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Theorem mbfposadd 33089
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 9984 . . . . 5 0 ∈ ℝ
3 ifcl 4102 . . . . 5 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
41, 2, 3sylancl 693 . . . 4 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5 mbfposadd.4 . . . . 5 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)
6 ifcl 4102 . . . . 5 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
75, 2, 6sylancl 693 . . . 4 ((𝜑𝑥𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
84, 7readdcld 10013 . . 3 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
9 eqid 2621 . . 3 (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
108, 9fmptd 6340 . 2 (𝜑 → (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))):𝐴⟶ℝ)
11 ssrab2 3666 . . . 4 {𝑥𝐴 ∣ 0 ≤ 𝐶} ⊆ 𝐴
12 fssres 6027 . . . 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)
1310, 11, 12sylancl 693 . . 3 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}):{𝑥𝐴 ∣ 0 ≤ 𝐶}⟶ℝ)
14 inss2 3812 . . . . . 6 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐶}
15 resabs1 5386 . . . . . 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})))
1614, 15ax-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}))
17 elin 3774 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
18 rabid 3106 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ↔ (𝑥𝐴 ∧ 0 ≤ 𝐵))
19 rabid 3106 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶} ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶))
2018, 19anbi12i 732 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
2117, 20bitri 264 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
22 iftrue 4064 . . . . . . . . . 10 (0 ≤ 𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 𝐵)
23 iftrue 4064 . . . . . . . . . 10 (0 ≤ 𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 𝐶)
2422, 23oveqan12d 6623 . . . . . . . . 9 ((0 ≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2524ad2ant2l 781 . . . . . . . 8 (((𝑥𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2621, 25sylbi 207 . . . . . . 7 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
2726mpteq2ia 4700 . . . . . 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))
28 inss1 3811 . . . . . . . 8 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐵}
29 ssrab2 3666 . . . . . . . 8 {𝑥𝐴 ∣ 0 ≤ 𝐵} ⊆ 𝐴
3028, 29sstri 3592 . . . . . . 7 ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴
31 resmpt 5408 . . . . . . . 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))))
32 nfcv 2761 . . . . . . . . . 10 𝑦(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
33 nfcsb1v 3530 . . . . . . . . . 10 𝑥𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
34 csbeq1a 3523 . . . . . . . . . 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)))
3532, 33, 34cbvmpt 4709 . . . . . . . . 9 (𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦𝐴𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
3635reseq1i 5352 . . . . . . . 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}))
37 nfv 1840 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
38 nfrab1 3111 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ 0 ≤ 𝐵}
39 nfrab1 3111 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ 0 ≤ 𝐶}
4038, 39nfin 3798 . . . . . . . . . . . 12 𝑥({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
4140nfcri 2755 . . . . . . . . . . 11 𝑥 𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
4233nfeq2 2776 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
4341, 42nfan 1825 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
44 eleq1 2686 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
4534eqeq2d 2631 . . . . . . . . . . 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))))
4644, 45anbi12d 746 . . . . . . . . . 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)))))
4737, 43, 46cbvopab1 4685 . . . . . . . . 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)))}
48 df-mpt 4675 . . . . . . . . 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)))}
49 df-mpt 4675 . . . . . . . . 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)))}
