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Theorem mbfmax 24813

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 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ℝ)
3		mbfmax.1	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4	3	ffvelrnda 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
5	2, 4	ifcld 4505	. . 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 6988	. 2 ⊢ (𝜑 → 𝐻:𝐴⟶ℝ)
8	3	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐹:𝐴⟶ℝ)
9	8	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℝ)
10	9	rexrd 11025	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℝ^)
11	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐺:𝐴⟶ℝ)
12	11	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ)
13	12	rexrd 11025	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ^)
14		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → 𝑦 ∈ ℝ^)
15		xrmaxle 12917	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦 ↔ ¬ ((𝐹‘𝑧) ≤ 𝑦 ∧ (𝐺‘𝑧) ≤ 𝑦)))
18		ianor 979	. . . . . . . . . 10 ⊢ (¬ ((𝐹‘𝑧) ≤ 𝑦 ∧ (𝐺‘𝑧) ≤ 𝑦) ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦))
19	17, 18	bitrdi 287	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦 ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦)))
20		pnfxr 11029	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
21		elioo2 13120	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 1095	. . . . . . . . . . . 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	bitrdi 287	. . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
26		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
27	25, 26	breq12d 5087	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≤ (𝐺‘𝑥) ↔ (𝐹‘𝑧) ≤ (𝐺‘𝑧)))
28	27, 26, 25	ifbieq12d 4487	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥)) = if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
29		fvex 6787	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘𝑧) ∈ V
30		fvex 6787	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑧) ∈ V
31	29, 30	ifex 4509	. . . . . . . . . . . . . 14 ⊢ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ V
32	28, 6, 31	fvmpt 6875	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (𝐻‘𝑧) = if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
33	32	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝐻‘𝑧) = if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
34	33	eleq1d 2823	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (𝑦(,)+∞)))
35	12, 9	ifcld 4505	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ)
36		ltpnf 12856	. . . . . . . . . . . . 13 ⊢ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞)
37	35, 36	jccir 522	. . . . . . . . . . . 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 532	. . . . . . . . . . 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 311	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ 𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧))))
40	35	rexrd 11025	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ^)
41		xrltnle 11042	. . . . . . . . . . 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 584	. . . . . . . . . 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 278	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦))
44		elioo2 13120	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞)))
45	14, 20, 44	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞)))
46		3anan12 1095	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞)))
47	45, 46	bitrdi 287	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))))
48		ltpnf 12856	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) ∈ ℝ → (𝐹‘𝑧) < +∞)
49	9, 48	jccir 522	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))
50	49	biantrud 532	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐹‘𝑧) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))))
51		xrltnle 11042	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ^* ∧ (𝐹‘𝑧) ∈ ℝ^*) → (𝑦 < (𝐹‘𝑧) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
52	14, 10, 51	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐹‘𝑧) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
53	47, 50, 52	3bitr2d 307	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
54		elioo2 13120	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞)))
55	14, 20, 54	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞)))
56		3anan12 1095	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞)))
57	55, 56	bitrdi 287	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))))
58		ltpnf 12856	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑧) ∈ ℝ → (𝐺‘𝑧) < +∞)
59	12, 58	jccir 522	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))
60	59	biantrud 532	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐺‘𝑧) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))))
61		xrltnle 11042	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ^* ∧ (𝐺‘𝑧) ∈ ℝ^*) → (𝑦 < (𝐺‘𝑧) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
62	14, 13, 61	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐺‘𝑧) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
63	57, 60, 62	3bitr2d 307	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
64	53, 63	orbi12d 916	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞)) ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦)))
65	19, 43, 64	3bitr4d 311	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞))))
66	65	pm5.32da 579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞)))))
67		andi 1005	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞))) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞)) ∨ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞))))
68	66, 67	bitrdi 287	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (𝑦(,)+∞)) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞)) ∨ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞)))))
69	7	ffnd 6601	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝐴)
70	69	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐻 Fn 𝐴)
71		elpreima 6935	. . . . . . 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 6601	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐹 Fn 𝐴)
74		elpreima 6935	. . . . . . . 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 6601	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐺 Fn 𝐴)
77		elpreima 6935	. . . . . . . 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 916	. . . . . 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 311	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ∨ 𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞)))))
