mbfposadd - Mathbox for Brendan Leahy

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Theorem mbfposadd 34954

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 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
mbfposadd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
mbfposadd.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
mbfposadd.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
mbfposadd.5	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)

**Assertion**
Ref	Expression
mbfposadd	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (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.

Step	Hyp	Ref	Expression
1		mbfposadd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
2		0re 10643	. . . . 5 ⊢ 0 ∈ ℝ
3		ifcl 4511	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
4	1, 2, 3	sylancl 588	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
5		mbfposadd.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
6		ifcl 4511	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
7	5, 2, 6	sylancl 588	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
8	4, 7	readdcld 10670	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
9	8	fmpttd 6879	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))):𝐴⟶ℝ)
10		ssrab2 4056	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ⊆ 𝐴
11		fssres 6544	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))):𝐴⟶ℝ ∧ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ⊆ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}):{𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}⟶ℝ)
12	9, 10, 11	sylancl 588	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}):{𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}⟶ℝ)
13		inss2 4206	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}
14		resabs1 5883	. . . . . 6 ⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})))
15	13, 14	ax-mp 5	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
16		elin 4169	. . . . . . . . 9 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
17		rabid 3378	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵))
18		rabid 3378	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶))
19	17, 18	anbi12i 628	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)))
20	16, 19	bitri 277	. . . . . . . 8 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)))
21		iftrue 4473	. . . . . . . . . 10 ⊢ (0 ≤ 𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 𝐵)
22		iftrue 4473	. . . . . . . . . 10 ⊢ (0 ≤ 𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 𝐶)
23	21, 22	oveqan12d 7175	. . . . . . . . 9 ⊢ ((0 ≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
24	23	ad2ant2l 744	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
25	20, 24	sylbi 219	. . . . . . 7 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶))
26	25	mpteq2ia 5157	. . . . . 6 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶))
27		inss1 4205	. . . . . . . 8 ⊢ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵}
28		ssrab2 4056	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ⊆ 𝐴
29	27, 28	sstri 3976	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴
30		resmpt 5905	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
31		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑦(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
32		nfcsb1v 3907	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
33		csbeq1a 3897	. . . . . . . . . 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)))
34	31, 32, 33	cbvmpt 5167	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
35	34	reseq1i 5849	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
36		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
37		nfrab1 3384	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵}
38		nfrab1 3384	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}
39	37, 38	nfin 4193	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})
40	39	nfcri 2971	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})
41	32	nfeq2 2995	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))
42	40, 41	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
43		eleq1w 2895	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})))
44	33	eqeq2d 2832	. . . . . . . . . . 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))))
45	43, 44	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))))
46	36, 42, 45	cbvopab1 5139	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
47		df-mpt 5147	. . . . . . . . 9 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
48		df-mpt 5147	. . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
49	46, 47, 48	3eqtr4i 2854	. . . . . . . 8 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
50	30, 35, 49	3eqtr4g 2881	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
51	29, 50	ax-mp 5	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
52		resmpt 5905	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)))
53		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐵 + 𝐶)
54		nfcsb1v 3907	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)
55		csbeq1a 3897	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐵 + 𝐶) = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))
56	53, 54, 55	cbvmpt 5167	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))
57	56	reseq1i 5849	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
58		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))
59	54	nfeq2 2995	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)
60	40, 59	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))
61	55	eqeq2d 2832	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑧 = (𝐵 + 𝐶) ↔ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)))
62	43, 61	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶)) ↔ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))))
63	58, 60, 62	cbvopab1 5139	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))}
64		df-mpt 5147	. . . . . . . . 9 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))}
65		df-mpt 5147	. . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))}
66	63, 64, 65	3eqtr4i 2854	. . . . . . . 8 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))
67	52, 57, 66	3eqtr4g 2881	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)))
