iblabsnclem - Mathbox for Brendan Leahy

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Theorem iblabsnclem 37212

Description: Lemma for iblabsnc 37213; cf. iblabslem 25773. (Contributed by Brendan Leahy, 7-Nov-2017.)

**Hypotheses**
Ref	Expression
iblabsnc.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
iblabsnc.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
iblabsnclem.1	⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0))
iblabsnclem.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ 𝐿¹)
iblabsnclem.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℝ)

**Assertion**
Ref	Expression
iblabsnclem	⊢ (𝜑 → (𝐺 ∈ MblFn ∧ (∫₂‘𝐺) ∈ ℝ))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝐹(𝑥) 𝐺(𝑥) 𝑉(𝑥)

Proof of Theorem iblabsnclem Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iblabsnclem.1	. . 3 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0))
2		iblabsnclem.2	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ 𝐿¹)
3		iblabsnclem.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℝ)
4	3	iblrelem 25736	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))) ∈ ℝ)))
5	2, 4	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))) ∈ ℝ))
6	5	simp1d 1139	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn)
7	6, 3	mbfdm2 25582	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom vol)
8		mblss 25476	. . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
10		rembl 25485	. . . . 5 ⊢ ℝ ∈ dom vol
11	10	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ dom vol)
12	3	recnd 11270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℂ)
13	12	abscld 15413	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐹‘𝐵)) ∈ ℝ)
14		0re 11244	. . . . 5 ⊢ 0 ∈ ℝ
15		ifcl 4569	. . . . 5 ⊢ (((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) ∈ ℝ)
16	13, 14, 15	sylancl 584	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) ∈ ℝ)
17		eldifn 4120	. . . . . 6 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
18	17	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
19		iffalse 4533	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) = 0)
20	18, 19	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) = 0)
21		iftrue 4530	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) = (abs‘(𝐹‘𝐵)))
22	21	mpteq2ia 5246	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) = (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵)))
23	13	fmpttd 7119	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))):𝐴⟶ℝ)
24	13	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐹‘𝐵)) ∈ ℝ)
25	24	biantrurd 531	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (abs‘(𝐹‘𝐵)) ↔ ((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ 𝑦 < (abs‘(𝐹‘𝐵)))))
26	3	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℝ)
27		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
28	26, 27	absled 15407	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ≤ 𝑦 ↔ (-𝑦 ≤ (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ 𝑦)))
29	28	notbid 317	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (¬ (abs‘(𝐹‘𝐵)) ≤ 𝑦 ↔ ¬ (-𝑦 ≤ (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ 𝑦)))
30	27, 24	ltnled 11389	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (abs‘(𝐹‘𝐵)) ↔ ¬ (abs‘(𝐹‘𝐵)) ≤ 𝑦))
31		renegcl 11551	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
32	31	rexrd 11292	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ^*)
33	32	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ^*)
34		elioomnf 13451	. . . . . . . . . . . . . . 15 ⊢ (-𝑦 ∈ ℝ^* → ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (𝐹‘𝐵) < -𝑦)))
35	33, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (𝐹‘𝐵) < -𝑦)))
36	26	biantrurd 531	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) < -𝑦 ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (𝐹‘𝐵) < -𝑦)))
37	27	renegcld 11669	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ)
38	26, 37	ltnled 11389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) < -𝑦 ↔ ¬ -𝑦 ≤ (𝐹‘𝐵)))
39	35, 36, 38	3bitr2d 306	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ↔ ¬ -𝑦 ≤ (𝐹‘𝐵)))
40		rexr 11288	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
41	40	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
42		elioopnf 13450	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → ((𝐹‘𝐵) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ 𝑦 < (𝐹‘𝐵))))
43	41, 42	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ 𝑦 < (𝐹‘𝐵))))
44	26	biantrurd 531	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (𝐹‘𝐵) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ 𝑦 < (𝐹‘𝐵))))
45	27, 26	ltnled 11389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (𝐹‘𝐵) ↔ ¬ (𝐹‘𝐵) ≤ 𝑦))
46	43, 44, 45	3bitr2d 306	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) ∈ (𝑦(,)+∞) ↔ ¬ (𝐹‘𝐵) ≤ 𝑦))
47	39, 46	orbi12d 916	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞)) ↔ (¬ -𝑦 ≤ (𝐹‘𝐵) ∨ ¬ (𝐹‘𝐵) ≤ 𝑦)))
48		ianor 979	. . . . . . . . . . . 12 ⊢ (¬ (-𝑦 ≤ (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ 𝑦) ↔ (¬ -𝑦 ≤ (𝐹‘𝐵) ∨ ¬ (𝐹‘𝐵) ≤ 𝑦))
49	47, 48	bitr4di 288	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞)) ↔ ¬ (-𝑦 ≤ (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ 𝑦)))
50	29, 30, 49	3bitr4rd 311	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞)) ↔ 𝑦 < (abs‘(𝐹‘𝐵))))
51		elioopnf 13450	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → ((abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞) ↔ ((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ 𝑦 < (abs‘(𝐹‘𝐵)))))
52	41, 51	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞) ↔ ((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ 𝑦 < (abs‘(𝐹‘𝐵)))))
53	25, 50, 52	3bitr4rd 311	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))))
54	53	rabbidva 3426	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞)} = {𝑥 ∈ 𝐴 ∣ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))})
55		eqid 2725	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) = (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵)))
56	55	mptpreima 6237	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (𝑦(,)+∞)) = {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞)}
57		eqid 2725	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))
58	57	mptpreima 6237	. . . . . . . . . 10 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-∞(,)-𝑦)}
