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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatcvr1 | Structured version Visualization version GIF version |
Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlatcvr1.j | β’ β¨ = (joinβπΎ) |
cvlatcvr1.c | β’ πΆ = ( β βπΎ) |
cvlatcvr1.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
cvlatcvr1 | 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))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp13 1205 | . . . 4 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β πΎ β CvLat) | |
2 | cvlatl 38281 | . . . 4 β’ (πΎ β CvLat β πΎ β AtLat) | |
3 | 1, 2 | syl 17 | . . 3 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β πΎ β AtLat) |
4 | eqid 2732 | . . . 4 β’ (meetβπΎ) = (meetβπΎ) | |
5 | eqid 2732 | . . . 4 β’ (0.βπΎ) = (0.βπΎ) | |
6 | cvlatcvr1.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
7 | 4, 5, 6 | atnem0 38274 | . . 3 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΄) β (π β π β (π(meetβπΎ)π) = (0.βπΎ))) |
8 | 3, 7 | syld3an1 1410 | . 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 (π(meetβπΎ)π) = (0.βπΎ))) |
9 | eqid 2732 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
10 | 9, 6 | atbase 38245 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
11 | cvlatcvr1.j | . . . 4 β’ β¨ = (joinβπΎ) | |
12 | cvlatcvr1.c | . . . 4 β’ πΆ = ( β βπΎ) | |
13 | 9, 11, 4, 5, 12, 6 | cvlcvrp 38296 | . . 3 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β (BaseβπΎ) β§ π β π΄) β ((π(meetβπΎ)π) = (0.βπΎ) β ππΆ(π β¨ π))) |
14 | 10, 13 | syl3an2 1164 | . 2 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β ((π(meetβπΎ)π) = (0.βπΎ) β ππΆ(π β¨ π))) |
15 | 8, 14 | bitrd 278 | 1 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))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 class class class wbr 5148 βcfv 6543 (class class class)co 7411 Basecbs 17146 joincjn 18266 meetcmee 18267 0.cp0 18378 CLatccla 18453 OMLcoml 38131 β ccvr 38218 Atomscatm 38219 AtLatcal 38220 CvLatclc 38221 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7367 df-ov 7414 df-oprab 7415 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-lat 18387 df-clat 18454 df-oposet 38132 df-ol 38134 df-oml 38135 df-covers 38222 df-ats 38223 df-atl 38254 df-cvlat 38278 |
This theorem is referenced by: cvlatcvr2 38298 atcvr1 38374 |
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