icombl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > icombl		Structured version Visualization version GIF version

Theorem icombl 24728

Description: A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

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

**Proof of Theorem icombl**
Step	Hyp	Ref	Expression
1		uncom 4087	. . . . 5 ⊢ ((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = ((𝐴[,)𝐵) ∪ (𝐵[,)+∞))
2		rexr 11021	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
3	2	ad2antrr 723	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
4		simplr 766	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
5		pnfxr 11029	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → +∞ ∈ ℝ^)
7		xrltle 12883	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
8	2, 7	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
9	8	imp 407	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
10		pnfge 12866	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
11	4, 10	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
12		icoun 13207	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ +∞)) → ((𝐴[,)𝐵) ∪ (𝐵[,)+∞)) = (𝐴[,)+∞))
13	3, 4, 6, 9, 11, 12	syl32anc 1377	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)𝐵) ∪ (𝐵[,)+∞)) = (𝐴[,)+∞))
14	1, 13	eqtrid 2790	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞))
15		ssun1 4106	. . . . . 6 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (𝐴[,)𝐵))
16	15, 14	sseqtrid 3973	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ (𝐴[,)+∞))
17		incom 4135	. . . . . 6 ⊢ ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ((𝐴[,)𝐵) ∩ (𝐵[,)+∞))
18		icodisj 13208	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
19	5, 18	mp3an3 1449	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
20	3, 4, 19	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
21	17, 20	eqtrid 2790	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ∅)
22		uneqdifeq 4423	. . . . 5 ⊢ (((𝐵[,)+∞) ⊆ (𝐴[,)+∞) ∧ ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞) ↔ ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵)))
23	16, 21, 22	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞) ↔ ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵)))
24	14, 23	mpbid 231	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵))
25		icombl1 24727	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
26	25	ad2antrr 723	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴[,)+∞) ∈ dom vol)
27		xrleloe 12878	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
28	4, 6, 27	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
29	11, 28	mpbid 231	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ ∨ 𝐵 = +∞))
30		simpr 485	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
31		xrre2 12904	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < +∞)) → 𝐵 ∈ ℝ)
32	31	expr 457	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
33	3, 4, 6, 30, 32	syl31anc 1372	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
34	33	orim1d 963	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵 < +∞ ∨ 𝐵 = +∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞)))
35	29, 34	mpd 15	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
36		icombl1 24727	. . . . . 6 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
37		oveq1 7282	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
38		pnfge 12866	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
39	5, 38	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
40		ico0 13125	. . . . . . . . . 10 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
41	5, 5, 40	mp2an 689	. . . . . . . . 9 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
42	39, 41	mpbir 230	. . . . . . . 8 ⊢ (+∞[,)+∞) = ∅
43	37, 42	eqtrdi 2794	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
44		0mbl 24703	. . . . . . 7 ⊢ ∅ ∈ dom vol
45	43, 44	eqeltrdi 2847	. . . . . 6 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
46	36, 45	jaoi 854	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
47	35, 46	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
48		difmbl 24707	. . . 4 ⊢ (((𝐴[,)+∞) ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) ∈ dom vol)
49	26, 47, 48	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) ∈ dom vol)
50	24, 49	eqeltrrd 2840	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴[,)𝐵) ∈ dom vol)
51		ico0 13125	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
52	2, 51	sylan 580	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
53		simpr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
54	2	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
55	53, 54	xrlenltd 11041	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
56	52, 55	bitrd 278	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ ¬ 𝐴 < 𝐵))
57	56	biimpar 478	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴[,)𝐵) = ∅)
58	57, 44	eqeltrdi 2847	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴[,)𝐵) ∈ dom vol)
59	50, 58	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 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 dom cdm 5589 (class class class)co 7275 ℝcr 10870 +∞cpnf 11006 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 [,)cico 13081 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-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: ioombl 24729 volicoff 43536 voliooicof 43537 icoresmbl 44081 hoiprodcl 44085 hoiprodcl3 44118 hoidmvcl 44120 hsphoidmvle2 44123 hsphoidmvle 44124 hoidmv1lelem1 44129 hoidmv1lelem2 44130 hoidmv1lelem3 44131 hoidifhspdmvle 44158 volicorege0 44175 ovolval5lem1 44190

W3C validator