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Theorem ismbfd 25598

Description: Deduction to prove measurability of a real function. The third hypothesis is not necessary, but the proof of this requires countable choice, so we derive this separately as ismbf3d 25613. (Contributed by Mario Carneiro, 18-Jun-2014.)

**Hypotheses**
Ref	Expression
ismbfd.1	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
ismbfd.2	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
ismbfd.3	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)

**Assertion**
Ref	Expression
ismbfd	⊢ (𝜑 → 𝐹 ∈ MblFn)

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

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

Step	Hyp	Ref	Expression
1		ioof 13365	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6662	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3		ovelrn 7534	. . . . 5 ⊢ ((,) Fn (ℝ^* × ℝ^) → (𝑧 ∈ ran (,) ↔ ∃𝑥 ∈ ℝ^ ∃𝑦 ∈ ℝ^* 𝑧 = (𝑥(,)𝑦)))
4	1, 2, 3	mp2b 10	. . . 4 ⊢ (𝑧 ∈ ran (,) ↔ ∃𝑥 ∈ ℝ^* ∃𝑦 ∈ ℝ^* 𝑧 = (𝑥(,)𝑦))
5		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → 𝑥 ∈ ℝ^)
6		pnfxr 11188	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → +∞ ∈ ℝ^)
8		mnfxr 11191	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → -∞ ∈ ℝ^)
10		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → 𝑦 ∈ ℝ^)
11		iooin 13297	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^) ∧ (-∞ ∈ ℝ^ ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)))
12	5, 7, 9, 10, 11	syl22anc 838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)))
13		ifcl 4525	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^) → if(𝑥 ≤ -∞, -∞, 𝑥) ∈ ℝ^)
14	8, 5, 13	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → if(𝑥 ≤ -∞, -∞, 𝑥) ∈ ℝ^)
15		mnfle 13051	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → -∞ ≤ 𝑥)
16		xrleid 13067	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
17		breq1 5101	. . . . . . . . . . . . . . 15 ⊢ (-∞ = if(𝑥 ≤ -∞, -∞, 𝑥) → (-∞ ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
18		breq1 5101	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = if(𝑥 ≤ -∞, -∞, 𝑥) → (𝑥 ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
19	17, 18	ifboth 4519	. . . . . . . . . . . . . 14 ⊢ ((-∞ ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
20	15, 16, 19	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ^* → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
21	20	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
22		xrmax1 13092	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → 𝑥 ≤ if(𝑥 ≤ -∞, -∞, 𝑥))
23	5, 8, 22	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝑥 ≤ if(𝑥 ≤ -∞, -∞, 𝑥))
24	14, 5, 21, 23	xrletrid 13071	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) = 𝑥)
25		ifcl 4525	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → if(+∞ ≤ 𝑦, +∞, 𝑦) ∈ ℝ^)
26	6, 10, 25	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → if(+∞ ≤ 𝑦, +∞, 𝑦) ∈ ℝ^)
27		xrmin2 13095	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(+∞ ≤ 𝑦, +∞, 𝑦) ≤ 𝑦)
28	6, 10, 27	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) ≤ 𝑦)
29		pnfge 13046	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
30		xrleid 13067	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ 𝑦)
31		breq2 5102	. . . . . . . . . . . . . . 15 ⊢ (+∞ = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ +∞ ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
32		breq2 5102	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ 𝑦 ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
33	31, 32	ifboth 4519	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≤ +∞ ∧ 𝑦 ≤ 𝑦) → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
34	29, 30, 33	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
35	34	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
36	26, 10, 28, 35	xrletrid 13071	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) = 𝑦)
37	24, 36	oveq12d 7376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)) = (𝑥(,)𝑦))
38	12, 37	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (𝑥(,)𝑦))
39	38	imaeq2d 6019	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ ((𝑥(,)+∞) ∩ (-∞(,)𝑦))) = (^◡𝐹 “ (𝑥(,)𝑦)))
40		ismbfd.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
41	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝐹:𝐴⟶ℝ)
42	41	ffund 6666	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → Fun 𝐹)
43		inpreima 7009	. . . . . . . . 9 ⊢ (Fun 𝐹 → (^◡𝐹 “ ((𝑥(,)+∞) ∩ (-∞(,)𝑦))) = ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))))
44	42, 43	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ ((𝑥(,)+∞) ∩ (-∞(,)𝑦))) = ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))))
45	39, 44	eqtr3d 2773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)𝑦)) = ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))))
46		ismbfd.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
47	46	adantrr 717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
48		ismbfd.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
49	48	ralrimiva 3128	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
50		oveq2 7366	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (-∞(,)𝑥) = (-∞(,)𝑦))
51	50	imaeq2d 6019	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ (-∞(,)𝑦)))
52	51	eleq1d 2821	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol))
53	52	rspccva 3575	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ∧ 𝑦 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
54	49, 53	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
55	54	adantrl 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
56		inmbl 25501	. . . . . . . 8 ⊢ (((^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ∧ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol) → ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))) ∈ dom vol)
57	47, 55, 56	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))) ∈ dom vol)
58	45, 57	eqeltrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol)
59		imaeq2 6015	. . . . . . 7 ⊢ (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) = (^◡𝐹 “ (𝑥(,)𝑦)))
60	59	eleq1d 2821	. . . . . 6 ⊢ (𝑧 = (𝑥(,)𝑦) → ((^◡𝐹 “ 𝑧) ∈ dom vol ↔ (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol))
61	58, 60	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) ∈ dom vol))
62	61	rexlimdvva 3193	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ^* ∃𝑦 ∈ ℝ^* 𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) ∈ dom vol))
63	4, 62	biimtrid 242	. . 3 ⊢ (𝜑 → (𝑧 ∈ ran (,) → (^◡𝐹 “ 𝑧) ∈ dom vol))
64	63	ralrimiv 3127	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol)
65		ismbf 25587	. . 3 ⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol))
66	40, 65	syl 17	. 2 ⊢ (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol))
67	64, 66	mpbird 257	1 ⊢ (𝜑 → 𝐹 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 ∩ cin 3900 ifcif 4479 𝒫 cpw 4554 class class class wbr 5098 × cxp 5622 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 “ cima 5627 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 (class class class)co 7358 ℝcr 11027 +∞cpnf 11165 -∞cmnf 11166 ℝ^*cxr 11167 ≤ cle 11169 (,)cioo 13263 volcvol 25422 MblFncmbf 25573

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 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 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-pm 8768 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-sup 9347 df-inf 9348 df-oi 9417 df-dju 9815 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-q 12864 df-rp 12908 df-xadd 13029 df-ioo 13267 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-xmet 21304 df-met 21305 df-ovol 25423 df-vol 25424 df-mbf 25578

This theorem is referenced by: ismbf2d 25599 mbfmax 25608

W3C validator