5047, 48, 493eqtr4i 2653 . . . . . . . 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)))
5131, 36, 503eqtr4g 2680 . . . . . . 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))))
5230, 51ax-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)))
53 resmpt 5408 . . . . . . . 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)))
54 nfcv 2761 . . . . . . . . . 10 𝑦(𝐵 + 𝐶)
55 nfcsb1v 3530 . . . . . . . . . 10 𝑥𝑦 / 𝑥(𝐵 + 𝐶)
56 csbeq1a 3523 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐵 + 𝐶) = 𝑦 / 𝑥(𝐵 + 𝐶))
5754, 55, 56cbvmpt 4709 . . . . . . . . 9 (𝑥𝐴 ↦ (𝐵 + 𝐶)) = (𝑦𝐴𝑦 / 𝑥(𝐵 + 𝐶))
5857reseq1i 5352 . . . . . . . 8 ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦𝐴𝑦 / 𝑥(𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
59 nfv 1840 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))
6055nfeq2 2776 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶)
6141, 60nfan 1825 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶))
6256eqeq2d 2631 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑧 = (𝐵 + 𝐶) ↔ 𝑧 = 𝑦 / 𝑥(𝐵 + 𝐶)))
6344, 62anbi12d 746 . . . . . . . . . 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))))
6459, 61, 63cbvopab1 4685 . . . . . . . . 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))}
65 df-mpt 4675 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))}
66 df-mpt 4675 . . . . . . . . 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))}
6764, 65, 663eqtr4i 2653 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = (𝑦 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥(𝐵 + 𝐶))
6853, 58, 673eqtr4g 2680 . . . . . . 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)))
6930, 68ax-mp 5 . . . . . 6 ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶))
7027, 52, 693eqtr4i 2653 . . . . 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}))
7116, 70eqtri 2643 . . . 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}))
72 mbfposadd.5 . . . . 5 (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)
731biantrurd 529 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)))
74 elrege0 12220 . . . . . . . . . 10 (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
7573, 74syl6bbr 278 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ 𝐵𝐵 ∈ (0[,)+∞)))
7675rabbidva 3176 . . . . . . . 8 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} = {𝑥𝐴𝐵 ∈ (0[,)+∞)})
77 0xr 10030 . . . . . . . . . . 11 0 ∈ ℝ*
78 pnfxr 10036 . . . . . . . . . . 11 +∞ ∈ ℝ*
79 0ltpnf 11900 . . . . . . . . . . 11 0 < +∞
80 snunioo 12240 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 < +∞) → ({0} ∪ (0(,)+∞)) = (0[,)+∞))
8177, 78, 79, 80mp3an 1421 . . . . . . . . . 10 ({0} ∪ (0(,)+∞)) = (0[,)+∞)
8281imaeq2i 5423 . . . . . . . . 9 ((𝑥𝐴𝐵) “ ({0} ∪ (0(,)+∞))) = ((𝑥𝐴𝐵) “ (0[,)+∞))
83 imaundi 5504 . . . . . . . . 9 ((𝑥𝐴𝐵) “ ({0} ∪ (0(,)+∞))) = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞)))
84 eqid 2621 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
8584mptpreima 5587 . . . . . . . . 9 ((𝑥𝐴𝐵) “ (0[,)+∞)) = {𝑥𝐴𝐵 ∈ (0[,)+∞)}
8682, 83, 853eqtr3ri 2652 . . . . . . . 8 {𝑥𝐴𝐵 ∈ (0[,)+∞)} = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞)))
8776, 86syl6eq 2671 . . . . . . 7 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} = (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))))
88 mbfposadd.1 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
891, 84fmptd 6340 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
90 mbfimasn 23307 . . . . . . . . . 10 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ ∧ 0 ∈ ℝ) → ((𝑥𝐴𝐵) “ {0}) ∈ dom vol)
912, 90mp3an3 1410 . . . . . . . . 9 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ {0}) ∈ dom vol)
92 mbfima 23305 . . . . . . . . 9 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ (0(,)+∞)) ∈ dom vol)
93 unmbl 23212 . . . . . . . . 9 ((((𝑥𝐴𝐵) “ {0}) ∈ dom vol ∧ ((𝑥𝐴𝐵) “ (0(,)+∞)) ∈ dom vol) → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9491, 92, 93syl2anc 692 . . . . . . . 8 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9588, 89, 94syl2anc 692 . . . . . . 7 (𝜑 → (((𝑥𝐴𝐵) “ {0}) ∪ ((𝑥𝐴𝐵) “ (0(,)+∞))) ∈ dom vol)
9687, 95eqeltrd 2698 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol)
