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| Mirrors > Home > MPE Home > Th. List > latjidm | Structured version Visualization version GIF version | ||
| Description: Lattice join is idempotent. Analogue of unidm 4147. (Contributed by NM, 8-Oct-2011.) |
| Ref | Expression |
|---|---|
| latjidm.b | β’ π΅ = (BaseβπΎ) |
| latjidm.j | β’ β¨ = (joinβπΎ) |
| Ref | Expression |
|---|---|
| latjidm | β’ ((πΎ β Lat β§ π β π΅) β (π β¨ π) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latjidm.b | . 2 β’ π΅ = (BaseβπΎ) | |
| 2 | eqid 2726 | . 2 β’ (leβπΎ) = (leβπΎ) | |
| 3 | simpl 482 | . 2 β’ ((πΎ β Lat β§ π β π΅) β πΎ β Lat) | |
| 4 | latjidm.j | . . . 4 β’ β¨ = (joinβπΎ) | |
| 5 | 1, 4 | latjcl 18401 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
| 6 | 5 | 3anidm23 1418 | . 2 β’ ((πΎ β Lat β§ π β π΅) β (π β¨ π) β π΅) |
| 7 | simpr 484 | . 2 β’ ((πΎ β Lat β§ π β π΅) β π β π΅) | |
| 8 | 1, 2 | latref 18403 | . . 3 β’ ((πΎ β Lat β§ π β π΅) β π(leβπΎ)π) |
| 9 | 1, 2, 4 | latjle12 18412 | . . . 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 β¨ π)(leβπΎ)π)) |
| 10 | 3, 7, 7, 7, 9 | syl13anc 1369 | . . 3 β’ ((πΎ β Lat β§ π β π΅) β ((π(leβπΎ)π β§ π(leβπΎ)π) β (π β¨ π)(leβπΎ)π)) |
| 11 | 8, 8, 10 | mpbi2and 709 | . 2 β’ ((πΎ β Lat β§ π β π΅) β (π β¨ π)(leβπΎ)π) |
| 12 | 1, 2, 4 | latlej1 18410 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π(leβπΎ)(π β¨ π)) |
| 13 | 12 | 3anidm23 1418 | . 2 β’ ((πΎ β Lat β§ π β π΅) β π(leβπΎ)(π β¨ π)) |
| 14 | 1, 2, 3, 6, 7, 11, 13 | latasymd 18407 | 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 395 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Basecbs 17150 lecple 17210 joincjn 18273 Latclat 18393 |
| 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 18257 df-poset 18275 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-lat 18394 |
| This theorem is referenced by: lubsn 18444 latjjdi 18453 latjjdir 18454 cvlsupr2 38725 hlatjidm 38751 cvrat3 38825 snatpsubN 39133 dalawlem7 39260 cdleme11 39653 cdleme23b 39733 cdlemg33a 40089 trljco 40123 doca2N 40509 djajN 40520 |
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