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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 32080 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 38720 | . 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 38703 | . 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 1160 | 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 1084 = wceq 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 (class class class)co 7402 Basecbs 17145 lecple 17205 joincjn 18268 CLatccla 18455 OMLcoml 38539 β ccvr 38626 Atomscatm 38627 CvLatclc 38629 HLchlt 38714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-lat 18389 df-clat 18456 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 |
This theorem is referenced by: cvr2N 38776 hlrelat3 38777 cvrval3 38778 cvrval4N 38779 cvrexchlem 38784 cvratlem 38786 cvrat3 38807 3dim0 38822 2dim 38835 1cvrjat 38840 llncvrlpln2 38922 lplnexllnN 38929 lplncvrlvol2 38980 lhp2lt 39366 |
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