ismbf2d - Metamath Proof Explorer

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Theorem ismbf2d 25589

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 13136	. . 3 ⊢ (𝑥 ∈ ℝ^* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
3		ismbf2d.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
4		oveq1 7433	. . . . . . . 8 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = (+∞(,)+∞))
5		iooid 13392	. . . . . . . 8 ⊢ (+∞(,)+∞) = ∅
6	4, 5	eqtrdi 2784	. . . . . . 7 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = ∅)
7	6	imaeq2d 6068	. . . . . 6 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (𝑥(,)+∞)) = (^◡𝐹 “ ∅))
8		ima0 6085	. . . . . . 7 ⊢ (^◡𝐹 “ ∅) = ∅
9		0mbl 25488	. . . . . . 7 ⊢ ∅ ∈ dom vol
10	8, 9	eqeltri 2825	. . . . . 6 ⊢ (^◡𝐹 “ ∅) ∈ dom vol
11	7, 10	eqeltrdi 2837	. . . . 5 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
12	11	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
13		fimacnv 6750	. . . . . . . 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 2829	. . . . . 6 ⊢ (𝜑 → (^◡𝐹 “ ℝ) ∈ dom vol)
17		oveq1 7433	. . . . . . . . 9 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = (-∞(,)+∞))
18		ioomax 13439	. . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ
19	17, 18	eqtrdi 2784	. . . . . . . 8 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = ℝ)
20	19	imaeq2d 6068	. . . . . . 7 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (𝑥(,)+∞)) = (^◡𝐹 “ ℝ))
21	20	eleq1d 2814	. . . . . 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 405	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
24	3, 12, 23	3jaodan 1427	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
25	2, 24	sylan2b 592	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
26		ismbf2d.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
27		oveq2 7434	. . . . . . . . 9 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = (-∞(,)+∞))
28	27, 18	eqtrdi 2784	. . . . . . . 8 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = ℝ)
29	28	imaeq2d 6068	. . . . . . 7 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ ℝ))
30	29	eleq1d 2814	. . . . . 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 405	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
33		oveq2 7434	. . . . . . . 8 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = (-∞(,)-∞))
34		iooid 13392	. . . . . . . 8 ⊢ (-∞(,)-∞) = ∅
35	33, 34	eqtrdi 2784	. . . . . . 7 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = ∅)
36	35	imaeq2d 6068	. . . . . 6 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ ∅))
37	36, 10	eqeltrdi 2837	. . . . 5 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
38	37	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
39	26, 32, 38	3jaodan 1427	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
40	2, 39	sylan2b 592	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
41	1, 25, 40	ismbfd 25588	1 ⊢ (𝜑 → 𝐹 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∨ w3o 1083 = wceq 1533 ∈ wcel 2098 ∅c0 4326 ^◡ccnv 5681 dom cdm 5682 “ cima 5685 ⟶wf 6549 (class class class)co 7426 ℝcr 11145 +∞cpnf 11283 -∞cmnf 11284 ℝ^*cxr 11285 (,)cioo 13364 volcvol 25412 MblFncmbf 25563

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-xmet 21279 df-met 21280 df-ovol 25413 df-vol 25414 df-mbf 25568

This theorem is referenced by: mbfres 25593 mbfmulc2lem 25596 mbfposr 25601 ismbf3d 25603 iblabsnclem 37189 ftc1anclem1 37199 ftc1anclem6 37204

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