59	57	mptpreima 6237	. . . . . . . . . 10 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (𝑦(,)+∞)}
60	58, 59	uneq12i 4154	. . . . . . . . 9 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) = ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-∞(,)-𝑦)} ∪ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (𝑦(,)+∞)})
61		unrab 4300	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-∞(,)-𝑦)} ∪ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (𝑦(,)+∞)}) = {𝑥 ∈ 𝐴 ∣ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))}
62	60, 61	eqtri 2753	. . . . . . . 8 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) = {𝑥 ∈ 𝐴 ∣ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))}
63	54, 56, 62	3eqtr4g 2790	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (𝑦(,)+∞)) = ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))))
64		iblmbf 25713	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn)
65	2, 64	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn)
66	3	fmpttd 7119	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ)
67		mbfima 25575	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∈ dom vol)
68		mbfima 25575	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞)) ∈ dom vol)
69		unmbl 25482	. . . . . . . . . 10 ⊢ (((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∈ dom vol ∧ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞)) ∈ dom vol) → ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) ∈ dom vol)
70	67, 68, 69	syl2anc 582	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ) → ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) ∈ dom vol)
71	65, 66, 70	syl2anc 582	. . . . . . . 8 ⊢ (𝜑 → ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) ∈ dom vol)
72	71	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) ∈ dom vol)
73	63, 72	eqeltrd 2825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (𝑦(,)+∞)) ∈ dom vol)
74		elioomnf 13451	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ ((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ (abs‘(𝐹‘𝐵)) < 𝑦)))
75	41, 74	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ ((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ (abs‘(𝐹‘𝐵)) < 𝑦)))
76	24	biantrurd 531	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) < 𝑦 ↔ ((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ (abs‘(𝐹‘𝐵)) < 𝑦)))
77	26, 27	absltd 15406	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) < 𝑦 ↔ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
78	75, 76, 77	3bitr2d 306	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
79	26	biantrurd 531	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦))))
80	78, 79	bitrd 278	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦))))
81		3anass 1092	. . . . . . . . . . 11 ⊢ (((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
82	80, 81	bitr4di 288	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
83		elioo2 13395	. . . . . . . . . . . 12 ⊢ ((-𝑦 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝐵) ∈ (-𝑦(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
84	32, 40, 83	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝐵) ∈ (-𝑦(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
85	84	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) ∈ (-𝑦(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
86	82, 85	bitr4d 281	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ (𝐹‘𝐵) ∈ (-𝑦(,)𝑦)))
87	86	rabbidva 3426	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-𝑦(,)𝑦)})
88	55	mptpreima 6237	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (-∞(,)𝑦)) = {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦)}
89	57	mptpreima 6237	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-𝑦(,)𝑦)}
90	87, 88, 89	3eqtr4g 2790	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (-∞(,)𝑦)) = (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)))
91		mbfima 25575	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)) ∈ dom vol)
92	65, 66, 91	syl2anc 582	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)) ∈ dom vol)
93	92	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)) ∈ dom vol)
94	90, 93	eqeltrd 2825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (-∞(,)𝑦)) ∈ dom vol)
95	23, 7, 73, 94	ismbf2d 25585	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) ∈ MblFn)
96	22, 95	eqeltrid 2829	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) ∈ MblFn)
97	9, 11, 16, 20, 96	mbfss 25591	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) ∈ MblFn)
98	1, 97	eqeltrid 2829	. 2 ⊢ (𝜑 → 𝐺 ∈ MblFn)
99		reex 11227	. . . . . . . . 9 ⊢ ℝ ∈ V
100	99	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
101		ifan 4577	. . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0)
102		ifcl 4569	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ ℝ)
103	3, 14, 102	sylancl 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ ℝ)
104		max1 13194	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝐵) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
105	14, 3, 104	sylancr 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
106		elrege0 13461	. . . . . . . . . . . 12 ⊢ (if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0)))
107	103, 105, 106	sylanbrc 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
108		0e0icopnf 13465	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,)+∞)
109	108	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
110	107, 109	ifclda 4559	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) ∈ (0[,)+∞))
111	101, 110	eqeltrid 2829	. . . . . . . . 9 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
112	111	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
113		ifan 4577	. . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)
114	3	renegcld 11669	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(𝐹‘𝐵) ∈ ℝ)
115		ifcl 4569	. . . . . . . . . . . . 13 ⊢ ((-(𝐹‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ ℝ)
116	114, 14, 115	sylancl 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ ℝ)
117		max1 13194	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝐵) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
118	14, 114, 117	sylancr 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
