Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > meetat2 | Structured version Visualization version GIF version | ||
| Description: The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-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 |
|---|---|
| meetat2 | β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β ((π β§ π) β π΄ β¨ (π β§ π) = 0 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | m.b | . . 3 β’ π΅ = (BaseβπΎ) | |
| 2 | m.m | . . 3 β’ β§ = (meetβπΎ) | |
| 3 | m.z | . . 3 β’ 0 = (0.βπΎ) | |
| 4 | m.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
| 5 | 1, 2, 3, 4 | meetat 38677 | . 2 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β ((π β§ π) = π β¨ (π β§ π) = 0 )) |
| 6 | eleq1a 2822 | . . . 4 β’ (π β π΄ β ((π β§ π) = π β (π β§ π) β π΄)) | |
| 7 | 6 | 3ad2ant3 1132 | . . 3 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β ((π β§ π) = π β (π β§ π) β π΄)) |
| 8 | 7 | orim1d 962 | . 2 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β (((π β§ π) = π β¨ (π β§ π) = 0 ) β ((π β§ π) β π΄ β¨ (π β§ π) = 0 ))) |
| 9 | 5, 8 | mpd 15 | 1 β’ ((πΎ β OL β§ π β π΅ β§ π β π΄) β ((π β§ π) β π΄ β¨ (π β§ π) = 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β¨ wo 844 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 Basecbs 17151 meetcmee 18275 0.cp0 18386 OLcol 38555 Atomscatm 38644 |
| 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 7721 |
| 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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-lat 18395 df-oposet 38557 df-ol 38559 df-covers 38647 df-ats 38648 |
| This theorem is referenced by: 2at0mat0 38907 atmod1i1m 39240 |
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