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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atcvr2 | Structured version Visualization version GIF version |
Description: An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.) |
Ref | Expression |
---|---|
atcvr1.j | β’ β¨ = (joinβπΎ) |
atcvr1.c | β’ πΆ = ( β βπΎ) |
atcvr1.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atcvr2 | β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β π β ππΆ(π β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmcv 38730 | . 2 β’ (πΎ β HL β (πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat)) | |
2 | atcvr1.j | . . 3 β’ β¨ = (joinβπΎ) | |
3 | atcvr1.c | . . 3 β’ πΆ = ( β βπΎ) | |
4 | atcvr1.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | 2, 3, 4 | cvlatcvr2 38716 | . 2 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β (π β π β ππΆ(π β¨ π))) |
6 | 1, 5 | syl3an1 1160 | 1 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β π β ππΆ(π β¨ π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 class class class wbr 5139 βcfv 6534 (class class class)co 7402 joincjn 18272 CLatccla 18459 OMLcoml 38549 β ccvr 38636 Atomscatm 38637 CvLatclc 38639 HLchlt 38724 |
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 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-lat 18393 df-clat 18460 df-oposet 38550 df-ol 38552 df-oml 38553 df-covers 38640 df-ats 38641 df-atl 38672 df-cvlat 38696 df-hlat 38725 |
This theorem is referenced by: atcvrj2b 38807 |
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