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 38772 | . 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 2823 | . . . 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 963 | . 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 845 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6551 (class class class)co 7424 Basecbs 17185 meetcmee 18309 0.cp0 18420 OLcol 38650 Atomscatm 38739 |
| 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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-lat 18429 df-oposet 38652 df-ol 38654 df-covers 38742 df-ats 38743 |
| This theorem is referenced by: 2at0mat0 39002 atmod1i1m 39335 |
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