iocmbl - Mathbox for Jon Pennant

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Theorem iocmbl 41044

Description: An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.)

**Assertion**
Ref	Expression
iocmbl	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ∈ dom vol)

**Proof of Theorem iocmbl**
Step	Hyp	Ref	Expression
1		rexr 11021	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
2		ioounsn 13209	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
3	1, 2	syl3an2 1163	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
4		ioombl 24729	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ dom vol
5		iccid 13124	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵[,]𝐵) = {𝐵})
6	1, 5	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (𝐵[,]𝐵) = {𝐵})
7		iccmbl 24730	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵[,]𝐵) ∈ dom vol)
8	7	anidms 567	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (𝐵[,]𝐵) ∈ dom vol)
9	6, 8	eqeltrrd 2840	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → {𝐵} ∈ dom vol)
10	9	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → {𝐵} ∈ dom vol)
11		unmbl 24701	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ {𝐵} ∈ dom vol) → ((𝐴(,)𝐵) ∪ {𝐵}) ∈ dom vol)
12	4, 10, 11	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴(,)𝐵) ∪ {𝐵}) ∈ dom vol)
13	12	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) ∈ dom vol)
14	3, 13	eqeltrrd 2840	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴(,]𝐵) ∈ dom vol)
15	14	3expa 1117	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 < 𝐵) → (𝐴(,]𝐵) ∈ dom vol)
16		id 22	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ∈ ℝ^*)
17		xrlenlt 11040	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
18	1, 16, 17	syl2anr 597	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
19	18	biimp3ar 1469	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐴 < 𝐵) → 𝐵 ≤ 𝐴)
20		ioc0 13126	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,]𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
21	20	biimp3ar 1469	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐴) → (𝐴(,]𝐵) = ∅)
22	1, 21	syl3an2 1163	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) → (𝐴(,]𝐵) = ∅)
23		0mbl 24703	. . . . 5 ⊢ ∅ ∈ dom vol
24	22, 23	eqeltrdi 2847	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) → (𝐴(,]𝐵) ∈ dom vol)
25	19, 24	syld3an3 1408	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐴 < 𝐵) → (𝐴(,]𝐵) ∈ dom vol)
26	25	3expa 1117	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,]𝐵) ∈ dom vol)
27	15, 26	pm2.61dan 810	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∪ cun 3885 ∅c0 4256 {csn 4561 class class class wbr 5074 dom cdm 5589 (class class class)co 7275 ℝcr 10870 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 (,)cioo 13079 (,]cioc 13080 [,]cicc 13082 volcvol 24627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xadd 12849 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-xmet 20590 df-met 20591 df-ovol 24628 df-vol 24629

This theorem is referenced by: (None)

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