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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlexch4N | Structured version Visualization version GIF version |
Description: An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cvlexch3.b | β’ π΅ = (BaseβπΎ) |
cvlexch3.l | β’ β€ = (leβπΎ) |
cvlexch3.j | β’ β¨ = (joinβπΎ) |
cvlexch3.m | β’ β§ = (meetβπΎ) |
cvlexch3.z | β’ 0 = (0.βπΎ) |
cvlexch3.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
cvlexch4N | β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅) β§ (π β§ π) = 0 ) β (π β€ (π β¨ π) β (π β¨ π) = (π β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlatl 38708 | . . . . 5 β’ (πΎ β CvLat β πΎ β AtLat) | |
2 | 1 | adantr 480 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅)) β πΎ β AtLat) |
3 | simpr1 1191 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅)) β π β π΄) | |
4 | simpr3 1193 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅)) β π β π΅) | |
5 | cvlexch3.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
6 | cvlexch3.l | . . . . 5 β’ β€ = (leβπΎ) | |
7 | cvlexch3.m | . . . . 5 β’ β§ = (meetβπΎ) | |
8 | cvlexch3.z | . . . . 5 β’ 0 = (0.βπΎ) | |
9 | cvlexch3.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
10 | 5, 6, 7, 8, 9 | atnle 38700 | . . . 4 β’ ((πΎ β AtLat β§ π β π΄ β§ π β π΅) β (Β¬ π β€ π β (π β§ π) = 0 )) |
11 | 2, 3, 4, 10 | syl3anc 1368 | . . 3 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅)) β (Β¬ π β€ π β (π β§ π) = 0 )) |
12 | cvlexch3.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
13 | 5, 6, 12, 9 | cvlexchb1 38713 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅) β§ Β¬ π β€ π) β (π β€ (π β¨ π) β (π β¨ π) = (π β¨ π))) |
14 | 13 | 3expia 1118 | . . 3 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅)) β (Β¬ π β€ π β (π β€ (π β¨ π) β (π β¨ π) = (π β¨ π)))) |
15 | 11, 14 | sylbird 260 | . 2 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅)) β ((π β§ π) = 0 β (π β€ (π β¨ π) β (π β¨ π) = (π β¨ π)))) |
16 | 15 | 3impia 1114 | 1 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅) β§ (π β§ π) = 0 ) β (π β€ (π β¨ π) β (π β¨ π) = (π β¨ π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 lecple 17213 joincjn 18276 meetcmee 18277 0.cp0 18388 Atomscatm 38646 AtLatcal 38647 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-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 |
This theorem is referenced by: hlexch4N 38776 |
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