iblabsnclem - Mathbox for Brendan Leahy

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Theorem iblabsnclem 38004

Description: Lemma for iblabsnc 38005; cf. iblabslem 25795. (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 25758	. . . . . . . 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 232	. . . . . . 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 1143	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn)
7	6, 3	mbfdm2 25604	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom vol)
8		mblss 25498	. . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
10		rembl 25507	. . . . 5 ⊢ ℝ ∈ dom vol
11	10	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ dom vol)
12	3	recnd 11173	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℂ)
13	12	abscld 15401	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐹‘𝐵)) ∈ ℝ)
14		0re 11146	. . . . 5 ⊢ 0 ∈ ℝ
15		ifcl 4512	. . . . 5 ⊢ (((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) ∈ ℝ)
16	13, 14, 15	sylancl 587	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) ∈ ℝ)
17		eldifn 4072	. . . . . 6 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
18	17	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
19		iffalse 4475	. . . . 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 4472	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) = (abs‘(𝐹‘𝐵)))
22	21	mpteq2ia 5180	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) = (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵)))
23	13	fmpttd 7067	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))):𝐴⟶ℝ)
24	13	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐹‘𝐵)) ∈ ℝ)
25	24	biantrurd 532	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (abs‘(𝐹‘𝐵)) ↔ ((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ 𝑦 < (abs‘(𝐹‘𝐵)))))
26	3	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℝ)
27		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
28	26, 27	absled 15395	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ≤ 𝑦 ↔ (-𝑦 ≤ (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ 𝑦)))
29	28	notbid 318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (¬ (abs‘(𝐹‘𝐵)) ≤ 𝑦 ↔ ¬ (-𝑦 ≤ (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ 𝑦)))
30	27, 24	ltnled 11293	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (abs‘(𝐹‘𝐵)) ↔ ¬ (abs‘(𝐹‘𝐵)) ≤ 𝑦))
31		renegcl 11457	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
32	31	rexrd 11195	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ^*)
33	32	ad2antlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ^*)
34		elioomnf 13397	. . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) < -𝑦 ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (𝐹‘𝐵) < -𝑦)))
37	27	renegcld 11577	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ)
38	26, 37	ltnled 11293	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) < -𝑦 ↔ ¬ -𝑦 ≤ (𝐹‘𝐵)))
39	35, 36, 38	3bitr2d 307	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ↔ ¬ -𝑦 ≤ (𝐹‘𝐵)))
40		rexr 11191	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
41	40	ad2antlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
42		elioopnf 13396	. . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (𝐹‘𝐵) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ 𝑦 < (𝐹‘𝐵))))
45	27, 26	ltnled 11293	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (𝐹‘𝐵) ↔ ¬ (𝐹‘𝐵) ≤ 𝑦))
46	43, 44, 45	3bitr2d 307	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) ∈ (𝑦(,)+∞) ↔ ¬ (𝐹‘𝐵) ≤ 𝑦))
47	39, 46	orbi12d 919	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞)) ↔ (¬ -𝑦 ≤ (𝐹‘𝐵) ∨ ¬ (𝐹‘𝐵) ≤ 𝑦)))
48		ianor 984	. . . . . . . . . . . 12 ⊢ (¬ (-𝑦 ≤ (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ 𝑦) ↔ (¬ -𝑦 ≤ (𝐹‘𝐵) ∨ ¬ (𝐹‘𝐵) ≤ 𝑦))
49	47, 48	bitr4di 289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞)) ↔ ¬ (-𝑦 ≤ (𝐹‘𝐵) ∧ (𝐹‘𝐵) ≤ 𝑦)))
50	29, 30, 49	3bitr4rd 312	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞)) ↔ 𝑦 < (abs‘(𝐹‘𝐵))))
51		elioopnf 13396	. . . . . . . . . . 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 312	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))))
54	53	rabbidva 3395	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞)} = {𝑥 ∈ 𝐴 ∣ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))})
55		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) = (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵)))
56	55	mptpreima 6202	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (𝑦(,)+∞)) = {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞)}
57		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))
58	57	mptpreima 6202	. . . . . . . . . 10 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-∞(,)-𝑦)}
59	57	mptpreima 6202	. . . . . . . . . 10 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (𝑦(,)+∞)}
60	58, 59	uneq12i 4106	. . . . . . . . 9 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) = ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-∞(,)-𝑦)} ∪ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (𝑦(,)+∞)})
61		unrab 4255	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-∞(,)-𝑦)} ∪ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (𝑦(,)+∞)}) = {𝑥 ∈ 𝐴 ∣ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))}
62	60, 61	eqtri 2759	. . . . . . . 8 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) = {𝑥 ∈ 𝐴 ∣ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))}
63	54, 56, 62	3eqtr4g 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (𝑦(,)+∞)) = ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))))
64		iblmbf 25734	. . . . . . . . . 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 7067	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ)
67		mbfima 25597	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∈ dom vol)
68		mbfima 25597	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞)) ∈ dom vol)
69		unmbl 25504	. . . . . . . . . 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 585	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝜑 → ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) ∈ dom vol)
72	71	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) ∈ dom vol)
73	63, 72	eqeltrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (𝑦(,)+∞)) ∈ dom vol)
74		elioomnf 13397	. . . . . . . . . . . . . 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 532	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) < 𝑦 ↔ ((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ (abs‘(𝐹‘𝐵)) < 𝑦)))
77	26, 27	absltd 15394	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) < 𝑦 ↔ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
78	75, 76, 77	3bitr2d 307	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
79	26	biantrurd 532	. . . . . . . . . . . 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 279	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦))))
81		3anass 1095	. . . . . . . . . . 11 ⊢ (((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ (-𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
82	80, 81	bitr4di 289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
83		elioo2 13339	. . . . . . . . . . . 12 ⊢ ((-𝑦 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝐵) ∈ (-𝑦(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
84	32, 40, 83	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝐵) ∈ (-𝑦(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