68	29, 67	ax-mp 5	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶))
69	26, 51, 68	3eqtr4i 2854	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
70	15, 69	eqtri 2844	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
71		mbfposadd.5	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)
72	1	biantrurd 535	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐵 ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)))
73		elrege0 12843	. . . . . . . . . 10 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
74	72, 73	syl6bbr 291	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐵 ↔ 𝐵 ∈ (0[,)+∞)))
75	74	rabbidva 3478	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (0[,)+∞)})
76		0xr 10688	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
77		pnfxr 10695	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
78		0ltpnf 12518	. . . . . . . . . . 11 ⊢ 0 < +∞
79		snunioo 12865	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 < +∞) → ({0} ∪ (0(,)+∞)) = (0[,)+∞))
80	76, 77, 78, 79	mp3an 1457	. . . . . . . . . 10 ⊢ ({0} ∪ (0(,)+∞)) = (0[,)+∞)
81	80	imaeq2i 5927	. . . . . . . . 9 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ ({0} ∪ (0(,)+∞))) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0[,)+∞))
82		imaundi 6008	. . . . . . . . 9 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ ({0} ∪ (0(,)+∞))) = ((^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞)))
83		eqid 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
84	83	mptpreima 6092	. . . . . . . . 9 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0[,)+∞)) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (0[,)+∞)}
85	81, 82, 84	3eqtr3ri 2853	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (0[,)+∞)} = ((^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞)))
86	75, 85	syl6eq 2872	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} = ((^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))))
87		mbfposadd.1	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
88	1	fmpttd 6879	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
89		mbfimasn 24233	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ ∧ 0 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∈ dom vol)
90	2, 89	mp3an3 1446	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∈ dom vol)
91		mbfima 24231	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞)) ∈ dom vol)
92		unmbl 24138	. . . . . . . . 9 ⊢ (((^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∈ dom vol ∧ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞)) ∈ dom vol) → ((^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))) ∈ dom vol)
93	90, 91, 92	syl2anc 586	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → ((^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))) ∈ dom vol)
94	87, 88, 93	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → ((^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))) ∈ dom vol)
95	86, 94	eqeltrd 2913	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol)
96	5	biantrurd 535	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐶 ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)))
97		elrege0 12843	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
98	96, 97	syl6bbr 291	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐶 ↔ 𝐶 ∈ (0[,)+∞)))
99	98	rabbidva 3478	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (0[,)+∞)})
100	80	imaeq2i 5927	. . . . . . . . 9 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ ({0} ∪ (0(,)+∞))) = (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0[,)+∞))
101		imaundi 6008	. . . . . . . . 9 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ ({0} ∪ (0(,)+∞))) = ((^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞)))
102		eqid 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
103	102	mptpreima 6092	. . . . . . . . 9 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0[,)+∞)) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (0[,)+∞)}
104	100, 101, 103	3eqtr3ri 2853	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (0[,)+∞)} = ((^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞)))
105	99, 104	syl6eq 2872	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} = ((^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))))
106		mbfposadd.3	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
107	5	fmpttd 6879	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ)
108		mbfimasn 24233	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ ∧ 0 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∈ dom vol)
109	2, 108	mp3an3 1446	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∈ dom vol)
110		mbfima 24231	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞)) ∈ dom vol)
111		unmbl 24138	. . . . . . . . 9 ⊢ (((^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∈ dom vol ∧ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞)) ∈ dom vol) → ((^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))) ∈ dom vol)
112	109, 110, 111	syl2anc 586	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) → ((^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))) ∈ dom vol)
113	106, 107, 112	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → ((^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))) ∈ dom vol)
114	105, 113	eqeltrd 2913	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol)
115		inmbl 24143	. . . . . 6 ⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
116	95, 114, 115	syl2anc 586	. . . . 5 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
117		mbfres 24245	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn ∧ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
118	71, 116, 117	syl2anc 586	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
119	70, 118	eqeltrid 2917	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
120		inss2 4206	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}
121		resabs1 5883	. . . . . 6 ⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})))
122	120, 121	ax-mp 5	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
123		rabid 3378	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐵))
124	123, 18	anbi12i 628	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)))
125		elin 4169	. . . . . . . . 9 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
