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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatcvr2 | 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 |
---|---|
cvlatcvr2 | 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 | cvlatcvr1.j | . . 3 β’ β¨ = (joinβπΎ) | |
2 | cvlatcvr1.c | . . 3 β’ πΆ = ( β βπΎ) | |
3 | cvlatcvr1.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | 1, 2, 3 | cvlatcvr1 38724 | . 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))) |
5 | simp13 1202 | . . . . 5 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β πΎ β CvLat) | |
6 | cvllat 38709 | . . . . 5 β’ (πΎ β CvLat β πΎ β Lat) | |
7 | 5, 6 | syl 17 | . . . 4 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β πΎ β Lat) |
8 | eqid 2726 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
9 | 8, 3 | atbase 38672 | . . . . 5 β’ (π β π΄ β π β (BaseβπΎ)) |
10 | 9 | 3ad2ant2 1131 | . . . 4 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β π β (BaseβπΎ)) |
11 | 8, 3 | atbase 38672 | . . . . 5 β’ (π β π΄ β π β (BaseβπΎ)) |
12 | 11 | 3ad2ant3 1132 | . . . 4 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β π β (BaseβπΎ)) |
13 | 8, 1 | latjcom 18412 | . . . 4 β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β (π β¨ π) = (π β¨ π)) |
14 | 7, 10, 12, 13 | syl3anc 1368 | . . 3 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β (π β¨ π) = (π β¨ π)) |
15 | 14 | breq2d 5153 | . 2 β’ (((πΎ β OML β§ πΎ β CLat β§ πΎ β CvLat) β§ π β π΄ β§ π β π΄) β (ππΆ(π β¨ π) β ππΆ(π β¨ π))) |
16 | 4, 15 | bitrd 279 | 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 1084 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 joincjn 18276 Latclat 18396 CLatccla 18463 OMLcoml 38558 β ccvr 38645 Atomscatm 38646 CvLatclc 38648 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 |
This theorem is referenced by: atcvr2 38802 |
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