6252 . . . 4 (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))):𝐴⟶ℝ ∧ {𝑥𝐴 ∣ 0 ≤ 𝐶} ⊆ 𝐴) → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}):{𝑥𝐴 ∣ 0 ≤ 𝐶}⟶ℝ)
129, 10, 11sylancl 580 . . 3 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}):{𝑥𝐴 ∣ 0 ≤ 𝐶}⟶ℝ)
13 inss2 3993 . . . . . 6 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐶}
14 resabs1 5602 . . . . . 6 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐶} → (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
1513, 14ax-mp 5 . . . . 5 (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
16 elin 3958 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
17 rabid 3263 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ↔ (𝑥𝐴 ∧ 0 ≤ 𝐵))
18 rabid 3263 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶} ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶))
1917, 18anbi12i 620 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
2016, 19bitri 266 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
21 iftrue 4249 . . . . . . . . . 10 (0 ≤ 𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 𝐵)
22 iftrue 4249 . . . . . . . . . 10 (0 ≤ 𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 𝐶)
2321, 22oveqan12d 6861 . . . . . . . . 9 ((0 ≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2423ad2ant2l 752 . . . . . . . 8 (((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2520, 24sylbi 208 . . . . . . 7 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2625mpteq2ia 4899 . . . . . 6 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶))
27 inss1 3992 . . . . . . . 8 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐵}
28 ssrab2 3847 . . . . . . . 8 {𝑥𝐴 ∣ 0 ≤ 𝐵} ⊆ 𝐴
2927, 28sstri 3770 . . . . . . 7 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴
30 resmpt 5626 . . . . . . . 8 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦𝐴𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
31 nfcv 2907 . . . . . . . . . 10 𝑦(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
32 nfcsb1v 3707 . . . . . . . . . 10 𝑥𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
33 csbeq1a 3700 . . . . . . . . . 10 (𝑥 = 𝑦 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
3431, 32, 33cbvmpt 4908 . . . . . . . . 9 (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦𝐴𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
3534reseq1i 5561 . . . . . . . 8 ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦𝐴𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
36 nfv 2009 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
37 nfrab1 3270 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ 0 ≤ 𝐵}
38 nfrab1 3270 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ 0 ≤ 𝐶}
3937, 38nfin 3980 . . . . . . . . . . . 12 𝑥({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
4039nfcri 2901 . . . . . . . . . . 11 𝑥 𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
4132nfeq2 2923 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
4240, 41nfan 1998 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
43 eleq1w 2827 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
4433eqeq2d 2775 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ↔ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
4543, 44anbi12d 624 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))))
4636, 42, 45cbvopab1 4882 . . . . . . . . 9 {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
47 df-mpt 4889 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
48 df-mpt 4889 . . . . . . . . 9 (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
4946, 47, 483eqtr4i 2797 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
5030, 35, 493eqtr4g 2824 . . . . . . 7 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
5129, 50ax-mp 5 . . . . . 6 ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
52 resmpt 5626 . . . . . . . 8 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦𝐴𝑦 / 𝑥(𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(𝐵 + 𝐶)))
53 nfcv 2907 . . . . . . . . . 10 𝑦(𝐵 + 𝐶)
54 nfcsb1v 3707 . . . . . . . . . 10 𝑥𝑦 / 𝑥(𝐵 + 𝐶)
55 csbeq1a 3700 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐵 + 𝐶) = 𝑦 / 𝑥(𝐵 + 𝐶))
5653, 54, 55cbvmpt 4908 . . . . . . . . 9 (𝑥𝐴 ↦ (𝐵 + 𝐶)) = (𝑦𝐴𝑦 / 𝑥(𝐵 + 𝐶))
5756reseq1i 5561 . . . . . . . 8 ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦𝐴𝑦 / 𝑥(𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
58 nfv 2009 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))
5954nfeq2 2923 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶)
6040, 59nfan 1998 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶))
6155eqeq2d 2775 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑧 = (𝐵 + 𝐶) ↔ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶)))
6243, 61anbi12d 624 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶)) ↔ (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶))))
