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 25623

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 4111	. . . . 5 ⊢ ((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = ((𝐴[,)𝐵) ∪ (𝐵[,)+∞))
2		rexr 11228	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
3	2	ad2antrr 736	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
4		simplr 778	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
5		pnfxr 11236	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → +∞ ∈ ℝ^)
7		xrltle 13151	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
8	2, 7	sylan 589	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
9	8	imp 410	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵)
10		pnfge 13132	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
11	4, 10	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
12		icoun 13479	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ +∞)) → ((𝐴[,)𝐵) ∪ (𝐵[,)+∞)) = (𝐴[,)+∞))
13	3, 4, 6, 9, 11, 12	syl32anc 1397	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)𝐵) ∪ (𝐵[,)+∞)) = (𝐴[,)+∞))
14	1, 13	eqtrid 2809	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞))
15		ssun1 4130	. . . . . 6 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (𝐴[,)𝐵))
16	15, 14	sseqtrid 3978	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ (𝐴[,)+∞))
17		incom 4161	. . . . . 6 ⊢ ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ((𝐴[,)𝐵) ∩ (𝐵[,)+∞))
18		icodisj 13480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
19	5, 18	mp3an3 1471	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
20	3, 4, 19	syl2anc 593	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
21	17, 20	eqtrid 2809	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ∅)
22		uneqdifeq 4446	. . . . 5 ⊢ (((𝐵[,)+∞) ⊆ (𝐴[,)+∞) ∧ ((𝐵[,)+∞) ∩ (𝐴[,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞) ↔ ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵)))
23	16, 21, 22	syl2anc 593	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (((𝐵[,)+∞) ∪ (𝐴[,)𝐵)) = (𝐴[,)+∞) ↔ ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵)))
24	14, 23	mpbid 234	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) = (𝐴[,)𝐵))
25		icombl1 25622	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol)
26	25	ad2antrr 736	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴[,)+∞) ∈ dom vol)
27		xrleloe 13146	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
28	4, 6, 27	syl2anc 593	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
29	11, 28	mpbid 234	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ ∨ 𝐵 = +∞))
30		simpr 488	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
31		xrre2 13173	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < +∞)) → 𝐵 ∈ ℝ)
32	31	expr 460	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
33	3, 4, 6, 30, 32	syl31anc 1392	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
34	33	orim1d 979	. . . . . 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 25622	. . . . . 6 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
37		oveq1 7403	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
38		pnfge 13132	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
39	5, 38	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
40		ico0 13395	. . . . . . . . . 10 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
41	5, 5, 40	mp2an 702	. . . . . . . . 9 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
42	39, 41	mpbir 233	. . . . . . . 8 ⊢ (+∞[,)+∞) = ∅
43	37, 42	eqtrdi 2813	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
44		0mbl 25598	. . . . . . 7 ⊢ ∅ ∈ dom vol
45	43, 44	eqeltrdi 2870	. . . . . 6 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
46	36, 45	jaoi 868	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
47	35, 46	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
48		difmbl 25602	. . . 4 ⊢ (((𝐴[,)+∞) ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) ∈ dom vol)
49	26, 47, 48	syl2anc 593	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴[,)+∞) ∖ (𝐵[,)+∞)) ∈ dom vol)
50	24, 49	eqeltrrd 2863	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴[,)𝐵) ∈ dom vol)
51		ico0 13395	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
52	2, 51	sylan 589	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
53		simpr 488	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
54	2	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
55	53, 54	xrlenltd 11248	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
56	52, 55	bitrd 281	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ ¬ 𝐴 < 𝐵))
57	56	biimpar 481	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴[,)𝐵) = ∅)
58	57, 44	eqeltrdi 2870	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴[,)𝐵) ∈ dom vol)
59	50, 58	pm2.61dan 822	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ∖ cdif 3901 ∪ cun 3902 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 class class class wbr 5100 dom cdm 5647 (class class class)co 7396 ℝcr 11072 +∞cpnf 11213 ℝ^*cxr 11215 < clt 11216 ≤ cle 11217 [,)cico 13351 volcvol 25522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9388 df-inf 9389 df-oi 9458 df-dju 9859 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-n0 12482 df-z 12569 df-uz 12840 df-q 12950 df-rp 12994 df-xadd 13115 df-ioo 13353 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-clim 15515 df-rlim 15516 df-sum 15714 df-xmet 21414 df-met 21415 df-ovol 25523 df-vol 25524

This theorem is referenced by: ioombl 25624 volicoff 46566 voliooicof 46567 icoresmbl 47114 hoiprodcl 47118 hoiprodcl3 47151 hoidmvcl 47153 hsphoidmvle2 47156 hsphoidmvle 47157 hoidmv1lelem1 47162 hoidmv1lelem2 47163 hoidmv1lelem3 47164 hoidifhspdmvle 47191 volicorege0 47208 ovolval5lem1 47223

W3C validator