mbfmax - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mbfmax		Structured version Visualization version GIF version

Theorem mbfmax 25641

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	ffvelcdmda 7032	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ℝ)
3		mbfmax.1	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4	3	ffvelcdmda 7032	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
5	2, 4	ifcld 4508	. . 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 7062	. 2 ⊢ (𝜑 → 𝐻:𝐴⟶ℝ)
8	3	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐹:𝐴⟶ℝ)
9	8	ffvelcdmda 7032	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℝ)
10	9	rexrd 11193	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℝ^)
11	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐺:𝐴⟶ℝ)
12	11	ffvelcdmda 7032	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ)
13	12	rexrd 11193	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ ℝ^)
14		simplr 774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → 𝑦 ∈ ℝ^)
15		xrmaxle 13133	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 319	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦 ↔ ¬ ((𝐹‘𝑧) ≤ 𝑦 ∧ (𝐺‘𝑧) ≤ 𝑦)))
18		ianor 989	. . . . . . . . . 10 ⊢ (¬ ((𝐹‘𝑧) ≤ 𝑦 ∧ (𝐺‘𝑧) ≤ 𝑦) ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦))
19	17, 18	bitrdi 288	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦 ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦)))
20		pnfxr 11197	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
21		elioo2 13337	. . . . . . . . . . . . 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 592	. . . . . . . . . . . 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 1101	. . . . . . . . . . . 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 288	. . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
26		fveq2 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
27	25, 26	breq12d 5092	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≤ (𝐺‘𝑥) ↔ (𝐹‘𝑧) ≤ (𝐺‘𝑧)))
28	27, 26, 25	ifbieq12d 4490	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥)) = if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
29		fvex 6847	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘𝑧) ∈ V
30		fvex 6847	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑧) ∈ V
31	29, 30	ifex 4512	. . . . . . . . . . . . . 14 ⊢ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ V
32	28, 6, 31	fvmpt 6942	. . . . . . . . . . . . 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 2825	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (𝑦(,)+∞)))
35	12, 9	ifcld 4508	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ)
36		ltpnf 13069	. . . . . . . . . . . . 13 ⊢ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < +∞)
37	35, 36	jccir 526	. . . . . . . . . . . 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 536	. . . . . . . . . . 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 312	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ 𝑦 < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧))))
40	35	rexrd 11193	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^) ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ^)
41		xrltnle 11210	. . . . . . . . . . 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 590	. . . . . . . . . 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 280	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ≤ 𝑦))
44		elioo2 13337	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞)))
45	14, 20, 44	sylancl 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞)))
46		3anan12 1101	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < +∞) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞)))
47	45, 46	bitrdi 288	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))))
48		ltpnf 13069	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) ∈ ℝ → (𝐹‘𝑧) < +∞)
49	9, 48	jccir 526	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))
50	49	biantrud 536	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐹‘𝑧) ↔ (𝑦 < (𝐹‘𝑧) ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) < +∞))))
51		xrltnle 11210	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ^* ∧ (𝐹‘𝑧) ∈ ℝ^*) → (𝑦 < (𝐹‘𝑧) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
52	14, 10, 51	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐹‘𝑧) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
53	47, 50, 52	3bitr2d 308	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ (𝐹‘𝑧) ≤ 𝑦))
54		elioo2 13337	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞)))
55	14, 20, 54	sylancl 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞)))
56		3anan12 1101	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑧) ∈ ℝ ∧ 𝑦 < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < +∞) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞)))
