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| Mirrors > Home > MPE Home > Th. List > latj12 | Structured version Visualization version GIF version | ||
| Description: Swap 1st and 2nd members of lattice join. (chj12 31052 analog.) (Contributed by NM, 4-Jun-2012.) |
| Ref | Expression |
|---|---|
| latjass.b | β’ π΅ = (BaseβπΎ) |
| latjass.j | β’ β¨ = (joinβπΎ) |
| Ref | Expression |
|---|---|
| latj12 | 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))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latjass.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 2 | latjass.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
| 3 | 1, 2 | latjcom 18406 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
| 4 | 3 | 3adant3r3 1182 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β¨ π) = (π β¨ π)) |
| 5 | 4 | oveq1d 7428 | . 2 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)) |
| 6 | 1, 2 | latjass 18442 | . 2 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))) |
| 7 | simpl 481 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β Lat) | |
| 8 | simpr2 1193 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 9 | simpr1 1192 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 10 | simpr3 1194 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 11 | 1, 2 | latjass 18442 | . . 3 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))) |
| 12 | 7, 8, 9, 10, 11 | syl13anc 1370 | . 2 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))) |
| 13 | 5, 6, 12 | 3eqtr3d 2778 | 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))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 βcfv 6544 (class class class)co 7413 Basecbs 17150 joincjn 18270 Latclat 18390 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 |
| 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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-proset 18254 df-poset 18272 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-lat 18391 |
| This theorem is referenced by: latj31 18446 latj4 18448 4atlem4b 38776 4atlem4c 38777 dalawlem3 39049 cdleme1 39403 cdleme5 39416 cdleme11g 39441 |
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