icombl - Metamath Proof Explorer

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Theorem icombl 25792

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 4105	. . . . 5 ⊢ ((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = ((𝐴[,)𝐵) ∪ (𝐵[,)+∞))
2		rexr 11279	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
3	2	ad2antrr 739	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
4		simplr 781	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
5		pnfxr 11287	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → +∞ ∈ ℝ^)
7		xrltle 13200	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
8	2, 7	sylan 592	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
9	8	imp 412	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
10		pnfge 13181	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
11	4, 10	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
12		icoun 13528	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ +∞)) → ((𝐴[,)𝐵) ∪ (𝐵[,)+∞)) = (𝐴[,)+∞))
13	3, 4, 6, 9, 11, 12	syl32anc 1405	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)𝐵) ∪ (𝐵[,)+∞)) = (𝐴[,)+∞))
14	1, 13	eqtrid 2807	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞))
15		ssun1 4124	. . . . . 6 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (𝐴[,)𝐵))
16	15, 14	sseqtrid 3973	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ (𝐴[,)+∞))
17		incom 4155	. . . . . 6 ⊢ ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ((𝐴[,)𝐵) ∩ (𝐵[,)+∞))
18		icodisj 13529	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
19	5, 18	mp3an3 1479	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
20	3, 4, 19	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
21	17, 20	eqtrid 2807	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ∅)
22		uneqdifeq 4448	. . . . 5 ⊢ (((𝐵[,)+∞) ⊆ (𝐴[,)+∞) ∧ ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞) ↔ ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵)))
23	16, 21, 22	syl2anc 596	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞) ↔ ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵)))
24	14, 23	mpbid 235	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵))
25		icombl1 25791	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
26	25	ad2antrr 739	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴[,)+∞) ∈ dom vol)
27		xrleloe 13195	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
28	4, 6, 27	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
29	11, 28	mpbid 235	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ ∨ 𝐵 = +∞))
30		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
31		xrre2 13222	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < +∞)) → 𝐵 ∈ ℝ)
32	31	expr 462	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
33	3, 4, 6, 30, 32	syl31anc 1400	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
34	33	orim1d 981	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵 < +∞ ∨ 𝐵 = +∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞)))
35	29, 34	mpd 16	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
36		icombl1 25791	. . . . . 6 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
37		oveq1 7420	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
38		pnfge 13181	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
39	5, 38	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
40		ico0 13444	. . . . . . . . . 10 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
41	5, 5, 40	mp2an 705	. . . . . . . . 9 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
42	39, 41	mpbir 234	. . . . . . . 8 ⊢ (+∞[,)+∞) = ∅
43	37, 42	eqtrdi 2811	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
44		0mbl 25767	. . . . . . 7 ⊢ ∅ ∈ dom vol
45	43, 44	eqeltrdi 2868	. . . . . 6 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
46	36, 45	jaoi 871	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
47	35, 46	syl 18	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
48		difmbl 25771	. . . 4 ⊢ (((𝐴[,)+∞) ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) ∈ dom vol)
49	26, 47, 48	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) ∈ dom vol)
50	24, 49	eqeltrrd 2861	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴[,)𝐵) ∈ dom vol)
51		ico0 13444	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
52	2, 51	sylan 592	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
53		simpr 490	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
54	2	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
55	53, 54	xrlenltd 11299	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
56	52, 55	bitrd 282	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ ¬ 𝐴 < 𝐵))
57	56	biimpar 483	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴[,)𝐵) = ∅)
58	57, 44	eqeltrdi 2868	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴[,)𝐵) ∈ dom vol)
59	50, 58	pm2.61dan 825	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 class class class wbr 5103 dom cdm 5655 (class class class)co 7413 ℝcr 11123 +∞cpnf 11264 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 [,)cico 13400 volcvol 25691

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-z 12616 df-uz 12888 df-q 12998 df-rp 13043 df-xadd 13164 df-ioo 13402 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-rlim 15576 df-sum 15774 df-xmet 21578 df-met 21579 df-ovol 25692 df-vol 25693

This theorem is used by: ioombl 25793 volicoff 46823 voliooicof 46824 icoresmbl 47371 hoiprodcl 47375 hoiprodcl3 47408 hoidmvcl 47410 hsphoidmvle2 47413 hsphoidmvle 47414 hoidmv1lelem1 47419 hoidmv1lelem2 47420 hoidmv1lelem3 47421 hoidifhspdmvle 47448 volicorege0 47465 ovolval5lem1 47480

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