975biantrurd 529 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 ≤ 𝐶 ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)))
98 elrege0 12220 . . . . . . . . . 10 (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
9997, 98syl6bbr 278 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ 𝐶𝐶 ∈ (0[,)+∞)))
10099rabbidva 3176 . . . . . . . 8 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = {𝑥𝐴𝐶 ∈ (0[,)+∞)})
10181imaeq2i 5423 . . . . . . . . 9 ((𝑥𝐴𝐶) “ ({0} ∪ (0(,)+∞))) = ((𝑥𝐴𝐶) “ (0[,)+∞))
102 imaundi 5504 . . . . . . . . 9 ((𝑥𝐴𝐶) “ ({0} ∪ (0(,)+∞))) = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞)))
103 eqid 2621 . . . . . . . . . 10 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
104103mptpreima 5587 . . . . . . . . 9 ((𝑥𝐴𝐶) “ (0[,)+∞)) = {𝑥𝐴𝐶 ∈ (0[,)+∞)}
105101, 102, 1043eqtr3ri 2652 . . . . . . . 8 {𝑥𝐴𝐶 ∈ (0[,)+∞)} = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞)))
106100, 105syl6eq 2671 . . . . . . 7 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))))
107 mbfposadd.3 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
1085, 103fmptd 6340 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐶):𝐴⟶ℝ)
109 mbfimasn 23307 . . . . . . . . . 10 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ ∧ 0 ∈ ℝ) → ((𝑥𝐴𝐶) “ {0}) ∈ dom vol)
1102, 109mp3an3 1410 . . . . . . . . 9 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ {0}) ∈ dom vol)
111 mbfima 23305 . . . . . . . . 9 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ (0(,)+∞)) ∈ dom vol)
112 unmbl 23212 . . . . . . . . 9 ((((𝑥𝐴𝐶) “ {0}) ∈ dom vol ∧ ((𝑥𝐴𝐶) “ (0(,)+∞)) ∈ dom vol) → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
113110, 111, 112syl2anc 692 . . . . . . . 8 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
114107, 108, 113syl2anc 692 . . . . . . 7 (𝜑 → (((𝑥𝐴𝐶) “ {0}) ∪ ((𝑥𝐴𝐶) “ (0(,)+∞))) ∈ dom vol)
115106, 114eqeltrd 2698 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol)
116 inmbl 23217 . . . . . 6 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
11796, 115, 116syl2anc 692 . . . . 5 (𝜑 → ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
118 mbfres 23317 . . . . 5 (((𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn ∧ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
11972, 117, 118syl2anc 692 . . . 4 (𝜑 → ((𝑥𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
12071, 119syl5eqel 2702 . . 3 (𝜑 → (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
121 inss2 3812 . . . . . 6 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ 0 ≤ 𝐶}
122 resabs1 5386 . . . . . 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})))
123121, 122ax-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}))
124 rabid 3106 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ↔ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐵))
125124, 19anbi12i 732 . . . . . . . . 9 ((𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
126 elin 3774 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
127 anandi 870 . . . . . . . . 9 ((𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
128125, 126, 1273bitr4i 292 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
129 iffalse 4067 . . . . . . . . . . 11 (¬ 0 ≤ 𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 0)
130129, 23oveqan12d 6623 . . . . . . . . . 10 ((¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶))
131130ad2antll 764 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶))
1325recnd 10012 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
133132addid2d 10181 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (0 + 𝐶) = 𝐶)
134133adantrr 752 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (0 + 𝐶) = 𝐶)
135131, 134eqtrd 2655 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶)
136128, 135sylan2b 492 . . . . . . 7 ((𝜑𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶)
137136mpteq2dva 4704 . . . . . 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))
138 inss1 3811 . . . . . . . 8 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}
139 ssrab2 3666 . . . . . . . 8 {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ⊆ 𝐴