126		anandi 674	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)))
127	124, 125, 126	3bitr4i 305	. . . . . . . 8 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)))
128		iffalse 4476	. . . . . . . . . . 11 ⊢ (¬ 0 ≤ 𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 0)
129	128, 22	oveqan12d 7175	. . . . . . . . . 10 ⊢ ((¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶))
130	129	ad2antll 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶))
131	5	recnd 10669	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
132	131	addid2d 10841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 + 𝐶) = 𝐶)
133	132	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (0 + 𝐶) = 𝐶)
134	130, 133	eqtrd 2856	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶)
135	127, 134	sylan2b 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶)
136	135	mpteq2dva 5161	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶))
137		inss1 4205	. . . . . . . 8 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}
138		ssrab2 4056	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ⊆ 𝐴
139	137, 138	sstri 3976	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴
140		resmpt 5905	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
141	34	reseq1i 5849	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
142		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
143		nfrab1 3384	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}
144	143, 38	nfin 4193	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})
145	144	nfcri 2971	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})
146	145, 41	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
147		eleq1w 2895	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})))
148	147, 44	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))))
149	142, 146, 148	cbvopab1 5139	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
150		df-mpt 5147	. . . . . . . . 9 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
151		df-mpt 5147	. . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
152	149, 150, 151	3eqtr4i 2854	. . . . . . . 8 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
153	140, 141, 152	3eqtr4g 2881	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
154	139, 153	ax-mp 5	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
155		resmpt 5905	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌𝐶))
156		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐶
157		nfcsb1v 3907	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
158		csbeq1a 3897	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
159	156, 157, 158	cbvmpt 5167	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
160	159	reseq1i 5849	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
161		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)
162	157	nfeq2 2995	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑧 = ⦋𝑦 / 𝑥⦌𝐶
163	145, 162	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶)
164	158	eqeq2d 2832	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶))
165	147, 164	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶)))
166	161, 163, 165	cbvopab1 5139	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶)}
167		df-mpt 5147	. . . . . . . . 9 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)}
168		df-mpt 5147	. . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶)}
169	166, 167, 168	3eqtr4i 2854	. . . . . . . 8 ⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌𝐶)
170	155, 160, 169	3eqtr4g 2881	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶))
171	139, 170	ax-mp 5	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶)
172	136, 154, 171	3eqtr4g 2881	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})))
173	122, 172	syl5eq 2868	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})))
174	83	mptpreima 6092	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)0)) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (-∞(,)0)}
175		elioomnf 12833	. . . . . . . . . . 11 ⊢ (0 ∈ ℝ^* → (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0)))
176	76, 175	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0))
177	1	biantrurd 535	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 < 0 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0)))
178		ltnle 10720	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
179	1, 2, 178	sylancl 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵))
180	177, 179	bitr3d 283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐵 ∈ ℝ ∧ 𝐵 < 0) ↔ ¬ 0 ≤ 𝐵))
181	176, 180	syl5bb 285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (-∞(,)0) ↔ ¬ 0 ≤ 𝐵))
182	181	rabbidva 3478	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (-∞(,)0)} = {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵})
183	174, 182	syl5eq 2868	. . . . . . 7 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)0)) = {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵})
184		mbfima 24231	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)0)) ∈ dom vol)
185	87, 88, 184	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)0)) ∈ dom vol)
186	183, 185	eqeltrrd 2914	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol)
187		inmbl 24143	. . . . . 6 ⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
188	186, 114, 187	syl2anc 586	. . . . 5 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol)
189		mbfres 24245	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
190	106, 188, 189	syl2anc 586	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
191	173, 190	eqeltrd 2913	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn)
192		ssid 3989	. . . . . 6 ⊢ 𝐴 ⊆ 𝐴
193		dfrab3ss 4281	. . . . . 6 ⊢ (𝐴 ⊆ 𝐴 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
194	192, 193	ax-mp 5	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})
195		rabxm 4340	. . . . . 6 ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵})
196	195	ineq1i 4185	. . . . 5 ⊢ (𝐴 ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})
197		indir 4252	. . . . 5 ⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))
198	194, 196, 197	3eqtrri 2849	. . . 4 ⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}
199	198	a1i 11	. . 3 ⊢ (𝜑 → (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})