85	84	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝐵) ∈ (-𝑦(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
86	82, 85	bitr4d 282	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦) ↔ (𝐹‘𝐵) ∈ (-𝑦(,)𝑦)))
87	86	rabbidva 3395	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-𝑦(,)𝑦)})
88	55	mptpreima 6202	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (-∞(,)𝑦)) = {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦)}
89	57	mptpreima 6202	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-𝑦(,)𝑦)}
90	87, 88, 89	3eqtr4g 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (-∞(,)𝑦)) = (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)))
91		mbfima 25597	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝜑 → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)) ∈ dom vol)
93	92	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-𝑦(,)𝑦)) ∈ dom vol)
94	90, 93	eqeltrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (-∞(,)𝑦)) ∈ dom vol)
95	23, 7, 73, 94	ismbf2d 25607	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) ∈ MblFn)
96	22, 95	eqeltrid 2840	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) ∈ MblFn)
97	9, 11, 16, 20, 96	mbfss 25613	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) ∈ MblFn)
98	1, 97	eqeltrid 2840	. 2 ⊢ (𝜑 → 𝐺 ∈ MblFn)
99		reex 11129	. . . . . . . . 9 ⊢ ℝ ∈ V
100	99	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
101		ifan 4520	. . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0)
102		ifcl 4512	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ ℝ)
103	3, 14, 102	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ ℝ)
104		max1 13137	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝐵) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
105	14, 3, 104	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
106		elrege0 13407	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
108		0e0icopnf 13411	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,)+∞)
109	108	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
110	107, 109	ifclda 4502	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) ∈ (0[,)+∞))
111	101, 110	eqeltrid 2840	. . . . . . . . 9 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
112	111	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
113		ifan 4520	. . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)
114	3	renegcld 11577	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(𝐹‘𝐵) ∈ ℝ)
115		ifcl 4512	. . . . . . . . . . . . 13 ⊢ ((-(𝐹‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ ℝ)
116	114, 14, 115	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ ℝ)
117		max1 13137	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝐵) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
118	14, 114, 117	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
119		elrege0 13407	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ (0[,)+∞))
121	120, 109	ifclda 4502	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) ∈ (0[,)+∞))
122	113, 121	eqeltrid 2840	. . . . . . . . 9 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0) ∈ (0[,)+∞))
123	122	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0) ∈ (0[,)+∞))
124		eqidd 2737	. . . . . . . 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 2737	. . . . . . . 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 7651	. . . . . . 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 7379	. . . . . . . . 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 15272	. . . . . . . . . . . . 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 4472	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) = if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
131	130	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) = if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
132		iftrue 4472	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) = if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
133	132	adantl 481	. . . . . . . . . . . . 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 7385	. . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) = (abs‘(𝐹‘𝐵)))
136	129, 134, 135	3eqtr4d 2781	. . . . . . . . . . 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 412	. . . . . . . . . 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 11321	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
139		iffalse 4475	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) = 0)
140		iffalse 4475	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) = 0)
141	139, 140	oveq12d 7385	. . . . . . . . . . 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 2797	. . . . . . . . . 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 2783	. . . . . . . 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 5179	. . . . . . 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 2771	. . . . . 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 2790	. . . . 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 6844	. . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
150	101, 139	eqtrid 2783	. . . . . . 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 528	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (0 ≤ (𝐹‘𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵))))
153	152	ifbid 4490	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))
154	153	mpteq2ia 5180	. . . . . . 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 25618	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0)) ∈ MblFn)
156	154, 155	eqeltrrid 2841	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∈ MblFn)
157	9, 11, 149, 151, 156	mbfss 25613	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∈ MblFn)
158	112	fmpttd 7067	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)):ℝ⟶(0[,)+∞))
159	5	simp2d 1144	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))) ∈ ℝ)
160	123	fmpttd 7067	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)):ℝ⟶(0[,)+∞))
161	5	simp3d 1145	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))) ∈ ℝ)
162	157, 158, 159, 160, 161	itg2addnc 37995	. . . 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 2771	. . 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 11174	. . 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 2836	. 2 ⊢ (𝜑 → (∫₂‘𝐺) ∈ ℝ)
166	98, 165	jca 511	1 ⊢ (𝜑 → (𝐺 ∈ MblFn ∧ (∫₂‘𝐺) ∈ ℝ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3389 Vcvv 3429 ∖ cdif 3886 ∪ cun 3887 ⊆ wss 3889 ifcif 4466 class class class wbr 5085 ↦ cmpt 5166 ^◡ccnv 5630 dom cdm 5631 “ cima 5634 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∘_f cof 7629 ℝcr 11037 0cc0 11038 + caddc 11041 +∞cpnf 11176 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 -cneg 11378 (,)cioo 13298 [,)cico 13300 abscabs 15196 volcvol 25430 MblFncmbf 25581 ∫₂citg2 25583 𝐿¹cibl 25584

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 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-rest 17385 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-top 22859 df-topon 22876 df-bases 22911 df-cmp 23352 df-ovol 25431 df-vol 25432 df-mbf 25586 df-itg1 25587 df-itg2 25588 df-ibl 25589 df-0p 25637

This theorem is referenced by: iblabsnc 38005 iblmulc2nc 38006

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