iblabsnclem - Mathbox for Brendan Leahy

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Theorem iblabsnclem 37684

Description: Lemma for iblabsnc 37685; cf. iblabslem 25736. (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 25699	. . . . . . . 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 1142	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn)
7	6, 3	mbfdm2 25545	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom vol)
8		mblss 25439	. . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
10		rembl 25448	. . . . 5 ⊢ ℝ ∈ dom vol
11	10	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ dom vol)
12	3	recnd 11209	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℂ)
13	12	abscld 15412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐹‘𝐵)) ∈ ℝ)
14		0re 11183	. . . . 5 ⊢ 0 ∈ ℝ
15		ifcl 4537	. . . . 5 ⊢ (((abs‘(𝐹‘𝐵)) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) ∈ ℝ)
16	13, 14, 15	sylancl 586	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) ∈ ℝ)
17		eldifn 4098	. . . . . 6 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
18	17	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
19		iffalse 4500	. . . . 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 4497	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0) = (abs‘(𝐹‘𝐵)))
22	21	mpteq2ia 5205	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) = (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵)))
23	13	fmpttd 7090	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))):𝐴⟶ℝ)
24	13	adantlr 715	. . . . . . . . . . 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 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℝ)
27		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
28	26, 27	absled 15406	. . . . . . . . . . . 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 11328	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑦 < (abs‘(𝐹‘𝐵)) ↔ ¬ (abs‘(𝐹‘𝐵)) ≤ 𝑦))
31		renegcl 11492	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
32	31	rexrd 11231	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ^*)
33	32	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ^*)
34		elioomnf 13412	. . . . . . . . . . . . . . 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 11612	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → -𝑦 ∈ ℝ)
38	26, 37	ltnled 11328	. . . . . . . . . . . . . 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 11227	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
41	40	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
42		elioopnf 13411	. . . . . . . . . . . . . . 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 11328	. . . . . . . . . . . . . 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 918	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞)) ↔ (¬ -𝑦 ≤ (𝐹‘𝐵) ∨ ¬ (𝐹‘𝐵) ≤ 𝑦)))
48		ianor 983	. . . . . . . . . . . 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 13411	. . . . . . . . . . 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 3415	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞)} = {𝑥 ∈ 𝐴 ∣ ((𝐹‘𝐵) ∈ (-∞(,)-𝑦) ∨ (𝐹‘𝐵) ∈ (𝑦(,)+∞))})
55		eqid 2730	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) = (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵)))
56	55	mptpreima 6214	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (𝑦(,)+∞)) = {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (𝑦(,)+∞)}
57		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))
58	57	mptpreima 6214	. . . . . . . . . 10 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-∞(,)-𝑦)}
59	57	mptpreima 6214	. . . . . . . . . 10 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (𝑦(,)+∞)}
60	58, 59	uneq12i 4132	. . . . . . . . 9 ⊢ ((^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∪ (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞))) = ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-∞(,)-𝑦)} ∪ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (𝑦(,)+∞)})
61		unrab 4281	. . . . . . . . 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 25675	. . . . . . . . . 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 7090	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ)
67		mbfima 25538	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (-∞(,)-𝑦)) ∈ dom vol)
68		mbfima 25538	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) “ (𝑦(,)+∞)) ∈ dom vol)
69		unmbl 25445	. . . . . . . . . 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 584	. . . . . . . . 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 584	. . . . . . . 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 2829	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (𝑦(,)+∞)) ∈ dom vol)
74		elioomnf 13412	. . . . . . . . . . . . . 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 15405	. . . . . . . . . . . . 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 1094	. . . . . . . . . . 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 13354	. . . . . . . . . . . 12 ⊢ ((-𝑦 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝐵) ∈ (-𝑦(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
84	32, 40, 83	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝐵) ∈ (-𝑦(,)𝑦) ↔ ((𝐹‘𝐵) ∈ ℝ ∧ -𝑦 < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < 𝑦)))
85	84	ad2antlr 727	. . . . . . . . . 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 3415	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝐵) ∈ (-𝑦(,)𝑦)})
88	55	mptpreima 6214	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (-∞(,)𝑦)) = {𝑥 ∈ 𝐴 ∣ (abs‘(𝐹‘𝐵)) ∈ (-∞(,)𝑦)}