140138, 139sstri 3592 . . . . . . 7 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴
141 resmpt 5408 . . . . . . . 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))))
14235reseq1i 5352 . . . . . . . 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}))
143 nfv 1840 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
144 nfrab1 3111 . . . . . . . . . . . . 13 𝑥{𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}
145144, 39nfin 3798 . . . . . . . . . . . 12 𝑥({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
146145nfcri 2755 . . . . . . . . . . 11 𝑥 𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
147146, 42nfan 1825 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
148 eleq1 2686 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})))
149148, 45anbi12d 746 . . . . . . . . . 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)))))
150143, 147, 149cbvopab1 4685 . . . . . . . . 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)))}
151 df-mpt 4675 . . . . . . . . 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)))}
152 df-mpt 4675 . . . . . . . . 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)))}
153150, 151, 1523eqtr4i 2653 . . . . . . . 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)))
154141, 142, 1533eqtr4g 2680 . . . . . . 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))))
155140, 154ax-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)))
156 resmpt 5408 . . . . . . . 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))
157 nfcv 2761 . . . . . . . . . 10 𝑦𝐶
158 nfcsb1v 3530 . . . . . . . . . 10 𝑥𝑦 / 𝑥𝐶
159 csbeq1a 3523 . . . . . . . . . 10 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
160157, 158, 159cbvmpt 4709 . . . . . . . . 9 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
161160reseq1i 5352 . . . . . . . 8 ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
162 nfv 1840 . . . . . . . . . 10 𝑦(𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)
163158nfeq2 2776 . . . . . . . . . . 11 𝑥 𝑧 = 𝑦 / 𝑥𝐶
164146, 163nfan 1825 . . . . . . . . . 10 𝑥(𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)
165159eqeq2d 2631 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑧 = 𝐶𝑧 = 𝑦 / 𝑥𝐶))
166148, 165anbi12d 746 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)))
167162, 164, 166cbvopab1 4685 . . . . . . . . 9 {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)}
168 df-mpt 4675 . . . . . . . . 9 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)}
169 df-mpt 4675 . . . . . . . . 9 (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝑦 / 𝑥𝐶)}
170167, 168, 1693eqtr4i 2653 . . . . . . . 8 (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = (𝑦 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝑦 / 𝑥𝐶)
171156, 161, 1703eqtr4g 2680 . . . . . . 7 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶))
172140, 171ax-mp 5 . . . . . 6 ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶)
173137, 155, 1723eqtr4g 2680 . . . . 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})))
174123, 173syl5eq 2667 . . . 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})))
17584mptpreima 5587 . . . . . . . 8 ((𝑥𝐴𝐵) “ (-∞(,)0)) = {𝑥𝐴𝐵 ∈ (-∞(,)0)}
176 elioomnf 12210 . . . . . . . . . . 11 (0 ∈ ℝ* → (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0)))
17777, 176ax-mp 5 . . . . . . . . . 10 (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0))
1781biantrurd 529 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 < 0 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0)))
179 ltnle 10061 . . . . . . . . . . . 12 ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
1801, 2, 179sylancl 693 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
181178, 180bitr3d 270 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝐵 ∈ ℝ ∧ 𝐵 < 0) ↔ ¬ 0 ≤ 𝐵))
182177, 181syl5bb 272 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐵 ∈ (-∞(,)0) ↔ ¬ 0 ≤ 𝐵))
183182rabbidva 3176 . . . . . . . 8 (𝜑 → {𝑥𝐴𝐵 ∈ (-∞(,)0)} = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
184175, 183syl5eq 2667 . . . . . . 7 (𝜑 → ((𝑥𝐴𝐵) “ (-∞(,)0)) = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
