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Theorem mbfmax 24252

Description: The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)

**Hypotheses**
Ref	Expression
mbfmax.1	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
mbfmax.2	⊢ (𝜑 → 𝐹 ∈ MblFn)
mbfmax.3	⊢ (𝜑 → 𝐺:𝐴⟶ℝ)
mbfmax.4	⊢ (𝜑 → 𝐺 ∈ MblFn)
mbfmax.5	⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥)))

**Assertion**
Ref	Expression
mbfmax	⊢ (𝜑 → 𝐻 ∈ MblFn)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐹 𝑥,𝐺 𝜑,𝑥 Allowed substitution hint: 𝐻(𝑥)

Proof of Theorem mbfmax Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmax.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)
2	1	ffvelrnda 6853	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ℝ)
3		mbfmax.1	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4	3	ffvelrnda 6853	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
5	2, 4	ifcld 4514	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥)) ∈ ℝ)
6		mbfmax.5	. . 3 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥)))
7	5, 6	fmptd 6880	. 2 ⊢ (𝜑 → 𝐻:𝐴⟶ℝ)
8	3	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐹:𝐴⟶ℝ)
9	8	ffvelrnda 6853	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℝ)
10	9	rexrd 10693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℝ^)
11	1	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐺:𝐴⟶ℝ)
12	11	ffvelrnda 6853	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ)
13	12	rexrd 10693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ^)
14		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → 𝑦 ∈ ℝ^)
15		xrmaxle 12579	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ ℝ^* ∧ (𝐺‘𝑧) ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦 ↔ ((𝐹‘𝑧) ≤ 𝑦 ∧ (𝐺‘𝑧) ≤ 𝑦)))
16	10, 13, 14, 15	syl3anc 1367	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦 ↔ ((𝐹‘𝑧) ≤ 𝑦 ∧ (𝐺‘𝑧) ≤ 𝑦)))
17	16	notbid 320	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦 ↔ ¬ ((𝐹‘𝑧) ≤ 𝑦 ∧ (𝐺‘𝑧) ≤ 𝑦)))
18		ianor 978	. . . . . . . . . 10 ⊢ (¬ ((𝐹‘𝑧) ≤ 𝑦 ∧ (𝐺‘𝑧) ≤ 𝑦) ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦))
19	17, 18	syl6bb 289	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦 ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦)))
20		pnfxr 10697	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
21		elioo2 12782	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (𝑦(,)+∞) ↔ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ 𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞)))
22	14, 20, 21	sylancl 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (𝑦(,)+∞) ↔ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ 𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞)))
23		3anan12 1092	. . . . . . . . . . . 12 ⊢ ((if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ 𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞) ↔ (𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞)))
24	22, 23	syl6bb 289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (𝑦(,)+∞) ↔ (𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞))))
25		fveq2 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
26		fveq2 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
27	25, 26	breq12d 5081	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≤ (𝐺‘𝑥) ↔ (𝐹‘𝑧) ≤ (𝐺‘𝑧)))
28	27, 26, 25	ifbieq12d 4496	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥)) = if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
29		fvex 6685	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘𝑧) ∈ V
30		fvex 6685	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑧) ∈ V
31	29, 30	ifex 4517	. . . . . . . . . . . . . 14 ⊢ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ V
32	28, 6, 31	fvmpt 6770	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (𝐻‘𝑧) = if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
33	32	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝐻‘𝑧) = if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
34	33	eleq1d 2899	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (𝑦(,)+∞)))
35	12, 9	ifcld 4514	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ)
36		ltpnf 12518	. . . . . . . . . . . . 13 ⊢ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞)
37	35, 36	jccir 524	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞))
38	37	biantrud 534	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ↔ (𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞))))
39	24, 34, 38	3bitr4d 313	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ 𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧))))
40	35	rexrd 10693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ^)
41		xrltnle 10710	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ^* ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ^*) → (𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ↔ ¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦))
42	14, 40, 41	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ↔ ¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦))
43	39, 42	bitrd 281	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦))
44		elioo2 12782	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞)))
45	14, 20, 44	sylancl 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞)))
46		3anan12 1092	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞)))
47	45, 46	syl6bb 289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))))
48		ltpnf 12518	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) ∈ ℝ → (𝐹‘𝑧) < +∞)
49	9, 48	jccir 524	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))
50	49	biantrud 534	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐹‘𝑧) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))))
