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

 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)

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfposadd.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
2 0re 10641 . . . . 5 0 ∈ ℝ
3 ifcl 4494 . . . . 5 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
41, 2, 3sylancl 589 . . . 4 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5 mbfposadd.4 . . . . 5 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
6 ifcl 4494 . . . . 5 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
75, 2, 6sylancl 589 . . . 4 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
84, 7readdcld 10668 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
98fmpttd 6870 . 2 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))):𝐴⟶ℝ)
10 ssrab2 4042 . . . 4 {𝑥𝐴 ∣ 0 ≤ 𝐶} ⊆ 𝐴
11 fssres 6534 . . . 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 589 . . 3 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}):{𝑥𝐴 ∣ 0 ≤ 𝐶}⟶ℝ)
13 inss2 4191 . . . . . 6 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐶}
14 resabs1 5870 . . . . . 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 3935 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
17 rabid 3369 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ↔ (𝑥𝐴 ∧ 0 ≤ 𝐵))
18 rabid 3369 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶} ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶))
1917, 18anbi12i 629 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
2016, 19bitri 278 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
21 iftrue 4456 . . . . . . . . . 10 (0 ≤ 𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 𝐵)
22 iftrue 4456 . . . . . . . . . 10 (0 ≤ 𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 𝐶)
2321, 22oveqan12d 7168 . . . . . . . . 9 ((0 ≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2423ad2ant2l 745 . . . . . . . 8 (((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2520, 24sylbi 220 . . . . . . 7 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2625mpteq2ia 5143 . . . . . 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 4190 . . . . . . . 8 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐵}
28 ssrab2 4042 . . . . . . . 8 {𝑥𝐴 ∣ 0 ≤ 𝐵} ⊆ 𝐴
2927, 28sstri 3962 . . . . . . 7 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴
30 resmpt 5892 . . . . . . . 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 2982 . . . . . . . . . 10 𝑦(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
32 nfcsb1v 3890 . . . . . . . . . 10 𝑥𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
33 csbeq1a 3880 . . . . . . . . . 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 5153 . . . . . . . . 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 5836 . . . . . . . 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 1916 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
37 nfrab1 3375 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ 0 ≤ 𝐵}
38 nfrab1 3375 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ 0 ≤ 𝐶}
3937, 38nfin 4178 . . . . . . . . . . . 12 𝑥({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
4039nfcri 2969 . . . . . . . . . . 11 𝑥 𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
4132nfeq2 2999 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
4240, 41nfan 1901 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
43 eleq1w 2898 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
4433eqeq2d 2835 . . . . . . . . . . 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 633 . . . . . . . . . 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 5125 . . . . . . . . 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 5133 . . . . . . . . 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 5133 . . . . . . . . 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 2857 . . . . . . . 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 2884 . . . . . . 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 5892 . . . . . . . 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 2982 . . . . . . . . . 10 𝑦(𝐵 + 𝐶)
54 nfcsb1v 3890 . . . . . . . . . 10 𝑥𝑦 / 𝑥(𝐵 + 𝐶)
55 csbeq1a 3880 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐵 + 𝐶) = 𝑦 / 𝑥(𝐵 + 𝐶))
5653, 54, 55cbvmpt 5153 . . . . . . . . 9 (𝑥𝐴 ↦ (𝐵 + 𝐶)) = (𝑦𝐴𝑦 / 𝑥(𝐵 + 𝐶))
5756reseq1i 5836 . . . . . . . 8 ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦𝐴𝑦 / 𝑥(𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
58 nfv 1916 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))
5954nfeq2 2999 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶)
6040, 59nfan 1901 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶))
6155eqeq2d 2835 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑧 = (𝐵 + 𝐶) ↔ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶)))
6243, 61anbi12d 633 . . . . . . . . . 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 5125 . . . . . . . . 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 5133 . . . . . . . . 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 5133 . . . . . . . . 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 2857 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(𝐵 + 𝐶))
6752, 57, 663eqtr4g 2884 . . . . . . 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 2857 . . . . 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 2847 . . . 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 536 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)))
73 elrege0 12841 . . . . . . . . . 10 (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