185 mbfima 23305 . . . . . . . 8 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑥𝐴𝐵):𝐴⟶ℝ) → ((𝑥𝐴𝐵) “ (-∞(,)0)) ∈ dom vol)
18688, 89, 185syl2anc 692 . . . . . . 7 (𝜑 → ((𝑥𝐴𝐵) “ (-∞(,)0)) ∈ dom vol)
187184, 186eqeltrrd 2699 . . . . . 6 (𝜑 → {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol)
188 inmbl 23217 . . . . . 6 (({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
189187, 115, 188syl2anc 692 . . . . 5 (𝜑 → ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
190 mbfres 23317 . . . . 5 (((𝑥𝐴𝐶) ∈ MblFn ∧ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
191107, 189, 190syl2anc 692 . . . 4 (𝜑 → ((𝑥𝐴𝐶) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
192174, 191eqeltrd 2698 . . 3 (𝜑 → (((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
193 ssid 3603 . . . . . 6 𝐴𝐴
194 dfrab3ss 3881 . . . . . 6 (𝐴𝐴 → {𝑥𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
195193, 194ax-mp 5 . . . . 5 {𝑥𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
196 rabxm 3935 . . . . . 6 𝐴 = ({𝑥𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵})
197196ineq1i 3788 . . . . 5 (𝐴 ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})
198 indir 3851 . . . . 5 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}))
199195, 197, 1983eqtrri 2648 . . . 4 (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = {𝑥𝐴 ∣ 0 ≤ 𝐶}
200199a1i 11 . . 3 (𝜑 → (({𝑥𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥𝐴 ∣ 0 ≤ 𝐶})) = {𝑥𝐴 ∣ 0 ≤ 𝐶})
20113, 120, 192, 200mbfres2 23318 . 2 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ 0 ≤ 𝐶}) ∈ MblFn)
202 rabid 3106 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐶))
203 iffalse 4067 . . . . . . . . 9 (¬ 0 ≤ 𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 0)
204203oveq2d 6620 . . . . . . . 8 (¬ 0 ≤ 𝐶 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (if(0 ≤ 𝐵, 𝐵, 0) + 0))
2054recnd 10012 . . . . . . . . 9 ((𝜑𝑥𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
206205addid1d 10180 . . . . . . . 8 ((𝜑𝑥𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + 0) = if(0 ≤ 𝐵, 𝐵, 0))
207204, 206sylan9eqr 2677 . . . . . . 7 (((𝜑𝑥𝐴) ∧ ¬ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
208207anasss 678 . . . . . 6 ((𝜑 ∧ (𝑥𝐴 ∧ ¬ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
209202, 208sylan2b 492 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
210209mpteq2dva 4704 . . . 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)))
211 ssrab2 3666 . . . . 5 {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴
212 resmpt 5408 . . . . . 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))))
21335reseq1i 5352 . . . . . 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})
214 nfv 1840 . . . . . . . 8 𝑦(𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
215 nfrab1 3111 . . . . . . . . . 10 𝑥{𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}
216215nfcri 2755 . . . . . . . . 9 𝑥 𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}
217216, 42nfan 1825 . . . . . . . 8 𝑥(𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
218 eleq1 2686 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}))
219218, 45anbi12d 746 . . . . . . . 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)))))
220214, 217, 219cbvopab1 4685 . . . . . . 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)))}
221 df-mpt 4675 . . . . . . 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)))}
222 df-mpt 4675 . . . . . . 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)))}
223220, 221, 2223eqtr4i 2653 . . . . . 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)))
224212, 213, 2233eqtr4g 2680 . . . . 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))))
225211, 224ax-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)))
226 resmpt 5408 . . . . . 6 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑦𝐴𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)))
227 nfcv 2761 . . . . . . . 8 𝑦if(0 ≤ 𝐵, 𝐵, 0)
228 nfcsb1v 3530 . . . . . . . 8 𝑥𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)
229 csbeq1a 3523 . . . . . . . 8 (𝑥 = 𝑦 → if(0 ≤ 𝐵, 𝐵, 0) = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
