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Theorem hlatj32 34461
Description: Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 15596 for atoms. (Contributed by NM, 21-Jul-2012.)
Hypotheses
Ref Expression
hlatjcom.j  |-  .\/  =  ( join `  K )
hlatjcom.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlatj32  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( P 
.\/  R )  .\/  Q ) )

Proof of Theorem hlatj32
StepHypRef Expression
1 hllat 34453 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
21adantr 465 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
3 simpr1 1002 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
4 eqid 2467 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
5 hlatjcom.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atbase 34379 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
73, 6syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  ( Base `  K
) )
8 simpr2 1003 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
94, 5atbase 34379 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
108, 9syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  ( Base `  K
) )
11 simpr3 1004 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
124, 5atbase 34379 . . 3  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1311, 12syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  ( Base `  K
) )
14 hlatjcom.j . . 3  |-  .\/  =  ( join `  K )
154, 14latj32 15596 . 2  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) ) )  -> 
( ( P  .\/  Q )  .\/  R )  =  ( ( P 
.\/  R )  .\/  Q ) )
162, 7, 10, 13, 15syl13anc 1230 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( P 
.\/  R )  .\/  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5593  (class class class)co 6294   Basecbs 14502   joincjn 15443   Latclat 15544   Atomscatm 34353   HLchlt 34440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-poset 15445  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-lat 15545  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441
This theorem is referenced by:  hlatjrot  34462  ps-2  34567  3atlem2  34573  3atlem6  34577  4atlem3b  34687  4atlem11  34698  2lplnja  34708  dalawlem5  34964  dalawlem7  34966  cdleme9  35342  cdleme20aN  35398  cdleme22e  35433  cdleme22eALTN  35434  dia2dimlem3  36156
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