6358, 60, 62cbvopab1 4882 . . . . . . . . 9 {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶))}
64 df-mpt 4889 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))}
65 df-mpt 4889 . . . . . . . . 9 (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(𝐵 + 𝐶)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶))}
6663, 64, 653eqtr4i 2797 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(𝐵 + 𝐶))
6752, 57, 663eqtr4g 2824 . . . . . . 7 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)))
6829, 67ax-mp 5 . . . . . 6 ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶))
6926, 51, 683eqtr4i 2797 . . . . 5 ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
7015, 69eqtri 2787 . . . 4 (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
71 mbfposadd.5 . . . . 5 (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)
721biantrurd 528 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)))
73 elrege0 12482 . . . . . . . . . 10 (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
7472, 73syl6bbr 280 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ 𝐵𝐵 ∈ (0[,)+∞)))
7574rabbidva 3337 . . . . . . . 8 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} = {𝑥𝐴𝐵 ∈ (0[,)+∞)})
76 0xr 10340 . . . . . . . . . . 11 0 ∈ ℝ*
77 pnfxr 10346 . . . . . . . . . . 11 +∞ ∈ ℝ*
78 0ltpnf 12156 . . . . . . . . . . 11 0 < +∞
79 snunioo 12505 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 < +∞) → ({0} ∪ (0(,)+∞)) = (0[,)+∞))
8076, 77, 78, 79mp3an 1585 . . . . . . . . . 10 ({0} ∪ (0(,)+∞)) = (0[,)+∞)
8180imaeq2i 5646 . . . . . . . . 9 ((𝑥𝐴𝐵) “ ({0} ∪ (0(,)+∞))) = ((𝑥𝐴𝐵) “ (0[,)+∞))
82 imaundi 5728 . . . . . . . . 9 ((𝑥𝐴𝐵) “ ({0} ∪ (0(,)+∞))) = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞)))
83 eqid 2765 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
8483mptpreima 5814 . . . . . . . . 9 ((𝑥𝐴𝐵) “ (0[,)+∞)) = {𝑥𝐴𝐵 ∈ (0[,)+∞)}
8581, 82, 843eqtr3ri 2796 . . . . . . . 8 {𝑥𝐴𝐵 ∈ (0[,)+∞)} = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞)))
8675, 85syl6eq 2815 . . . . . . 7 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))))
87 mbfposadd.1 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
881fmpttd 6575 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
89 mbfimasn 23690 . . . . . . . . . 10 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ ∧ 0 ∈ ℝ) → ((𝑥𝐴𝐵) “ {0}) ∈ dom vol)
902, 89mp3an3 1574 . . . . . . . . 9 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ {0}) ∈ dom vol)
91 mbfima 23688 . . . . . . . . 9 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ (0(,)+∞)) ∈ dom vol)
92 unmbl 23595 . . . . . . . . 9 ((((𝑥𝐴𝐵) “ {0}) ∈ dom vol ∧ ((𝑥𝐴𝐵) “ (0(,)+∞)) ∈ dom vol) → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9390, 91, 92syl2anc 579 . . . . . . . 8 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9487, 88, 93syl2anc 579 . . . . . . 7 (𝜑 → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9586, 94eqeltrd 2844 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol)
965biantrurd 528 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 ≤ 𝐶 ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)))
97 elrege0 12482 . . . . . . . . . 10 (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
9896, 97syl6bbr 280 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ 𝐶𝐶 ∈ (0[,)+∞)))
9998rabbidva 3337 . . . . . . . 8 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = {𝑥𝐴𝐶 ∈ (0[,)+∞)})
10080imaeq2i 5646 . . . . . . . . 9 ((𝑥𝐴𝐶) “ ({0} ∪ (0(,)+∞))) = ((𝑥𝐴𝐶) “ (0[,)+∞))
101 imaundi 5728 . . . . . . . . 9 ((𝑥𝐴𝐶) “ ({0} ∪ (0(,)+∞))) = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞)))
102 eqid 2765 . . . . . . . . . 10 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
103102mptpreima 5814 . . . . . . . . 9 ((𝑥𝐴𝐶) “ (0[,)+∞)) = {𝑥𝐴𝐶 ∈ (0[,)+∞)}
104100, 101, 1033eqtr3ri 2796 . . . . . . . 8 {𝑥𝐴𝐶 ∈ (0[,)+∞)} = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞)))
10599, 104syl6eq 2815 . . . . . . 7 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))))
106 mbfposadd.3 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
1075fmpttd 6575 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶):𝐴⟶ℝ)
108 mbfimasn 23690 . . . . . . . . . 10 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ ∧ 0 ∈ ℝ) → ((𝑥𝐴𝐶) “ {0}) ∈ dom vol)
1092, 108mp3an3 1574 . . . . . . . . 9 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ {0}) ∈ dom vol)
110 mbfima 23688 . . . . . . . . 9 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ (0(,)+∞)) ∈ dom vol)
111 unmbl 23595 . . . . . . . . 9 ((((𝑥𝐴𝐶) “ {0}) ∈ dom vol ∧ ((𝑥𝐴𝐶) “ (0(,)+∞)) ∈ dom vol) → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
112109, 110, 111syl2anc 579 . . . . . . . 8 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
113106, 107, 112syl2anc 579 . . . . . . 7 (𝜑 → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