7472, 73syl6bbr 292 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ 𝐵𝐵 ∈ (0[,)+∞)))
7574rabbidva 3463 . . . . . . . 8 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} = {𝑥𝐴𝐵 ∈ (0[,)+∞)})
76 0xr 10686 . . . . . . . . . . 11 0 ∈ ℝ*
77 pnfxr 10693 . . . . . . . . . . 11 +∞ ∈ ℝ*
78 0ltpnf 12514 . . . . . . . . . . 11 0 < +∞
79 snunioo 12865 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 < +∞) → ({0} ∪ (0(,)+∞)) = (0[,)+∞))
8076, 77, 78, 79mp3an 1458 . . . . . . . . . 10 ({0} ∪ (0(,)+∞)) = (0[,)+∞)
8180imaeq2i 5914 . . . . . . . . 9 ((𝑥𝐴𝐵) “ ({0} ∪ (0(,)+∞))) = ((𝑥𝐴𝐵) “ (0[,)+∞))
82 imaundi 5995 . . . . . . . . 9 ((𝑥𝐴𝐵) “ ({0} ∪ (0(,)+∞))) = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞)))
83 eqid 2824 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
8483mptpreima 6079 . . . . . . . . 9 ((𝑥𝐴𝐵) “ (0[,)+∞)) = {𝑥𝐴𝐵 ∈ (0[,)+∞)}
8581, 82, 843eqtr3ri 2856 . . . . . . . 8 {𝑥𝐴𝐵 ∈ (0[,)+∞)} = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞)))
8675, 85syl6eq 2875 . . . . . . 7 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))))
87 mbfposadd.1 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
881fmpttd 6870 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
89 mbfimasn 24243 . . . . . . . . . 10 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ ∧ 0 ∈ ℝ) → ((𝑥𝐴𝐵) “ {0}) ∈ dom vol)
902, 89mp3an3 1447 . . . . . . . . 9 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ {0}) ∈ dom vol)
91 mbfima 24241 . . . . . . . . 9 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ (0(,)+∞)) ∈ dom vol)
92 unmbl 24148 . . . . . . . . 9 ((((𝑥𝐴𝐵) “ {0}) ∈ dom vol ∧ ((𝑥𝐴𝐵) “ (0(,)+∞)) ∈ dom vol) → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9390, 91, 92syl2anc 587 . . . . . . . 8 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9487, 88, 93syl2anc 587 . . . . . . 7 (𝜑 → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9586, 94eqeltrd 2916 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol)
965biantrurd 536 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 ≤ 𝐶 ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)))
97 elrege0 12841 . . . . . . . . . 10 (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
9896, 97syl6bbr 292 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ 𝐶𝐶 ∈ (0[,)+∞)))
9998rabbidva 3463 . . . . . . . 8 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = {𝑥𝐴𝐶 ∈ (0[,)+∞)})
10080imaeq2i 5914 . . . . . . . . 9 ((𝑥𝐴𝐶) “ ({0} ∪ (0(,)+∞))) = ((𝑥𝐴𝐶) “ (0[,)+∞))
101 imaundi 5995 . . . . . . . . 9 ((𝑥𝐴𝐶) “ ({0} ∪ (0(,)+∞))) = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞)))
102 eqid 2824 . . . . . . . . . 10 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
103102mptpreima 6079 . . . . . . . . 9 ((𝑥𝐴𝐶) “ (0[,)+∞)) = {𝑥𝐴𝐶 ∈ (0[,)+∞)}
104100, 101, 1033eqtr3ri 2856 . . . . . . . 8 {𝑥𝐴𝐶 ∈ (0[,)+∞)} = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞)))
10599, 104syl6eq 2875 . . . . . . 7 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))))
106 mbfposadd.3 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
1075fmpttd 6870 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶):𝐴⟶ℝ)
108 mbfimasn 24243 . . . . . . . . . 10 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ ∧ 0 ∈ ℝ) → ((𝑥𝐴𝐶) “ {0}) ∈ dom vol)
1092, 108mp3an3 1447 . . . . . . . . 9 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ {0}) ∈ dom vol)
110 mbfima 24241 . . . . . . . . 9 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ (0(,)+∞)) ∈ dom vol)
111 unmbl 24148 . . . . . . . . 9 ((((𝑥𝐴𝐶) “ {0}) ∈ dom vol ∧ ((𝑥𝐴𝐶) “ (0(,)+∞)) ∈ dom vol) → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
112109, 110, 111syl2anc 587 . . . . . . . 8 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
113106, 107, 112syl2anc 587 . . . . . . 7 (𝜑 → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
114105, 113eqeltrd 2916 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol)
115 inmbl 24153 . . . . . 6 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
11695, 114, 115syl2anc 587 . . . . 5 (𝜑 → ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
117 mbfres 24255 . . . . 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 587 . . . 4 (𝜑 → ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
11970, 118eqeltrid 2920 . . 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 4191 . . . . . 6 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐶}
121 resabs1 5870 . . . . . 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 3369 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ↔ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐵))
124123, 18anbi12i 629 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
125 elin 3935 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
126 anandi 675 . . . . . . . . 9 ((𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
127124, 125, 1263bitr4i 306 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
128 iffalse 4459 . . . . . . . . . . 11 (¬ 0 ≤ 𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 0)
129128, 22oveqan12d 7168 . . . . . . . . . 10 ((¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶))
130129ad2antll 728 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶))
1315recnd 10667 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
132131addid2d 10839 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 + 𝐶) = 𝐶)
133132adantrr 716 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (0 + 𝐶) = 𝐶)
134130, 133eqtrd 2859 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶)