81		elun 4083	. . . . 5 ⊢ (𝑧 ∈ ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ↔ (𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ∨ 𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞))))
82	80, 81	bitr4di 289	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (𝑦(,)+∞)) ↔ 𝑧 ∈ ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞)))))
83	82	eqrdv 2736	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (𝑦(,)+∞)) = ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))))
84		mbfmax.2	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ MblFn)
85		mbfima 24794	. . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol)
86	84, 3, 85	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol)
87		mbfmax.4	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ MblFn)
88		mbfima 24794	. . . . . 6 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:𝐴⟶ℝ) → (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol)
89	87, 1, 88	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol)
90		unmbl 24701	. . . . 5 ⊢ (((^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol ∧ (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol) → ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ∈ dom vol)
91	86, 89, 90	syl2anc 584	. . . 4 ⊢ (𝜑 → ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ∈ dom vol)
92	91	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ∈ dom vol)
93	83, 92	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (𝑦(,)+∞)) ∈ dom vol)
94		xrmaxlt 12915	. . . . . . . . . 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 1370	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦 ↔ ((𝐹‘𝑧) < 𝑦 ∧ (𝐺‘𝑧) < 𝑦)))
96		mnfxr 11032	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
97		elioo2 13120	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 1088	. . . . . . . . . . 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	bitrdi 287	. . . . . . . . . 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 2823	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (-∞(,)𝑦)))
102		mnflt 12859	. . . . . . . . . . . 12 ⊢ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ → -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
103	35, 102	jccir 522	. . . . . . . . . . 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 533	. . . . . . . . . 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 311	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦))
106		mnflt 12859	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ ℝ → -∞ < (𝐹‘𝑧))
107	9, 106	jccir 522	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)))
108		elioo2 13120	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦)))
109	96, 14, 108	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦)))
110		df-3an 1088	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦) ↔ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)) ∧ (𝐹‘𝑧) < 𝑦))
111	109, 110	bitrdi 287	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)) ∧ (𝐹‘𝑧) < 𝑦)))
112	107, 111	mpbirand 704	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ (𝐹‘𝑧) < 𝑦))
113		mnflt 12859	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) ∈ ℝ → -∞ < (𝐺‘𝑧))
114	12, 113	jccir 522	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)))
115		elioo2 13120	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦)))
116	96, 14, 115	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦)))
117		df-3an 1088	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦) ↔ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)) ∧ (𝐺‘𝑧) < 𝑦))
118	116, 117	bitrdi 287	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)) ∧ (𝐺‘𝑧) < 𝑦)))
119	114, 118	mpbirand 704	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ (𝐺‘𝑧) < 𝑦))
120	112, 119	anbi12d 631	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)) ↔ ((𝐹‘𝑧) < 𝑦 ∧ (𝐺‘𝑧) < 𝑦)))
121	95, 105, 120	3bitr4d 311	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))))
122	121	pm5.32da 579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)))))
123		anandi 673	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))))
124	122, 123	bitrdi 287	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (-∞(,)𝑦)) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)))))
125		elpreima 6935	. . . . . . 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 6935	. . . . . . . 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 6935	. . . . . . . 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 631	. . . . . 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 311	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ∧ 𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦)))))
133		elin 3903	. . . . 5 ⊢ (𝑧 ∈ ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ↔ (𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ∧ 𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦))))
134	132, 133	bitr4di 289	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (-∞(,)𝑦)) ↔ 𝑧 ∈ ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦)))))
135	134	eqrdv 2736	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (-∞(,)𝑦)) = ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))))
136		mbfima 24794	. . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
137	84, 3, 136	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
138		mbfima 24794	. . . . . 6 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:𝐴⟶ℝ) → (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol)
139	87, 1, 138	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol)
140		inmbl 24706	. . . . 5 ⊢ (((^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol ∧ (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol) → ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ∈ dom vol)
141	137, 139, 140	syl2anc 584	. . . 4 ⊢ (𝜑 → ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ∈ dom vol)
142	141	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ∈ dom vol)
143	135, 142	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (-∞(,)𝑦)) ∈ dom vol)
144	7, 93, 143	ismbfd 24803	1 ⊢ (𝜑 → 𝐻 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∪ cun 3885 ∩ cin 3886 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 ^◡ccnv 5588 dom cdm 5589 “ cima 5592 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 (,)cioo 13079 volcvol 24627 MblFncmbf 24778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xadd 12849 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-xmet 20590 df-met 20591 df-ovol 24628 df-vol 24629 df-mbf 24783

This theorem is referenced by: mbfpos 24815

W3C validator