icombl - Metamath Proof Explorer

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Theorem icombl 25531

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 4098	. . . . 5 ⊢ ((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = ((𝐴[,)𝐵) ∪ (𝐵[,)+∞))
2		rexr 11191	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
3	2	ad2antrr 727	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
4		simplr 769	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
5		pnfxr 11199	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → +∞ ∈ ℝ^)
7		xrltle 13100	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
8	2, 7	sylan 581	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
9	8	imp 406	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
10		pnfge 13081	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
11	4, 10	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
12		icoun 13428	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ +∞)) → ((𝐴[,)𝐵) ∪ (𝐵[,)+∞)) = (𝐴[,)+∞))
13	3, 4, 6, 9, 11, 12	syl32anc 1381	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)𝐵) ∪ (𝐵[,)+∞)) = (𝐴[,)+∞))
14	1, 13	eqtrid 2783	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞))
15		ssun1 4118	. . . . . 6 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (𝐴[,)𝐵))
16	15, 14	sseqtrid 3964	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ (𝐴[,)+∞))
17		incom 4149	. . . . . 6 ⊢ ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ((𝐴[,)𝐵) ∩ (𝐵[,)+∞))
18		icodisj 13429	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
19	5, 18	mp3an3 1453	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
20	3, 4, 19	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
21	17, 20	eqtrid 2783	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ∅)
22		uneqdifeq 4432	. . . . 5 ⊢ (((𝐵[,)+∞) ⊆ (𝐴[,)+∞) ∧ ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞) ↔ ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵)))
23	16, 21, 22	syl2anc 585	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞) ↔ ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵)))
24	14, 23	mpbid 232	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵))
25		icombl1 25530	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
26	25	ad2antrr 727	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴[,)+∞) ∈ dom vol)
27		xrleloe 13095	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
28	4, 6, 27	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
29	11, 28	mpbid 232	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ ∨ 𝐵 = +∞))
30		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
31		xrre2 13122	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < +∞)) → 𝐵 ∈ ℝ)
32	31	expr 456	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
33	3, 4, 6, 30, 32	syl31anc 1376	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
34	33	orim1d 968	. . . . . 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 25530	. . . . . 6 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
37		oveq1 7374	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
38		pnfge 13081	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
39	5, 38	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
40		ico0 13344	. . . . . . . . . 10 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
41	5, 5, 40	mp2an 693	. . . . . . . . 9 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
42	39, 41	mpbir 231	. . . . . . . 8 ⊢ (+∞[,)+∞) = ∅
43	37, 42	eqtrdi 2787	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
44		0mbl 25506	. . . . . . 7 ⊢ ∅ ∈ dom vol
45	43, 44	eqeltrdi 2844	. . . . . 6 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
46	36, 45	jaoi 858	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
47	35, 46	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
48		difmbl 25510	. . . 4 ⊢ (((𝐴[,)+∞) ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) ∈ dom vol)
49	26, 47, 48	syl2anc 585	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) ∈ dom vol)
50	24, 49	eqeltrrd 2837	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴[,)𝐵) ∈ dom vol)
51		ico0 13344	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
52	2, 51	sylan 581	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
53		simpr 484	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
54	2	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
55	53, 54	xrlenltd 11211	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
56	52, 55	bitrd 279	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ ¬ 𝐴 < 𝐵))
57	56	biimpar 477	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴[,)𝐵) = ∅)
58	57, 44	eqeltrdi 2844	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴[,)𝐵) ∈ dom vol)
59	50, 58	pm2.61dan 813	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∖ cdif 3886 ∪ cun 3887 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 class class class wbr 5085 dom cdm 5631 (class class class)co 7367 ℝcr 11037 +∞cpnf 11176 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 [,)cico 13300 volcvol 25430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xadd 13064 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-xmet 21345 df-met 21346 df-ovol 25431 df-vol 25432

This theorem is referenced by: ioombl 25532 volicoff 46423 voliooicof 46424 icoresmbl 46971 hoiprodcl 46975 hoiprodcl3 47008 hoidmvcl 47010 hsphoidmvle2 47013 hsphoidmvle 47014 hoidmv1lelem1 47019 hoidmv1lelem2 47020 hoidmv1lelem3 47021 hoidifhspdmvle 47048 volicorege0 47065 ovolval5lem1 47080

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