114105, 113eqeltrd 2844 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol)
115 inmbl 23600 . . . . . 6 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
11695, 114, 115syl2anc 579 . . . . 5 (𝜑 → ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
117 mbfres 23702 . . . . 5 (((𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn ∧ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
11871, 116, 117syl2anc 579 . . . 4 (𝜑 → ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
11970, 118syl5eqel 2848 . . 3 (𝜑 → (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
120 inss2 3993 . . . . . 6 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐶}
121 resabs1 5602 . . . . . 6 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐶} → (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
122120, 121ax-mp 5 . . . . 5 (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
123 rabid 3263 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ↔ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐵))
124123, 18anbi12i 620 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
125 elin 3958 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
126 anandi 666 . . . . . . . . 9 ((𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
127124, 125, 1263bitr4i 294 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
128 iffalse 4252 . . . . . . . . . . 11 (¬ 0 ≤ 𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 0)
129128, 22oveqan12d 6861 . . . . . . . . . 10 ((¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶))
130129ad2antll 720 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶))
1315recnd 10322 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
132131addid2d 10491 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 + 𝐶) = 𝐶)
133132adantrr 708 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (0 + 𝐶) = 𝐶)
134130, 133eqtrd 2799 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶)
135127, 134sylan2b 587 . . . . . . 7 ((𝜑𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶)
136135mpteq2dva 4903 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶))
137 inss1 3992 . . . . . . . 8 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}
138 ssrab2 3847 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ⊆ 𝐴
139137, 138sstri 3770 . . . . . . 7 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴
140 resmpt 5626 . . . . . . . 8 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦𝐴𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
14134reseq1i 5561 . . . . . . . 8 ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦𝐴𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
142 nfv 2009 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
143 nfrab1 3270 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}
144143, 38nfin 3980 . . . . . . . . . . . 12 𝑥({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
145144nfcri 2901 . . . . . . . . . . 11 𝑥 𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
146145, 41nfan 1998 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
147 eleq1w 2827 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
148147, 44anbi12d 624 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))))
149142, 146, 148cbvopab1 4882 . . . . . . . . 9 {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
150 df-mpt 4889 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
151 df-mpt 4889 . . . . . . . . 9 (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
152149, 150, 1513eqtr4i 2797 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
153140, 141, 1523eqtr4g 2824 . . . . . . 7 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
154139, 153ax-mp 5 . . . . . 6 ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
155 resmpt 5626 . . . . . . . 8 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥𝐶))
156 nfcv 2907 . . . . . . . . . 10 𝑦𝐶
157 nfcsb1v 3707 . . . . . . . . . 10 𝑥𝑦 / 𝑥𝐶
158 csbeq1a 3700 . . . . . . . . . 10 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
159156, 157, 158cbvmpt 4908 . . . . . . . . 9 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
160159reseq1i 5561 . . . . . . . 8 ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
161 nfv 2009 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)
162157nfeq2 2923 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥𝐶
163145, 162nfan 1998 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)
164158eqeq2d 2775 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑧 = 𝐶𝑧 = 𝑦 / 𝑥𝐶))
165147, 164anbi12d 624 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)))
166161, 163, 165cbvopab1 4882 . . . . . . . . 9 {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)}
167 df-mpt 4889 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)}
168 df-mpt 4889 . . . . . . . . 9 (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)}
