Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > meetat | Structured version Visualization version GIF version | ||
| Description: The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.) |
| Ref | Expression |
|---|---|
| m.b | β’ π΅ = (BaseβπΎ) |
| m.m | β’ β§ = (meetβπΎ) |
| m.z | β’ 0 = (0.βπΎ) |
| m.a | β’ π΄ = (AtomsβπΎ) |
| Ref | Expression |
|---|---|
| meetat | β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β ((π β§ π) = π β¨ (π β§ π) = 0 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ollat 38388 | . . . 4 β’ (πΎ β OL β πΎ β Lat) | |
| 2 | 1 | 3ad2ant1 1131 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β πΎ β Lat) |
| 3 | simp2 1135 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β π β π΅) | |
| 4 | simp3 1136 | . . . 4 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β π β π΄) | |
| 5 | m.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 6 | m.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
| 7 | 5, 6 | atbase 38464 | . . . 4 β’ (π β π΄ β π β π΅) |
| 8 | 4, 7 | syl 17 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β π β π΅) |
| 9 | eqid 2730 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
| 10 | m.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 11 | 5, 9, 10 | latmle2 18424 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π)(leβπΎ)π) |
| 12 | 2, 3, 8, 11 | syl3anc 1369 | . 2 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β (π β§ π)(leβπΎ)π) |
| 13 | olop 38389 | . . . 4 β’ (πΎ β OL β πΎ β OP) | |
| 14 | 13 | 3ad2ant1 1131 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β πΎ β OP) |
| 15 | 5, 10 | latmcl 18399 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) β π΅) |
| 16 | 2, 3, 8, 15 | syl3anc 1369 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β (π β§ π) β π΅) |
| 17 | m.z | . . . 4 β’ 0 = (0.βπΎ) | |
| 18 | 5, 9, 17, 6 | leatb 38467 | . . 3 β’ ((πΎ β OP β§ (π β§ π) β π΅ β§ π β π΄) β ((π β§ π)(leβπΎ)π β ((π β§ π) = π β¨ (π β§ π) = 0 ))) |
| 19 | 14, 16, 4, 18 | syl3anc 1369 | . 2 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β ((π β§ π)(leβπΎ)π β ((π β§ π) = π β¨ (π β§ π) = 0 ))) |
| 20 | 12, 19 | mpbid 231 | 1 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β ((π β§ π) = π β¨ (π β§ π) = 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β¨ wo 843 β§ w3a 1085 = wceq 1539 β wcel 2104 class class class wbr 5149 βcfv 6544 (class class class)co 7413 Basecbs 17150 lecple 17210 meetcmee 18271 0.cp0 18382 Latclat 18390 OPcops 38347 OLcol 38349 Atomscatm 38438 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-proset 18254 df-poset 18272 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-lat 18391 df-oposet 38351 df-ol 38353 df-covers 38441 df-ats 38442 |
| This theorem is referenced by: meetat2 38472 |
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