57	55, 56	bitrdi 288	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))))
58		ltpnf 13069	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑧) ∈ ℝ → (𝐺‘𝑧) < +∞)
59	12, 58	jccir 526	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))
60	59	biantrud 536	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐺‘𝑧) ↔ (𝑦 < (𝐺‘𝑧) ∧ ((𝐺‘𝑧) ∈ ℝ ∧ (𝐺‘𝑧) < +∞))))
61		xrltnle 11210	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ^* ∧ (𝐺‘𝑧) ∈ ℝ^*) → (𝑦 < (𝐺‘𝑧) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
62	14, 13, 61	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (𝑦 < (𝐺‘𝑧) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
63	57, 60, 62	3bitr2d 308	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (𝑦(,)+∞) ↔ ¬ (𝐺‘𝑧) ≤ 𝑦))
64	53, 63	orbi12d 924	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞)) ↔ (¬ (𝐹‘𝑧) ≤ 𝑦 ∨ ¬ (𝐺‘𝑧) ≤ 𝑦)))
65	19, 43, 64	3bitr4d 312	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (𝑦(,)+∞) ↔ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞))))
66	65	pm5.32da 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (𝑦(,)+∞)) ↔ (𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞)))))
67		andi 1015	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (𝑦(,)+∞) ∨ (𝐺‘𝑧) ∈ (𝑦(,)+∞))) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞)) ∨ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞))))
68	66, 67	bitrdi 288	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (𝑦(,)+∞)) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (𝑦(,)+∞)) ∨ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (𝑦(,)+∞)))))
69	7	ffnd 6663	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝐴)
70	69	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐻 Fn 𝐴)
71		elpreima 7006	. . . . . . 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 6663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐹 Fn 𝐴)
74		elpreima 7006	. . . . . . . 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 6663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → 𝐺 Fn 𝐴)
77		elpreima 7006	. . . . . . . 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 924	. . . . . 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 312	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (𝑦(,)+∞)) ↔ (𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ∨ 𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞)))))
81		elun 4090	. . . . 5 ⊢ (𝑧 ∈ ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ↔ (𝑧 ∈ (^◡𝐹 “ (𝑦(,)+∞)) ∨ 𝑧 ∈ (^◡𝐺 “ (𝑦(,)+∞))))
82	80, 81	bitr4di 290	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (𝑦(,)+∞)) ↔ 𝑧 ∈ ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞)))))
83	82	eqrdv 2738	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (𝑦(,)+∞)) = ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))))
84		mbfmax.2	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ MblFn)
85		mbfima 25622	. . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol)
86	84, 3, 85	syl2anc 590	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol)
87		mbfmax.4	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ MblFn)
88		mbfima 25622	. . . . . 6 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:𝐴⟶ℝ) → (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol)
89	87, 1, 88	syl2anc 590	. . . . 5 ⊢ (𝜑 → (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol)
90		unmbl 25529	. . . . 5 ⊢ (((^◡𝐹 “ (𝑦(,)+∞)) ∈ dom vol ∧ (^◡𝐺 “ (𝑦(,)+∞)) ∈ dom vol) → ((^◡𝐹 “ (𝑦(,)+∞)) ∪ (^◡𝐺 “ (𝑦(,)+∞))) ∈ dom vol)
91	86, 89, 90	syl2anc 590	. . . 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 2840	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (𝑦(,)+∞)) ∈ dom vol)
94		xrmaxlt 13131	. . . . . . . . . 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 1379	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦 ↔ ((𝐹‘𝑧) < 𝑦 ∧ (𝐺‘𝑧) < 𝑦)))
96		mnfxr 11200	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
97		elioo2 13337	. . . . . . . . . . . 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 593	. . . . . . . . . . 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 1094	. . . . . . . . . . 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 288	. . . . . . . . . 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 2825	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ (-∞(,)𝑦)))
102		mnflt 13072	. . . . . . . . . . . 12 ⊢ (if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) ∈ ℝ → -∞ < if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)))
103	35, 102	jccir 526	. . . . . . . . . . 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 537	. . . . . . . . . 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 312	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ if((𝐹‘𝑧) ≤ (𝐺‘𝑧), (𝐺‘𝑧), (𝐹‘𝑧)) < 𝑦))