89	57	mptpreima 6214	. . . . . . . 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 25538	. . . . . . . . 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 584	. . . . . . . 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 2829	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (^◡(𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) “ (-∞(,)𝑦)) ∈ dom vol)
95	23, 7, 73, 94	ismbf2d 25548	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘(𝐹‘𝐵))) ∈ MblFn)
96	22, 95	eqeltrid 2833	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) ∈ MblFn)
97	9, 11, 16, 20, 96	mbfss 25554	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0)) ∈ MblFn)
98	1, 97	eqeltrid 2833	. 2 ⊢ (𝜑 → 𝐺 ∈ MblFn)
99		reex 11166	. . . . . . . . 9 ⊢ ℝ ∈ V
100	99	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
101		ifan 4545	. . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0)
102		ifcl 4537	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ ℝ)
103	3, 14, 102	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ ℝ)
104		max1 13152	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝐵) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
105	14, 3, 104	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0))
106		elrege0 13422	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) ∈ (0[,)+∞))
108		0e0icopnf 13426	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,)+∞)
109	108	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
110	107, 109	ifclda 4527	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) ∈ (0[,)+∞))
111	101, 110	eqeltrid 2833	. . . . . . . . 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 4545	. . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0)
114	3	renegcld 11612	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(𝐹‘𝐵) ∈ ℝ)
115		ifcl 4537	. . . . . . . . . . . . 13 ⊢ ((-(𝐹‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ ℝ)
116	114, 14, 115	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ ℝ)
117		max1 13152	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝐵) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
118	14, 114, 117	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0))
119		elrege0 13422	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0) ∈ (0[,)+∞))
121	120, 109	ifclda 4527	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) ∈ (0[,)+∞))
122	113, 121	eqeltrid 2833	. . . . . . . . 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 2731	. . . . . . . 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 2731	. . . . . . . 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 7676	. . . . . . 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 7402	. . . . . . . . 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 15283	. . . . . . . . . . . . 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 4497	. . . . . . . . . . . . . 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 4497	. . . . . . . . . . . . . 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 7408	. . . . . . . . . . . 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 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 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 11356	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
139		iffalse 4500	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0), 0) = 0)
140		iffalse 4500	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -(𝐹‘𝐵), -(𝐹‘𝐵), 0), 0) = 0)
141	139, 140	oveq12d 7408	. . . . . . . . . . 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 5204	. . . . . . 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 6865	. . . 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 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 528	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (0 ≤ (𝐹‘𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵))))
153	152	ifbid 4515	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))
154	153	mpteq2ia 5205	. . . . . . 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 25559	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐹‘𝐵), (𝐹‘𝐵), 0)) ∈ MblFn)
156	154, 155	eqeltrrid 2834	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∈ MblFn)
157	9, 11, 149, 151, 156	mbfss 25554	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)) ∈ MblFn)
158	112	fmpttd 7090	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0)):ℝ⟶(0[,)+∞))
159	5	simp2d 1143	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (𝐹‘𝐵)), (𝐹‘𝐵), 0))) ∈ ℝ)
160	123	fmpttd 7090	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0)):ℝ⟶(0[,)+∞))
161	5	simp3d 1144	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(𝐹‘𝐵)), -(𝐹‘𝐵), 0))) ∈ ℝ)
162	157, 158, 159, 160, 161	itg2addnc 37675	. . . 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 11210	. . 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 2829	. 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 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {crab 3408 Vcvv 3450 ∖ cdif 3914 ∪ cun 3915 ⊆ wss 3917 ifcif 4491 class class class wbr 5110 ↦ cmpt 5191 ^◡ccnv 5640 dom cdm 5641 “ cima 5644 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∘_f cof 7654 ℝcr 11074 0cc0 11075 + caddc 11078 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 -cneg 11413 (,)cioo 13313 [,)cico 13315 abscabs 15207 volcvol 25371 MblFncmbf 25522 ∫₂citg2 25524 𝐿¹cibl 25525

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-ofr 7657 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-rest 17392 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-top 22788 df-topon 22805 df-bases 22840 df-cmp 23281 df-ovol 25372 df-vol 25373 df-mbf 25527 df-itg1 25528 df-itg2 25529 df-ibl 25530 df-0p 25578

This theorem is referenced by: iblabsnc 37685 iblmulc2nc 37686

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