ismbf2d - Metamath Proof Explorer

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Theorem ismbf2d 25524

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 13102	. . 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 13358	. . . . . . . 8 ⊢ (+∞(,)+∞) = ∅
6	4, 5	eqtrdi 2782	. . . . . . 7 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = ∅)
7	6	imaeq2d 6053	. . . . . 6 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (𝑥(,)+∞)) = (^◡𝐹 “ ∅))
8		ima0 6070	. . . . . . 7 ⊢ (^◡𝐹 “ ∅) = ∅
9		0mbl 25423	. . . . . . 7 ⊢ ∅ ∈ dom vol
10	8, 9	eqeltri 2823	. . . . . 6 ⊢ (^◡𝐹 “ ∅) ∈ dom vol
11	7, 10	eqeltrdi 2835	. . . . 5 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
12	11	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
13		fimacnv 6733	. . . . . . . 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 2827	. . . . . 6 ⊢ (𝜑 → (^◡𝐹 “ ℝ) ∈ dom vol)
17		oveq1 7412	. . . . . . . . 9 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = (-∞(,)+∞))
18		ioomax 13405	. . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ
19	17, 18	eqtrdi 2782	. . . . . . . 8 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = ℝ)
20	19	imaeq2d 6053	. . . . . . 7 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (𝑥(,)+∞)) = (^◡𝐹 “ ℝ))
21	20	eleq1d 2812	. . . . . 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 406	. . . 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 593	. 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 2782	. . . . . . . 8 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = ℝ)
29	28	imaeq2d 6053	. . . . . . 7 ⊢ (𝑥 = +∞ → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ ℝ))
30	29	eleq1d 2812	. . . . . 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 406	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
33		oveq2 7413	. . . . . . . 8 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = (-∞(,)-∞))
34		iooid 13358	. . . . . . . 8 ⊢ (-∞(,)-∞) = ∅
35	33, 34	eqtrdi 2782	. . . . . . 7 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = ∅)
36	35	imaeq2d 6053	. . . . . 6 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (-∞(,)𝑥)) = (^◡𝐹 “ ∅))
37	36, 10	eqeltrdi 2835	. . . . 5 ⊢ (𝑥 = -∞ → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
38	37	adantl 481	. . . 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 593	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ^*) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
41	1, 25, 40	ismbfd 25523	1 ⊢ (𝜑 → 𝐹 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1083 = wceq 1533 ∈ wcel 2098 ∅c0 4317 ^◡ccnv 5668 dom cdm 5669 “ cima 5672 ⟶wf 6533 (class class class)co 7405 ℝcr 11111 +∞cpnf 11249 -∞cmnf 11250 ℝ^*cxr 11251 (,)cioo 13330 volcvol 25347 MblFncmbf 25498

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-xadd 13099 df-ioo 13334 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-xmet 21233 df-met 21234 df-ovol 25348 df-vol 25349 df-mbf 25503

This theorem is referenced by: mbfres 25528 mbfmulc2lem 25531 mbfposr 25536 ismbf3d 25538 iblabsnclem 37064 ftc1anclem1 37074 ftc1anclem6 37079

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