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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlexch3 | Structured version Visualization version GIF version |
Description: An atomic covering lattice has the exchange property. (atexch 31365 analog.) (Contributed by NM, 5-Nov-2012.) |
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 |
---|---|
cvlexch3 | 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 β¨ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlatl 37833 | . . . . 5 β’ (πΎ β CvLat β πΎ β AtLat) | |
2 | 1 | adantr 482 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅)) β πΎ β AtLat) |
3 | simpr1 1195 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΅)) β π β π΄) | |
4 | simpr3 1197 | . . . 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 37825 | . . . 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 1372 | . . 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 | cvlexch1 37836 | . . . 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 β¨ π))) |
14 | 13 | 3expia 1122 | . . 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 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 β¨ π)))) |
16 | 15 | 3impia 1118 | 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 β¨ π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17088 lecple 17145 joincjn 18205 meetcmee 18206 0.cp0 18317 Atomscatm 37771 AtLatcal 37772 CvLatclc 37773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-p0 18319 df-lat 18326 df-covers 37774 df-ats 37775 df-atl 37806 df-cvlat 37830 |
This theorem is referenced by: hlexch3 37900 |
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