106		mnflt 13072	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑧) ∈ ℝ → -∞ < (𝐹‘𝑧))
107	9, 106	jccir 526	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)))
108		elioo2 13337	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦)))
109	96, 14, 108	sylancr 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦)))
110		df-3an 1094	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧) ∧ (𝐹‘𝑧) < 𝑦) ↔ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)) ∧ (𝐹‘𝑧) < 𝑦))
111	109, 110	bitrdi 288	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ (((𝐹‘𝑧) ∈ ℝ ∧ -∞ < (𝐹‘𝑧)) ∧ (𝐹‘𝑧) < 𝑦)))
112	107, 111	mpbirand 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ↔ (𝐹‘𝑧) < 𝑦))
113		mnflt 13072	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) ∈ ℝ → -∞ < (𝐺‘𝑧))
114	12, 113	jccir 526	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)))
115		elioo2 13337	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦)))
116	96, 14, 115	sylancr 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦)))
117		df-3an 1094	. . . . . . . . . . . 12 ⊢ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧) ∧ (𝐺‘𝑧) < 𝑦) ↔ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)) ∧ (𝐺‘𝑧) < 𝑦))
118	116, 117	bitrdi 288	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ (((𝐺‘𝑧) ∈ ℝ ∧ -∞ < (𝐺‘𝑧)) ∧ (𝐺‘𝑧) < 𝑦)))
119	114, 118	mpbirand 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐺‘𝑧) ∈ (-∞(,)𝑦) ↔ (𝐺‘𝑧) < 𝑦))
120	112, 119	anbi12d 638	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → (((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)) ↔ ((𝐹‘𝑧) < 𝑦 ∧ (𝐺‘𝑧) < 𝑦)))
121	95, 105, 120	3bitr4d 312	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ^*) ∧ 𝑧 ∈ 𝐴) → ((𝐻‘𝑧) ∈ (-∞(,)𝑦) ↔ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))))
122	121	pm5.32da 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (-∞(,)𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)))))
123		anandi 682	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ ((𝐹‘𝑧) ∈ (-∞(,)𝑦) ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦))))
124	122, 123	bitrdi 288	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → ((𝑧 ∈ 𝐴 ∧ (𝐻‘𝑧) ∈ (-∞(,)𝑦)) ↔ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ (-∞(,)𝑦)) ∧ (𝑧 ∈ 𝐴 ∧ (𝐺‘𝑧) ∈ (-∞(,)𝑦)))))
125		elpreima 7006	. . . . . . 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 7006	. . . . . . . 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 7006	. . . . . . . 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 638	. . . . . 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 312	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (-∞(,)𝑦)) ↔ (𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ∧ 𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦)))))
133		elin 3906	. . . . 5 ⊢ (𝑧 ∈ ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ↔ (𝑧 ∈ (^◡𝐹 “ (-∞(,)𝑦)) ∧ 𝑧 ∈ (^◡𝐺 “ (-∞(,)𝑦))))
134	132, 133	bitr4di 290	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (^◡𝐻 “ (-∞(,)𝑦)) ↔ 𝑧 ∈ ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦)))))
135	134	eqrdv 2738	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (-∞(,)𝑦)) = ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))))
136		mbfima 25622	. . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶ℝ) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
137	84, 3, 136	syl2anc 590	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
138		mbfima 25622	. . . . . 6 ⊢ ((𝐺 ∈ MblFn ∧ 𝐺:𝐴⟶ℝ) → (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol)
139	87, 1, 138	syl2anc 590	. . . . 5 ⊢ (𝜑 → (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol)
140		inmbl 25534	. . . . 5 ⊢ (((^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol ∧ (^◡𝐺 “ (-∞(,)𝑦)) ∈ dom vol) → ((^◡𝐹 “ (-∞(,)𝑦)) ∩ (^◡𝐺 “ (-∞(,)𝑦))) ∈ dom vol)
141	137, 139, 140	syl2anc 590	. . . 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 2840	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐻 “ (-∞(,)𝑦)) ∈ dom vol)
144	7, 93, 143	ismbfd 25631	1 ⊢ (𝜑 → 𝐻 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∪ cun 3888 ∩ cin 3889 ifcif 4461 class class class wbr 5079 ↦ cmpt 5160 ^◡ccnv 5624 dom cdm 5625 “ cima 5628 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℝcr 11035 +∞cpnf 11174 -∞cmnf 11175 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 (,)cioo 13296 volcvol 25455 MblFncmbf 25606

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9352 df-inf 9353 df-oi 9422 df-dju 9823 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-n0 12436 df-z 12523 df-uz 12787 df-q 12897 df-rp 12941 df-xadd 13062 df-ioo 13300 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-clim 15448 df-sum 15647 df-xmet 21347 df-met 21348 df-ovol 25456 df-vol 25457 df-mbf 25611

This theorem is referenced by: mbfpos 25643

W3C validator