Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > latmidm | Structured version Visualization version GIF version | ||
| Description: Lattice meet is idempotent. Analogue of inidm 4217. (Contributed by NM, 8-Nov-2011.) |
| Ref | Expression |
|---|---|
| latmidm.b | β’ π΅ = (BaseβπΎ) |
| latmidm.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| latmidm | β’ ((πΎ β Lat β§ π β π΅) β (π β§ π) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latmidm.b | . 2 β’ π΅ = (BaseβπΎ) | |
| 2 | eqid 2730 | . 2 β’ (leβπΎ) = (leβπΎ) | |
| 3 | simpl 481 | . 2 β’ ((πΎ β Lat β§ π β π΅) β πΎ β Lat) | |
| 4 | latmidm.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 5 | 1, 4 | latmcl 18397 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) β π΅) |
| 6 | 5 | 3anidm23 1419 | . 2 β’ ((πΎ β Lat β§ π β π΅) β (π β§ π) β π΅) |
| 7 | simpr 483 | . 2 β’ ((πΎ β Lat β§ π β π΅) β π β π΅) | |
| 8 | 1, 2, 4 | latmle1 18421 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π)(leβπΎ)π) |
| 9 | 8 | 3anidm23 1419 | . 2 β’ ((πΎ β Lat β§ π β π΅) β (π β§ π)(leβπΎ)π) |
| 10 | 1, 2 | latref 18398 | . . 3 β’ ((πΎ β Lat β§ π β π΅) β π(leβπΎ)π) |
| 11 | 1, 2, 4 | latlem12 18423 | . . . 4 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π(leβπΎ)π β§ π(leβπΎ)π) β π(leβπΎ)(π β§ π))) |
| 12 | 3, 7, 7, 7, 11 | syl13anc 1370 | . . 3 β’ ((πΎ β Lat β§ π β π΅) β ((π(leβπΎ)π β§ π(leβπΎ)π) β π(leβπΎ)(π β§ π))) |
| 13 | 10, 10, 12 | mpbi2and 708 | . 2 β’ ((πΎ β Lat β§ π β π΅) β π(leβπΎ)(π β§ π)) |
| 14 | 1, 2, 3, 6, 7, 9, 13 | latasymd 18402 | 1 β’ ((πΎ β Lat β§ π β π΅) β (π β§ π) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 class class class wbr 5147 βcfv 6542 (class class class)co 7411 Basecbs 17148 lecple 17208 meetcmee 18269 Latclat 18388 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
| 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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-proset 18252 df-poset 18270 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-lat 18389 |
| This theorem is referenced by: latmmdiN 38407 latmmdir 38408 2llnm3N 38743 |
| Copyright terms: Public domain | W3C validator |