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| Mirrors > Home > MPE Home > Th. List > latjjdir | Structured version Visualization version GIF version | ||
| Description: Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.) |
| Ref | Expression |
|---|---|
| latjass.b | β’ π΅ = (BaseβπΎ) |
| latjass.j | β’ β¨ = (joinβπΎ) |
| Ref | Expression |
|---|---|
| latjjdir | β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β¨ π) β¨ π) = ((π β¨ π) β¨ (π β¨ π))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latjass.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 2 | latjass.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
| 3 | 1, 2 | latjidm 18420 | . . . 4 β’ ((πΎ β Lat β§ π β π΅) β (π β¨ π) = π) |
| 4 | 3 | 3ad2antr3 1189 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β¨ π) = π) |
| 5 | 4 | oveq2d 7428 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β¨ π) β¨ (π β¨ π)) = ((π β¨ π) β¨ π)) |
| 6 | simpl 482 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Lat) | |
| 7 | simpr1 1193 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 8 | simpr2 1194 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 9 | simpr3 1195 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 10 | 1, 2 | latj4 18447 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅) β§ (π β π΅ β§ π β π΅)) β ((π β¨ π) β¨ (π β¨ π)) = ((π β¨ π) β¨ (π β¨ π))) |
| 11 | 6, 7, 8, 9, 9, 10 | syl122anc 1378 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β¨ π) β¨ (π β¨ π)) = ((π β¨ π) β¨ (π β¨ π))) |
| 12 | 5, 11 | eqtr3d 2773 | 1 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β¨ π) β¨ π) = ((π β¨ π) β¨ (π β¨ π))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6543 (class class class)co 7412 Basecbs 17149 joincjn 18269 Latclat 18389 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-proset 18253 df-poset 18271 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-lat 18390 |
| This theorem is referenced by: dalem38 38885 cdleme23b 39525 |
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