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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvr1 | Structured version Visualization version GIF version |
Description: A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 32159 analog.) (Contributed by NM, 17-Nov-2011.) |
Ref | Expression |
---|---|
cvr1.b | β’ π΅ = (BaseβπΎ) |
cvr1.l | β’ β€ = (leβπΎ) |
cvr1.j | β’ β¨ = (joinβπΎ) |
cvr1.c | β’ πΆ = ( β βπΎ) |
cvr1.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
cvr1 | β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (Β¬ π β€ π β ππΆ(π β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmcv 38823 | . 2 β’ (πΎ β HL β (πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat)) | |
2 | cvr1.b | . . 3 β’ π΅ = (BaseβπΎ) | |
3 | cvr1.l | . . 3 β’ β€ = (leβπΎ) | |
4 | cvr1.j | . . 3 β’ β¨ = (joinβπΎ) | |
5 | cvr1.c | . . 3 β’ πΆ = ( β βπΎ) | |
6 | cvr1.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
7 | 2, 3, 4, 5, 6 | cvlcvr1 38806 | . 2 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΅ β§ π β π΄) β (Β¬ π β€ π β ππΆ(π β¨ π))) |
8 | 1, 7 | syl3an1 1161 | 1 β’ ((πΎ β HL β§ π β π΅ β§ π β π΄) β (Β¬ π β€ π β ππΆ(π β¨ π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5143 βcfv 6543 (class class class)co 7415 Basecbs 17174 lecple 17234 joincjn 18297 CLatccla 18484 OMLcoml 38642 β ccvr 38729 Atomscatm 38730 CvLatclc 38732 HLchlt 38817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-lat 18418 df-clat 18485 df-oposet 38643 df-ol 38645 df-oml 38646 df-covers 38733 df-ats 38734 df-atl 38765 df-cvlat 38789 df-hlat 38818 |
This theorem is referenced by: cvr2N 38879 hlrelat3 38880 cvrval3 38881 cvrval4N 38882 cvrexchlem 38887 cvratlem 38889 cvrat3 38910 3dim0 38925 2dim 38938 1cvrjat 38943 llncvrlpln2 39025 lplnexllnN 39032 lplncvrlvol2 39083 lhp2lt 39469 |
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