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Theorem hlatj4 33321
Description: Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 15370 for atoms. (Contributed by NM, 9-Aug-2012.)
Hypotheses
Ref Expression
hlatjcom.j  |-  .\/  =  ( join `  K )
hlatjcom.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlatj4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( P 
.\/  R )  .\/  ( Q  .\/  S ) ) )

Proof of Theorem hlatj4
StepHypRef Expression
1 hllat 33311 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1009 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
3 simp2l 1014 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  A )
4 eqid 2451 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
5 hlatjcom.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atbase 33237 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
73, 6syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  ( Base `  K ) )
8 simp2r 1015 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  A )
94, 5atbase 33237 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
108, 9syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  ( Base `  K ) )
11 simp3l 1016 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  A )
124, 5atbase 33237 . . 3  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1311, 12syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  ( Base `  K ) )
14 simp3r 1017 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  S  e.  A )
154, 5atbase 33237 . . 3  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1614, 15syl 16 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  S  e.  ( Base `  K ) )
17 hlatjcom.j . . 3  |-  .\/  =  ( join `  K )
184, 17latj4 15370 . 2  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K ) )  /\  ( R  e.  ( Base `  K )  /\  S  e.  ( Base `  K ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  R ) 
.\/  ( Q  .\/  S ) ) )
192, 7, 10, 13, 16, 18syl122anc 1228 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( P 
.\/  R )  .\/  ( Q  .\/  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5513  (class class class)co 6187   Basecbs 14273   joincjn 15213   Latclat 15314   Atomscatm 33211   HLchlt 33298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-poset 15215  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-lat 15315  df-ats 33215  df-atl 33246  df-cvlat 33270  df-hlat 33299
This theorem is referenced by:  4atlem4b  33547  4atlem11  33556  dalem2  33608  dalem23  33643  cdleme16c  34227
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