169166, 167, 1683eqtr4i 2797 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥𝐶)
170155, 160, 1693eqtr4g 2824 . . . . . . 7 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶))
171139, 170ax-mp 5 . . . . . 6 ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶)
172136, 154, 1713eqtr4g 2824 . . . . 5 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
173122, 172syl5eq 2811 . . . 4 (𝜑 → (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
17483mptpreima 5814 . . . . . . . 8 ((𝑥𝐴𝐵) “ (-∞(,)0)) = {𝑥𝐴𝐵 ∈ (-∞(,)0)}
175 elioomnf 12471 . . . . . . . . . . 11 (0 ∈ ℝ* → (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0)))
17676, 175ax-mp 5 . . . . . . . . . 10 (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0))
1771biantrurd 528 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 < 0 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0)))
178 ltnle 10371 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
1791, 2, 178sylancl 580 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
180177, 179bitr3d 272 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝐵 ∈ ℝ ∧ 𝐵 < 0) ↔ ¬ 0 ≤ 𝐵))
181176, 180syl5bb 274 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 ∈ (-∞(,)0) ↔ ¬ 0 ≤ 𝐵))
182181rabbidva 3337 . . . . . . . 8 (𝜑 → {𝑥𝐴𝐵 ∈ (-∞(,)0)} = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
183174, 182syl5eq 2811 . . . . . . 7 (𝜑 → ((𝑥𝐴𝐵) “ (-∞(,)0)) = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
184 mbfima 23688 . . . . . . . 8 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ (-∞(,)0)) ∈ dom vol)
18587, 88, 184syl2anc 579 . . . . . . 7 (𝜑 → ((𝑥𝐴𝐵) “ (-∞(,)0)) ∈ dom vol)
186183, 185eqeltrrd 2845 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol)
187 inmbl 23600 . . . . . 6 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
188186, 114, 187syl2anc 579 . . . . 5 (𝜑 → ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
189 mbfres 23702 . . . . 5 (((𝑥𝐴𝐶) ∈ MblFn ∧ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
190106, 188, 189syl2anc 579 . . . 4 (𝜑 → ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
191173, 190eqeltrd 2844 . . 3 (𝜑 → (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
192 ssid 3783 . . . . . 6 𝐴𝐴
193 dfrab3ss 4069 . . . . . 6 (𝐴𝐴 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
194192, 193ax-mp 5 . . . . 5 {𝑥𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
195 rabxm 4123 . . . . . 6 𝐴 = ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
196195ineq1i 3972 . . . . 5 (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
197 indir 4040 . . . . 5 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
198194, 196, 1973eqtrri 2792 . . . 4 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = {𝑥𝐴 ∣ 0 ≤ 𝐶}
199198a1i 11 . . 3 (𝜑 → (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = {𝑥𝐴 ∣ 0 ≤ 𝐶})
20012, 119, 191, 199mbfres2 23703 . 2 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ MblFn)
201 rabid 3263 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐶))
202 iffalse 4252 . . . . . . . . 9 (¬ 0 ≤ 𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 0)
203202oveq2d 6858 . . . . . . . 8 (¬ 0 ≤ 𝐶 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (if(0 ≤ 𝐵, 𝐵, 0) + 0))
2044recnd 10322 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
205204addid1d 10490 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + 0) = if(0 ≤ 𝐵, 𝐵, 0))
206203, 205sylan9eqr 2821 . . . . . . 7 (((𝜑𝑥𝐴) ∧ ¬ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
207206anasss 458 . . . . . 6 ((𝜑 ∧ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
208201, 207sylan2b 587 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
209208mpteq2dva 4903 . . . 4 (𝜑 → (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)))
210 ssrab2 3847 . . . . 5 {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴
211 resmpt 5626 . . . . . 6 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑦𝐴𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
21234reseq1i 5561 . . . . . 6 ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑦𝐴𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
213 nfv 2009 . . . . . . . 8 𝑦(𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
214 nfrab1 3270 . . . . . . . . . 10 𝑥{𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}
215214nfcri 2901 . . . . . . . . 9 𝑥 𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}
216215, 41nfan 1998 . . . . . . . 8 𝑥(𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
217 eleq1w 2827 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}))
218217, 44anbi12d 624 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))))
219213, 216, 218cbvopab1 4882 . . . . . . 7 {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
220 df-mpt 4889 . . . . . . 7 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