119		elrege0 13461	. . . . . . . . . . . 12 ⊢ (if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0)))
120	116, 118, 119	sylanbrc 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ (0[,)+∞))
121	120, 109	ifclda 4559	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) ∈ (0[,)+∞))
122	113, 121	eqeltrid 2829	. . . . . . . . 9 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0) ∈ (0[,)+∞))
123	122	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0) ∈ (0[,)+∞))
124		eqidd 2726	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)))
125		eqidd 2726	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)))
126	100, 112, 123, 124, 125	offval2 7701	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))) = (𝑥 ∈ ℝ ↦ (if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))))
127	101, 113	oveq12i 7427	. . . . . . . . 9 ⊢ (if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)) = (if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) + if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0))
128		max0add 15287	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝐵) ∈ ℝ → (if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) + if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0)) = (abs‘(𝐹‘𝐵)))
129	3, 128	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) + if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0)) = (abs‘(𝐹‘𝐵)))
130		iftrue 4530	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) = if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
131	130	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) = if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
132		iftrue 4530	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) = if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
133	132	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) = if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
134	131, 133	oveq12d 7433	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) + if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)) = (if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) + if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0)))
135	21	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) = (abs‘(𝐹‘𝐵)))
136	129, 134, 135	3eqtr4d 2775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) + if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0))
137	136	ex 411	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) + if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)))
138		00id 11417	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
139		iffalse 4533	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) = 0)
140		iffalse 4533	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) = 0)
141	139, 140	oveq12d 7433	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) + if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)) = (0 + 0))
142	138, 141, 19	3eqtr4a 2791	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) + if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0))
143	137, 142	pm2.61d1 180	. . . . . . . . 9 ⊢ (𝜑 → (if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) + if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0))
144	127, 143	eqtrid 2777	. . . . . . . 8 ⊢ (𝜑 → (if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0))
145	144	mpteq2dv 5245	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)))
146	126, 145	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)))
147	1, 146	eqtr4id 2784	. . . . 5 ⊢ (𝜑 → 𝐺 = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))))
148	147	fveq2d 6895	. . . 4 ⊢ (𝜑 → (∫₂‘𝐺) = (∫₂‘((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)))))
149	111	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
150	101, 139	eqtrid 2777	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) = 0)
151	18, 150	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) = 0)
152		ibar 527	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (0 ≤ (𝐹‘𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵))))
153	152	ifbid 4547	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))
154	153	mpteq2ia 5246	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0)) = (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))
155	3, 6	mbfpos 25596	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0)) ∈ MblFn)
156	154, 155	eqeltrrid 2830	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∈ MblFn)
157	9, 11, 149, 151, 156	mbfss 25591	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∈ MblFn)
158	112	fmpttd 7119	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)):ℝ⟶(0[,)+∞))
159	5	simp2d 1140	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))) ∈ ℝ)
160	123	fmpttd 7119	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)):ℝ⟶(0[,)+∞))
161	5	simp3d 1141	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))) ∈ ℝ)
162	157, 158, 159, 160, 161	itg2addnc 37203	. . . 4 ⊢ (𝜑 → (∫₂‘((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)))) = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)))))
163	148, 162	eqtrd 2765	. . 3 ⊢ (𝜑 → (∫₂‘𝐺) = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)))))
164	159, 161	readdcld 11271	. . 3 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)))) ∈ ℝ)
165	163, 164	eqeltrd 2825	. 2 ⊢ (𝜑 → (∫₂‘𝐺) ∈ ℝ)
166	98, 165	jca 510	1 ⊢ (𝜑 → (𝐺 ∈ MblFn ∧ (∫₂‘𝐺) ∈ ℝ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ∖ cdif 3937 ∪ cun 3938 ⊆ wss 3940 ifcif 4524 class class class wbr 5143 ↦ cmpt 5226 ^◡ccnv 5671 dom cdm 5672 “ cima 5675 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 ∘_f cof 7679 ℝcr 11135 0cc0 11136 + caddc 11139 +∞cpnf 11273 -∞cmnf 11274 ℝ^*cxr 11275 < clt 11276 ≤ cle 11277 -cneg 11473 (,)cioo 13354 [,)cico 13356 abscabs 15211 volcvol 25408 MblFncmbf 25559 ∫₂citg2 25561 𝐿¹cibl 25562

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-ofr 7682 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-dju 9922 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-rest 17401 df-topgen 17422 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-top 22812 df-topon 22829 df-bases 22865 df-cmp 23307 df-ovol 25409 df-vol 25410 df-mbf 25564 df-itg1 25565 df-itg2 25566 df-ibl 25567 df-0p 25615

This theorem is referenced by: iblabsnc 37213 iblmulc2nc 37214

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