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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatj32 | Structured version Visualization version GIF version |
Description: Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 18434 for atoms. (Contributed by NM, 21-Jul-2012.) |
Ref | Expression |
---|---|
hlatjcom.j | β’ β¨ = (joinβπΎ) |
hlatjcom.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlatj32 | β’ ((πΎ β HL 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 | hllat 38221 | . . 3 β’ (πΎ β HL β πΎ β Lat) | |
2 | 1 | adantr 481 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β πΎ β Lat) |
3 | simpr1 1194 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
4 | eqid 2732 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
5 | hlatjcom.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | 4, 5 | atbase 38147 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
7 | 3, 6 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
8 | simpr2 1195 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
9 | 4, 5 | atbase 38147 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
10 | 8, 9 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
11 | simpr3 1196 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β π΄) | |
12 | 4, 5 | atbase 38147 | . . 3 β’ (π β π΄ β π β (BaseβπΎ)) |
13 | 11, 12 | syl 17 | . 2 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
14 | hlatjcom.j | . . 3 β’ β¨ = (joinβπΎ) | |
15 | 4, 14 | latj32 18434 | . 2 β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ))) β ((π β¨ π) β¨ π ) = ((π β¨ π ) β¨ π)) |
16 | 2, 7, 10, 13, 15 | syl13anc 1372 | 1 β’ ((πΎ β HL 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6540 (class class class)co 7405 Basecbs 17140 joincjn 18260 Latclat 18380 Atomscatm 38121 HLchlt 38208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18244 df-poset 18262 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-lat 18381 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 |
This theorem is referenced by: hlatjrot 38231 ps-2 38337 3atlem2 38343 3atlem6 38347 4atlem3b 38457 4atlem11 38468 2lplnja 38478 dalawlem5 38734 dalawlem7 38736 cdleme9 39112 cdleme20aN 39168 cdleme22e 39203 cdleme22eALTN 39204 dia2dimlem3 39925 |
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