135127, 134sylan2b 596 . . . . . . 7 ((𝜑𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶)
136135mpteq2dva 5147 . . . . . 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 4190 . . . . . . . 8 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}
138 ssrab2 4042 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ⊆ 𝐴
139137, 138sstri 3962 . . . . . . 7 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴
140 resmpt 5892 . . . . . . . 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 5836 . . . . . . . 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 1916 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
143 nfrab1 3375 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}
144143, 38nfin 4178 . . . . . . . . . . . 12 𝑥({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
145144nfcri 2969 . . . . . . . . . . 11 𝑥 𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
146145, 41nfan 1901 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
147 eleq1w 2898 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
148147, 44anbi12d 633 . . . . . . . . . 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 5125 . . . . . . . . 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 5133 . . . . . . . . 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 5133 . . . . . . . . 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 2857 . . . . . . . 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 2884 . . . . . . 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 5892 . . . . . . . 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 2982 . . . . . . . . . 10 𝑦𝐶
157 nfcsb1v 3890 . . . . . . . . . 10 𝑥𝑦 / 𝑥𝐶
158 csbeq1a 3880 . . . . . . . . . 10 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
159156, 157, 158cbvmpt 5153 . . . . . . . . 9 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
160159reseq1i 5836 . . . . . . . 8 ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
161 nfv 1916 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)
162157nfeq2 2999 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥𝐶
163145, 162nfan 1901 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)
164158eqeq2d 2835 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑧 = 𝐶𝑧 = 𝑦 / 𝑥𝐶))
165147, 164anbi12d 633 . . . . . . . . . 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 5125 . . . . . . . . 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 5133 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)}
168 df-mpt 5133 . . . . . . . . 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 2857 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥𝐶)
170155, 160, 1693eqtr4g 2884 . . . . . . 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 2884 . . . . 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 2871 . . . 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 6079 . . . . . . . 8 ((𝑥𝐴𝐵) “ (-∞(,)0)) = {𝑥𝐴𝐵 ∈ (-∞(,)0)}
175 elioomnf 12831 . . . . . . . . . . 11 (0 ∈ ℝ* → (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0)))
17676, 175ax-mp 5 . . . . . . . . . 10 (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0))
1771biantrurd 536 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 < 0 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0)))
178 ltnle 10718 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
1791, 2, 178sylancl 589 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
180177, 179bitr3d 284 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝐵 ∈ ℝ ∧ 𝐵 < 0) ↔ ¬ 0 ≤ 𝐵))
181176, 180syl5bb 286 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 ∈ (-∞(,)0) ↔ ¬ 0 ≤ 𝐵))
182181rabbidva 3463 . . . . . . . 8 (𝜑 → {𝑥𝐴𝐵 ∈ (-∞(,)0)} = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
183174, 182syl5eq 2871 . . . . . . 7 (𝜑 → ((𝑥𝐴𝐵) “ (-∞(,)0)) = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
184 mbfima 24241 . . . . . . . 8 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ (-∞(,)0)) ∈ dom vol)
18587, 88, 184syl2anc 587 . . . . . . 7 (𝜑 → ((𝑥𝐴𝐵) “ (-∞(,)0)) ∈ dom vol)
186183, 185eqeltrrd 2917 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol)
187 inmbl 24153 . . . . . 6 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
188186, 114, 187syl2anc 587 . . . . 5 (𝜑 → ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
189 mbfres 24255 . . . . 5 (((𝑥𝐴𝐶) ∈ MblFn ∧ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
190106, 188, 189syl2anc 587 . . . 4 (𝜑 → ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
191173, 190eqeltrd 2916 . . 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 3975 . . . . . 6 𝐴𝐴
193 dfrab3ss 4266 . . . . . 6 (𝐴𝐴 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
194192, 193ax-mp 5 . . . . 5 {𝑥𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
195 rabxm 4323 . . . . . 6 𝐴 = ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
196195ineq1i 4170 . . . . 5 (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
197 indir 4237 . . . . 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 2852 . . . 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 24256 . 2 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ MblFn)
201 rabid 3369 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐶))
202 iffalse 4459 . . . . . . . . 9 (¬ 0 ≤ 𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 0)
203202oveq2d 7165 . . . . . . . 8 (¬ 0 ≤ 𝐶 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (if(0 ≤ 𝐵, 𝐵, 0) + 0))
2044recnd 10667 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
205204addid1d 10838 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + 0) = if(0 ≤ 𝐵, 𝐵, 0))