51		xrltnle 10710	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ^* ∧ (𝐹‘𝑧) ∈ ℝ^*) → (𝑦 < (𝐹‘𝑧) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
52	14, 10, 51	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐹‘𝑧) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
53	47, 50, 52	3bitr2d 309	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
54		elioo2 12782	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞)))
55	14, 20, 54	sylancl 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞)))
56		3anan12 1092	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞)))
57	55, 56	syl6bb 289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))))
58		ltpnf 12518	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑧) ∈ ℝ → (𝐺‘𝑧) < +∞)
59	12, 58	jccir 524	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))
60	59	biantrud 534	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐺‘𝑧) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))))
61		xrltnle 10710	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ^* ∧ (𝐺‘𝑧) ∈ ℝ^*) → (𝑦 < (𝐺‘𝑧) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
62	14, 13, 61	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐺‘𝑧) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
63	57, 60, 62	3bitr2d 309	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
64	53, 63	orbi12d 915	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞)) ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦)))
65	19, 43, 64	3bitr4d 313	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞))))
66	65	pm5.32da 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞)))))
67		andi 1004	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞))) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞)) ∨ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞))))
68	66, 67	syl6bb 289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (𝑦(,)+∞)) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞)) ∨ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞)))))
69	7	ffnd 6517	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝐴)
70	69	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐻 Fn 𝐴)
71		elpreima 6830	. . . . . . 7 ⊢ (𝐻 Fn 𝐴 → (𝑧 ∈ (^◡𝐻 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (𝑦(,)+∞))))
72	70, 71	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (𝑦(,)+∞))))
73	8	ffnd 6517	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐹 Fn 𝐴)
74		elpreima 6830	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞))))
75	73, 74	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞))))
76	11	ffnd 6517	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐺 Fn 𝐴)
77		elpreima 6830	. . . . . . . 8 ⊢ (𝐺 Fn 𝐴 → (𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞))))
78	76, 77	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞))))
79	75, 78	orbi12d 915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ∨ 𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞))) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞)) ∨ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞)))))
80	68, 72, 79	3bitr4d 313	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ∨ 𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞)))))
81		elun 4127	. . . . 5 ⊢ (𝑧 ∈ ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ↔ (𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ∨ 𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞))))
82	80, 81	syl6bbr 291	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (𝑦(,)+∞)) ↔ 𝑧 ∈ ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞)))))
83	82	eqrdv 2821	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (𝑦(,)+∞)) = ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))))
84		mbfmax.2	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ MblFn)
85		mbfima 24233	. . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol)
86	84, 3, 85	syl2anc 586	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol)
87		mbfmax.4	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ MblFn)
88		mbfima 24233	. . . . . 6 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:𝐴⟶ℝ) → (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol)
89	87, 1, 88	syl2anc 586	. . . . 5 ⊢ (𝜑 → (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol)
90		unmbl 24140	. . . . 5 ⊢ (((^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol ∧ (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol) → ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ∈ dom vol)
91	86, 89, 90	syl2anc 586	. . . 4 ⊢ (𝜑 → ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ∈ dom vol)
92	91	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ∈ dom vol)
93	83, 92	eqeltrd 2915	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (𝑦(,)+∞)) ∈ dom vol)
94		xrmaxlt 12577	. . . . . . . . . 10 ⊢ (((𝐹‘𝑧) ∈ ℝ^* ∧ (𝐺‘𝑧) ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦 ↔ ((𝐹‘𝑧) < 𝑦 ∧ (𝐺‘𝑧) < 𝑦)))
95	10, 13, 14, 94	syl3anc 1367	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦 ↔ ((𝐹‘𝑧) < 𝑦 ∧ (𝐺‘𝑧) < 𝑦)))
96		mnfxr 10700	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
97		elioo2 12782	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (-∞(,)𝑦) ↔ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦)))
98	96, 14, 97	sylancr 589	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (-∞(,)𝑦) ↔ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦)))
99		df-3an 1085	. . . . . . . . . . 11 ⊢ ((if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦) ↔ ((if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧))) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦))
