ismbf2d - Metamath Proof Explorer

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Theorem ismbf2d 25148

Description: Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.)

**Hypotheses**
Ref	Expression
ismbf2d.1	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
ismbf2d.2	⊢ (𝜑 → 𝐴 ∈ dom vol)
ismbf2d.3	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
ismbf2d.4	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)

**Assertion**
Ref	Expression
ismbf2d	⊢ (𝜑 → 𝐹 ∈ MblFn)

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

**Proof of Theorem ismbf2d**
Step	Hyp	Ref	Expression
1		ismbf2d.1	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
2		elxr 13092	. . 3 ⊢ (𝑥 ∈ ℝ^* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
3		ismbf2d.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
4		oveq1 7412	. . . . . . . 8 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = (+∞(,)+∞))
5		iooid 13348	. . . . . . . 8 ⊢ (+∞(,)+∞) = ∅
6	4, 5	eqtrdi 2788	. . . . . . 7 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = ∅)
7	6	imaeq2d 6057	. . . . . 6 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (𝑥(,)+∞)) = (^◡𝐹 “ ∅))
8		ima0 6073	. . . . . . 7 ⊢ (^◡𝐹 “ ∅) = ∅
9		0mbl 25047	. . . . . . 7 ⊢ ∅ ∈ dom vol
10	8, 9	eqeltri 2829	. . . . . 6 ⊢ (^◡𝐹 “ ∅) ∈ dom vol
11	7, 10	eqeltrdi 2841	. . . . 5 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
12	11	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
13		fimacnv 6736	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → (^◡𝐹 “ ℝ) = 𝐴)
14	1, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → (^◡𝐹 “ ℝ) = 𝐴)
15		ismbf2d.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ dom vol)
16	14, 15	eqeltrd 2833	. . . . . 6 ⊢ (𝜑 → (^◡𝐹 “ ℝ) ∈ dom vol)
17		oveq1 7412	. . . . . . . . 9 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = (-∞(,)+∞))
18		ioomax 13395	. . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ
19	17, 18	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = ℝ)
20	19	imaeq2d 6057	. . . . . . 7 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (𝑥(,)+∞)) = (^◡𝐹 “ ℝ))
21	20	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = -∞ → ((^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ↔ (^◡𝐹 “ ℝ) ∈ dom vol))
22	16, 21	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → (𝑥 = -∞ → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol))
23	22	imp 407	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
24	3, 12, 23	3jaodan 1430	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
25	2, 24	sylan2b 594	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
26		ismbf2d.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
27		oveq2 7413	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = (-∞(,)+∞))
28	27, 18	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = ℝ)
29	28	imaeq2d 6057	. . . . . . 7 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ ℝ))
30	29	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = +∞ → ((^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (^◡𝐹 “ ℝ) ∈ dom vol))
31	16, 30	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → (𝑥 = +∞ → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol))
32	31	imp 407	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
33		oveq2 7413	. . . . . . . 8 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = (-∞(,)-∞))
34		iooid 13348	. . . . . . . 8 ⊢ (-∞(,)-∞) = ∅
35	33, 34	eqtrdi 2788	. . . . . . 7 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = ∅)
36	35	imaeq2d 6057	. . . . . 6 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ ∅))
37	36, 10	eqeltrdi 2841	. . . . 5 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
38	37	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
39	26, 32, 38	3jaodan 1430	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
40	2, 39	sylan2b 594	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
41	1, 25, 40	ismbfd 25147	1 ⊢ (𝜑 → 𝐹 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ∅c0 4321 ^◡ccnv 5674 dom cdm 5675 “ cima 5678 ⟶wf 6536 (class class class)co 7405 ℝcr 11105 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 (,)cioo 13320 volcvol 24971 MblFncmbf 25122

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xadd 13089 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-xmet 20929 df-met 20930 df-ovol 24972 df-vol 24973 df-mbf 25127

This theorem is referenced by: mbfres 25152 mbfmulc2lem 25155 mbfposr 25160 ismbf3d 25162 iblabsnclem 36539 ftc1anclem1 36549 ftc1anclem6 36554

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