206203, 205sylan9eqr 2881 . . . . . . 7 (((𝜑𝑥𝐴) ∧ ¬ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
207206anasss 470 . . . . . 6 ((𝜑 ∧ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
208201, 207sylan2b 596 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
209208mpteq2dva 5147 . . . 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 4042 . . . . 5 {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴
211 resmpt 5892 . . . . . 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 5836 . . . . . 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 1916 . . . . . . . 8 𝑦(𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
214 nfrab1 3375 . . . . . . . . . 10 𝑥{𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}
215214nfcri 2969 . . . . . . . . 9 𝑥 𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}
216215, 41nfan 1901 . . . . . . . 8 𝑥(𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
217 eleq1w 2898 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}))
218217, 44anbi12d 633 . . . . . . . 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 5125 . . . . . . 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 5133 . . . . . . 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 5133 . . . . . . 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 2857 . . . . . 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 2884 . . . . 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 5892 . . . . . 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 2982 . . . . . . . 8 𝑦if(0 ≤ 𝐵, 𝐵, 0)
227 nfcsb1v 3890 . . . . . . . 8 𝑥𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)
228 csbeq1a 3880 . . . . . . . 8 (𝑥 = 𝑦 → if(0 ≤ 𝐵, 𝐵, 0) = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
229226, 227, 228cbvmpt 5153 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦𝐴𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
230229reseq1i 5836 . . . . . 6 ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑦𝐴𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
231 nfv 1916 . . . . . . . 8 𝑦(𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))
232227nfeq2 2999 . . . . . . . . 9 𝑥 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)
233215, 232nfan 1901 . . . . . . . 8 𝑥(𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
234228eqeq2d 2835 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑧 = if(0 ≤ 𝐵, 𝐵, 0) ↔ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)))
235217, 234anbi12d 633 . . . . . . . 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 5125 . . . . . . 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 5133 . . . . . . 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 5133 . . . . . . 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 2857 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
240225, 230, 2393eqtr4g 2884 . . . . 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 2884 . . 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 24262 . . . 4 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
244102mptpreima 6079 . . . . . 6 ((𝑥𝐴𝐶) “ (-∞(,)0)) = {𝑥𝐴𝐶 ∈ (-∞(,)0)}
245 elioomnf 12831 . . . . . . . . 9 (0 ∈ ℝ* → (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0)))
24676, 245ax-mp 5 . . . . . . . 8 (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0))
2475biantrurd 536 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐶 < 0 ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0)))
248 ltnle 10718 . . . . . . . . . 10 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
2495, 2, 248sylancl 589 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
250247, 249bitr3d 284 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝐶 ∈ ℝ ∧ 𝐶 < 0) ↔ ¬ 0 ≤ 𝐶))
251246, 250syl5bb 286 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 ∈ (-∞(,)0) ↔ ¬ 0 ≤ 𝐶))
252251rabbidva 3463 . . . . . 6 (𝜑 → {𝑥𝐴𝐶 ∈ (-∞(,)0)} = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
253244, 252syl5eq 2871 . . . . 5 (𝜑 → ((𝑥𝐴𝐶) “ (-∞(,)0)) = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
254 mbfima 24241 . . . . . 6 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ (-∞(,)0)) ∈ dom vol)
255106, 107, 254syl2anc 587 . . . . 5 (𝜑 → ((𝑥𝐴𝐶) “ (-∞(,)0)) ∈ dom vol)
256253, 255eqeltrrd 2917 . . . 4 (𝜑 → {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol)
257 mbfres 24255 . . . 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 587 . . 3 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
259242, 258eqeltrd 2916 . 2 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
260 rabxm 4323 . . . 4 𝐴 = ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
261260eqcomi 2833 . . 3 ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴
262261a1i 11 . 2 (𝜑 → ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴)
2639, 200, 259, 262mbfres2 24256 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {crab 3137  ⦋csb 3866   ∪ cun 3917   ∩ cin 3918   ⊆ wss 3919  ifcif 4450  {csn 4550   class class class wbr 5052  {copab 5114   ↦ cmpt 5132  ◡ccnv 5541  dom cdm 5542   ↾ cres 5544   “ cima 5545  ⟶wf 6339  (class class class)co 7149  ℝcr 10534  0cc0 10535   + caddc 10538  +∞cpnf 10670  -∞cmnf 10671  ℝ*cxr 10672   < clt 10673   ≤ cle 10674  (,)cioo 12735  [,)cico 12737  volcvol 24074  MblFncmbf 24225 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-xadd 12505  df-ioo 12739  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043  df-xmet 20091  df-met 20092  df-ovol 24075  df-vol 24076  df-mbf 24230 This theorem is referenced by:  itgaddnclem2  35065
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