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Theorem ismbfd 24708

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 24723. (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 13108	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2		ffn 6584	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
3		ovelrn 7426	. . . . 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 767	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → 𝑥 ∈ ℝ^)
6		pnfxr 10960	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → +∞ ∈ ℝ^)
8		mnfxr 10963	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → -∞ ∈ ℝ^)
10		simprr 769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → 𝑦 ∈ ℝ^)
11		iooin 13042	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^) ∧ (-∞ ∈ ℝ^ ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)))
12	5, 7, 9, 10, 11	syl22anc 835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)))
13		ifcl 4501	. . . . . . . . . . . . 13 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^) → if(𝑥 ≤ -∞, -∞, 𝑥) ∈ ℝ^)
14	8, 5, 13	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → if(𝑥 ≤ -∞, -∞, 𝑥) ∈ ℝ^)
15		mnfle 12799	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → -∞ ≤ 𝑥)
16		xrleid 12814	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ^* → 𝑥 ≤ 𝑥)
17		breq1 5073	. . . . . . . . . . . . . . 15 ⊢ (-∞ = if(𝑥 ≤ -∞, -∞, 𝑥) → (-∞ ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
18		breq1 5073	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = if(𝑥 ≤ -∞, -∞, 𝑥) → (𝑥 ≤ 𝑥 ↔ if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥))
19	17, 18	ifboth 4495	. . . . . . . . . . . . . 14 ⊢ ((-∞ ≤ 𝑥 ∧ 𝑥 ≤ 𝑥) → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
20	15, 16, 19	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ^* → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
21	20	ad2antrl 724	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) ≤ 𝑥)
22		xrmax1 12838	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → 𝑥 ≤ if(𝑥 ≤ -∞, -∞, 𝑥))
23	5, 8, 22	sylancl 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝑥 ≤ if(𝑥 ≤ -∞, -∞, 𝑥))
24	14, 5, 21, 23	xrletrid 12818	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(𝑥 ≤ -∞, -∞, 𝑥) = 𝑥)
25		ifcl 4501	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) → if(+∞ ≤ 𝑦, +∞, 𝑦) ∈ ℝ^)
26	6, 10, 25	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^)) → if(+∞ ≤ 𝑦, +∞, 𝑦) ∈ ℝ^)
27		xrmin2 12841	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → if(+∞ ≤ 𝑦, +∞, 𝑦) ≤ 𝑦)
28	6, 10, 27	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) ≤ 𝑦)
29		pnfge 12795	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
30		xrleid 12814	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ 𝑦)
31		breq2 5074	. . . . . . . . . . . . . . 15 ⊢ (+∞ = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ +∞ ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
32		breq2 5074	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = if(+∞ ≤ 𝑦, +∞, 𝑦) → (𝑦 ≤ 𝑦 ↔ 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦)))
33	31, 32	ifboth 4495	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ≤ +∞ ∧ 𝑦 ≤ 𝑦) → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
34	29, 30, 33	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
35	34	ad2antll 725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → 𝑦 ≤ if(+∞ ≤ 𝑦, +∞, 𝑦))
36	26, 10, 28, 35	xrletrid 12818	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → if(+∞ ≤ 𝑦, +∞, 𝑦) = 𝑦)
37	24, 36	oveq12d 7273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (if(𝑥 ≤ -∞, -∞, 𝑥)(,)if(+∞ ≤ 𝑦, +∞, 𝑦)) = (𝑥(,)𝑦))
38	12, 37	eqtrd 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((𝑥(,)+∞) ∩ (-∞(,)𝑦)) = (𝑥(,)𝑦))
39	38	imaeq2d 5958	. . . . . . . 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 6588	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → Fun 𝐹)
43		inpreima 6923	. . . . . . . . 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 2780	. . . . . . 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 713	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
48		ismbfd.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
49	48	ralrimiva 3107	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
50		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (-∞(,)𝑥) = (-∞(,)𝑦))
51	50	imaeq2d 5958	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ (-∞(,)𝑦)))
52	51	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol))
53	52	rspccva 3551	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ^* (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ∧ 𝑦 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
54	49, 53	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
55	54	adantrl 712	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol)
56		inmbl 24611	. . . . . . . 8 ⊢ (((^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ∧ (^◡𝐹 “ (-∞(,)𝑦)) ∈ dom vol) → ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))) ∈ dom vol)
57	47, 55, 56	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → ((^◡𝐹 “ (𝑥(,)+∞)) ∩ (^◡𝐹 “ (-∞(,)𝑦))) ∈ dom vol)
58	45, 57	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol)
59		imaeq2 5954	. . . . . . 7 ⊢ (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) = (^◡𝐹 “ (𝑥(,)𝑦)))
60	59	eleq1d 2823	. . . . . 6 ⊢ (𝑧 = (𝑥(,)𝑦) → ((^◡𝐹 “ 𝑧) ∈ dom vol ↔ (^◡𝐹 “ (𝑥(,)𝑦)) ∈ dom vol))
61	58, 60	syl5ibrcom 246	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*)) → (𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) ∈ dom vol))
62	61	rexlimdvva 3222	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ^* ∃𝑦 ∈ ℝ^* 𝑧 = (𝑥(,)𝑦) → (^◡𝐹 “ 𝑧) ∈ dom vol))
63	4, 62	syl5bi 241	. . 3 ⊢ (𝜑 → (𝑧 ∈ ran (,) → (^◡𝐹 “ 𝑧) ∈ dom vol))
64	63	ralrimiv 3106	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (,)(^◡𝐹 “ 𝑧) ∈ dom vol)
65		ismbf 24697	. . 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 256	1 ⊢ (𝜑 → 𝐹 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ∩ cin 3882 ifcif 4456 𝒫 cpw 4530 class class class wbr 5070 × cxp 5578 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 “ cima 5583 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 (class class class)co 7255 ℝcr 10801 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 ≤ cle 10941 (,)cioo 13008 volcvol 24532 MblFncmbf 24683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xadd 12778 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-xmet 20503 df-met 20504 df-ovol 24533 df-vol 24534 df-mbf 24688

This theorem is referenced by: ismbf2d 24709 mbfmax 24718

W3C validator