100	98, 99	syl6bb 289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (-∞(,)𝑦) ↔ ((if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧))) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦)))
101	33	eleq1d 2899	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (-∞(,)𝑦)))
102		mnflt 12521	. . . . . . . . . . . 12 ⊢ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ → -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
103	35, 102	jccir 524	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧))))
104	103	biantrurd 535	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦 ↔ ((if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ ∧ -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧))) ∧ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦)))
105	100, 101, 104	3bitr4d 313	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦))
106		mnflt 12521	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ ℝ → -∞ < (𝐹‘𝑧))
107	9, 106	jccir 524	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)))
108		elioo2 12782	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦)))
109	96, 14, 108	sylancr 589	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦)))
110		df-3an 1085	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦) ↔ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)) ∧ (𝐹‘𝑧) < 𝑦))
111	109, 110	syl6bb 289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)) ∧ (𝐹‘𝑧) < 𝑦)))
112	107, 111	mpbirand 705	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ (𝐹‘𝑧) < 𝑦))
113		mnflt 12521	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) ∈ ℝ → -∞ < (𝐺‘𝑧))
114	12, 113	jccir 524	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)))
115		elioo2 12782	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦)))
116	96, 14, 115	sylancr 589	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦)))
117		df-3an 1085	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦) ↔ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)) ∧ (𝐺‘𝑧) < 𝑦))
118	116, 117	syl6bb 289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)) ∧ (𝐺‘𝑧) < 𝑦)))
119	114, 118	mpbirand 705	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ (𝐺‘𝑧) < 𝑦))
120	112, 119	anbi12d 632	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)) ↔ ((𝐹‘𝑧) < 𝑦 ∧ (𝐺‘𝑧) < 𝑦)))
121	95, 105, 120	3bitr4d 313	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))))
122	121	pm5.32da 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)))))
123		anandi 674	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))))
124	122, 123	syl6bb 289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (-∞(,)𝑦)) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)))))
125		elpreima 6830	. . . . . . 7 ⊢ (𝐻 Fn 𝐴 → (𝑧 ∈ (^◡𝐻 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (-∞(,)𝑦))))
126	70, 125	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (-∞(,)𝑦))))
127		elpreima 6830	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦))))
128	73, 127	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦))))
129		elpreima 6830	. . . . . . . 8 ⊢ (𝐺 Fn 𝐴 → (𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))))
130	76, 129	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))))
131	128, 130	anbi12d 632	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ∧ 𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦))) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)))))
132	124, 126, 131	3bitr4d 313	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ∧ 𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦)))))
133		elin 4171	. . . . 5 ⊢ (𝑧 ∈ ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ↔ (𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ∧ 𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦))))
134	132, 133	syl6bbr 291	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (-∞(,)𝑦)) ↔ 𝑧 ∈ ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦)))))
135	134	eqrdv 2821	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (-∞(,)𝑦)) = ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))))
136		mbfima 24233	. . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
137	84, 3, 136	syl2anc 586	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
138		mbfima 24233	. . . . . 6 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:𝐴⟶ℝ) → (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol)
139	87, 1, 138	syl2anc 586	. . . . 5 ⊢ (𝜑 → (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol)
140		inmbl 24145	. . . . 5 ⊢ (((^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol ∧ (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol) → ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ∈ dom vol)
141	137, 139, 140	syl2anc 586	. . . 4 ⊢ (𝜑 → ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ∈ dom vol)
142	141	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ∈ dom vol)
143	135, 142	eqeltrd 2915	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (-∞(,)𝑦)) ∈ dom vol)
144	7, 93, 143	ismbfd 24242	1 ⊢ (𝜑 → 𝐻 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ∩ cin 3937 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ^◡ccnv 5556 dom cdm 5557 “ cima 5560 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 +∞cpnf 10674 -∞cmnf 10675 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 (,)cioo 12741 volcvol 24066 MblFncmbf 24217

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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xadd 12511 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-xmet 20540 df-met 20541 df-ovol 24067 df-vol 24068 df-mbf 24222

This theorem is referenced by: mbfpos 24254

W3C validator