221 df-mpt 4889 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
222219, 220, 2213eqtr4i 2797 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
223211, 212, 2223eqtr4g 2824 . . . . 5 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
224210, 223ax-mp 5 . . . 4 ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
225 resmpt 5626 . . . . . 6 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑦𝐴𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)))
226 nfcv 2907 . . . . . . . 8 𝑦if(0 ≤ 𝐵, 𝐵, 0)
227 nfcsb1v 3707 . . . . . . . 8 𝑥𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)
228 csbeq1a 3700 . . . . . . . 8 (𝑥 = 𝑦 → if(0 ≤ 𝐵, 𝐵, 0) = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
229226, 227, 228cbvmpt 4908 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦𝐴𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
230229reseq1i 5561 . . . . . 6 ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑦𝐴𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
231 nfv 2009 . . . . . . . 8 𝑦(𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))
232227nfeq2 2923 . . . . . . . . 9 𝑥 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)
233215, 232nfan 1998 . . . . . . . 8 𝑥(𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
234228eqeq2d 2775 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑧 = if(0 ≤ 𝐵, 𝐵, 0) ↔ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)))
235217, 234anbi12d 624 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0)) ↔ (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))))
236231, 233, 235cbvopab1 4882 . . . . . . 7 {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))}
237 df-mpt 4889 . . . . . . 7 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))}
238 df-mpt 4889 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))}
239236, 237, 2383eqtr4i 2797 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
240225, 230, 2393eqtr4g 2824 . . . . 5 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)))
241210, 240ax-mp 5 . . . 4 ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0))
242209, 224, 2413eqtr4g 2824 . . 3 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}))
2431, 87mbfpos 23709 . . . 4 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
244102mptpreima 5814 . . . . . 6 ((𝑥𝐴𝐶) “ (-∞(,)0)) = {𝑥𝐴𝐶 ∈ (-∞(,)0)}
245 elioomnf 12471 . . . . . . . . 9 (0 ∈ ℝ* → (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0)))
24676, 245ax-mp 5 . . . . . . . 8 (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0))
2475biantrurd 528 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐶 < 0 ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0)))
248 ltnle 10371 . . . . . . . . . 10 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
2495, 2, 248sylancl 580 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
250247, 249bitr3d 272 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝐶 ∈ ℝ ∧ 𝐶 < 0) ↔ ¬ 0 ≤ 𝐶))
251246, 250syl5bb 274 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 ∈ (-∞(,)0) ↔ ¬ 0 ≤ 𝐶))
252251rabbidva 3337 . . . . . 6 (𝜑 → {𝑥𝐴𝐶 ∈ (-∞(,)0)} = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
253244, 252syl5eq 2811 . . . . 5 (𝜑 → ((𝑥𝐴𝐶) “ (-∞(,)0)) = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
254 mbfima 23688 . . . . . 6 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ (-∞(,)0)) ∈ dom vol)
255106, 107, 254syl2anc 579 . . . . 5 (𝜑 → ((𝑥𝐴𝐶) “ (-∞(,)0)) ∈ dom vol)
256253, 255eqeltrrd 2845 . . . 4 (𝜑 → {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol)
257 mbfres 23702 . . . 4 (((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol) → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
258243, 256, 257syl2anc 579 . . 3 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
259242, 258eqeltrd 2844 . 2 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
260 rabxm 4123 . . . 4 𝐴 = ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
261260eqcomi 2774 . . 3 ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴
262261a1i 11 . 2 (𝜑 → ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴)
2639, 200, 259, 262mbfres2 23703 1 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {crab 3059  csb 3691  cun 3730  cin 3731  wss 3732  ifcif 4243  {csn 4334   class class class wbr 4809  {copab 4871  cmpt 4888  ccnv 5276  dom cdm 5277  cres 5279  cima 5280  wf 6064  (class class class)co 6842  cr 10188  0cc0 10189   + caddc 10192  +∞cpnf 10325  -∞cmnf 10326  *cxr 10327   < clt 10328  cle 10329  (,)cioo 12377  [,)cico 12379  volcvol 23521  MblFncmbf 23672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-xadd 12147  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-sum 14704  df-xmet 20012  df-met 20013  df-ovol 23522  df-vol 23523  df-mbf 23677
This theorem is referenced by:  itgaddnclem2  33824
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