230227, 228, 229cbvmpt 4709 . . . . . . 7 (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦𝐴𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
231230reseq1i 5352 . . . . . 6 ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑦𝐴𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
232 nfv 1840 . . . . . . . 8 𝑦(𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))
233228nfeq2 2776 . . . . . . . . 9 𝑥 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)
234216, 233nfan 1825 . . . . . . . 8 𝑥(𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
235229eqeq2d 2631 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑧 = if(0 ≤ 𝐵, 𝐵, 0) ↔ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)))
236218, 235anbi12d 746 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0)) ↔ (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))))
237232, 234, 236cbvopab1 4685 . . . . . . 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))}
238 df-mpt 4675 . . . . . . 7 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))}
239 df-mpt 4675 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))}
240237, 238, 2393eqtr4i 2653 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ 𝑦 / 𝑥if(0 ≤ 𝐵, 𝐵, 0))
241226, 231, 2403eqtr4g 2680 . . . . 5 ({𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)))
242211, 241ax-mp 5 . . . 4 ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0))
243210, 225, 2423eqtr4g 2680 . . 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}))
2441, 88mbfpos 23324 . . . 4 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
245103mptpreima 5587 . . . . . 6 ((𝑥𝐴𝐶) “ (-∞(,)0)) = {𝑥𝐴𝐶 ∈ (-∞(,)0)}
246 elioomnf 12210 . . . . . . . . 9 (0 ∈ ℝ* → (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0)))
24777, 246ax-mp 5 . . . . . . . 8 (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0))
2485biantrurd 529 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐶 < 0 ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0)))
249 ltnle 10061 . . . . . . . . . 10 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
2505, 2, 249sylancl 693 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
251248, 250bitr3d 270 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝐶 ∈ ℝ ∧ 𝐶 < 0) ↔ ¬ 0 ≤ 𝐶))
252247, 251syl5bb 272 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 ∈ (-∞(,)0) ↔ ¬ 0 ≤ 𝐶))
253252rabbidva 3176 . . . . . 6 (𝜑 → {𝑥𝐴𝐶 ∈ (-∞(,)0)} = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
254245, 253syl5eq 2667 . . . . 5 (𝜑 → ((𝑥𝐴𝐶) “ (-∞(,)0)) = {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
255 mbfima 23305 . . . . . 6 (((𝑥𝐴𝐶) ∈ MblFn ∧ (𝑥𝐴𝐶):𝐴⟶ℝ) → ((𝑥𝐴𝐶) “ (-∞(,)0)) ∈ dom vol)
256107, 108, 255syl2anc 692 . . . . 5 (𝜑 → ((𝑥𝐴𝐶) “ (-∞(,)0)) ∈ dom vol)
257254, 256eqeltrrd 2699 . . . 4 (𝜑 → {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol)
258 mbfres 23317 . . . 4 (((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol) → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
259244, 257, 258syl2anc 692 . . 3 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
260243, 259eqeltrd 2698 . 2 (𝜑 → ((𝑥𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
261 rabxm 3935 . . . 4 𝐴 = ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶})
262261eqcomi 2630 . . 3 ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴
263262a1i 11 . 2 (𝜑 → ({𝑥𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴)
26410, 201, 260, 263mbfres2 23318 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 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  csb 3514  cun 3553  cin 3554  wss 3555  ifcif 4058  {csn 4148   class class class wbr 4613  {copab 4672  cmpt 4673  ccnv 5073  dom cdm 5074  cres 5076  cima 5077  wf 5843  (class class class)co 6604  cr 9879  0cc0 9880   + caddc 9883  +∞cpnf 10015  -∞cmnf 10016  *cxr 10017   < clt 10018  cle 10019  (,)cioo 12117  [,)cico 12119  volcvol 23139  MblFncmbf 23289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xadd 11891  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-xmet 19658  df-met 19659  df-ovol 23140  df-vol 23141  df-mbf 23294
This theorem is referenced by:  itgaddnclem2  33101
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