200	12, 119, 191, 199	mbfres2 24246	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ MblFn)
201		rabid 3378	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐶))
202		iffalse 4476	. . . . . . . . 9 ⊢ (¬ 0 ≤ 𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 0)
203	202	oveq2d 7172	. . . . . . . 8 ⊢ (¬ 0 ≤ 𝐶 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (if(0 ≤ 𝐵, 𝐵, 0) + 0))
204	4	recnd 10669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
205	204	addid1d 10840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + 0) = if(0 ≤ 𝐵, 𝐵, 0))
206	203, 205	sylan9eqr 2878	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
207	206	anasss 469	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
208	201, 207	sylan2b 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0))
209	208	mpteq2dva 5161	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)))
210		ssrab2 4056	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴
211		resmpt 5905	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
212	34	reseq1i 5849	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶})
213		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
214		nfrab1 3384	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}
215	214	nfcri 2971	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}
216	215, 41	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
217		eleq1w 2895	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}))
218	217, 44	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))))
219	213, 216, 218	cbvopab1 5139	. . . . . . 7 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
220		df-mpt 5147	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
221		df-mpt 5147	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))}
222	219, 220, 221	3eqtr4i 2854	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
223	211, 212, 222	3eqtr4g 2881	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
224	210, 223	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
225		resmpt 5905	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)))
226		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑦if(0 ≤ 𝐵, 𝐵, 0)
227		nfcsb1v 3907	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)
228		csbeq1a 3897	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → if(0 ≤ 𝐵, 𝐵, 0) = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))
229	226, 227, 228	cbvmpt 5167	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))
230	229	reseq1i 5849	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶})
231		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))
232	227	nfeq2 2995	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)
233	215, 232	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))
234	228	eqeq2d 2832	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑧 = if(0 ≤ 𝐵, 𝐵, 0) ↔ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)))
235	217, 234	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0)) ↔ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))))
236	231, 233, 235	cbvopab1 5139	. . . . . . 7 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))}
237		df-mpt 5147	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))}
238		df-mpt 5147	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))}
239	236, 237, 238	3eqtr4i 2854	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))
240	225, 230, 239	3eqtr4g 2881	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)))
241	210, 240	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0))
242	209, 224, 241	3eqtr4g 2881	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}))
243	1, 87	mbfpos 24252	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
244	102	mptpreima 6092	. . . . . 6 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (-∞(,)0)) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (-∞(,)0)}
245		elioomnf 12833	. . . . . . . . 9 ⊢ (0 ∈ ℝ^* → (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0)))
246	76, 245	ax-mp 5	. . . . . . . 8 ⊢ (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0))
247	5	biantrurd 535	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 < 0 ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0)))
248		ltnle 10720	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
249	5, 2, 248	sylancl 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶))
250	247, 249	bitr3d 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐶 ∈ ℝ ∧ 𝐶 < 0) ↔ ¬ 0 ≤ 𝐶))
251	246, 250	syl5bb 285	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ (-∞(,)0) ↔ ¬ 0 ≤ 𝐶))
252	251	rabbidva 3478	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (-∞(,)0)} = {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶})
253	244, 252	syl5eq 2868	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (-∞(,)0)) = {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶})
254		mbfima 24231	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (-∞(,)0)) ∈ dom vol)
255	106, 107, 254	syl2anc 586	. . . . 5 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (-∞(,)0)) ∈ dom vol)
256	253, 255	eqeltrrd 2914	. . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol)
257		mbfres 24245	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
258	243, 256, 257	syl2anc 586	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
259	242, 258	eqeltrd 2913	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn)
260		rabxm 4340	. . . 4 ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶})
261	260	eqcomi 2830	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴
262	261	a1i 11	. 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴)
263	9, 200, 259, 262	mbfres2 24246	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 ⦋csb 3883 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ifcif 4467 {csn 4567 class class class wbr 5066 {copab 5128 ↦ cmpt 5146 ^◡ccnv 5554 dom cdm 5555 ↾ cres 5557 “ cima 5558 ⟶wf 6351 (class class class)co 7156 ℝcr 10536 0cc0 10537 + caddc 10540 +∞cpnf 10672 -∞cmnf 10673 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 (,)cioo 12739 [,)cico 12741 volcvol 24064 MblFncmbf 24215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xadd 12509 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-xmet 20538 df-met 20539 df-ovol 24065 df-vol 24066 df-mbf 24220

This theorem is referenced by: itgaddnclem2 34966

W3C validator