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Theorem 4atlem4b 29714
Description: Lemma for 4at 29727. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)
Hypotheses
Ref Expression
4at.l  |-  .<_  =  ( le `  K )
4at.j  |-  .\/  =  ( join `  K )
4at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
4atlem4b  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( Q  .\/  (
( P  .\/  R
)  .\/  S )
) )

Proof of Theorem 4atlem4b
StepHypRef Expression
1 simpl1 960 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  K  e.  HL )
2 simpl2 961 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  P  e.  A )
3 simpl3 962 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  Q  e.  A )
4 simprl 733 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  R  e.  A )
5 simprr 734 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  S  e.  A )
6 4at.j . . . 4  |-  .\/  =  ( join `  K )
7 4at.a . . . 4  |-  A  =  ( Atoms `  K )
86, 7hlatj4 29488 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( P 
.\/  R )  .\/  ( Q  .\/  S ) ) )
91, 2, 3, 4, 5, 8syl122anc 1193 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  R )  .\/  ( Q 
.\/  S ) ) )
10 hllat 29478 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
111, 10syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  K  e.  Lat )
12 eqid 2387 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
1312, 6, 7hlatjcl 29481 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
141, 2, 4, 13syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
1512, 7atbase 29404 . . . 4  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
163, 15syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  Q  e.  ( Base `  K
) )
1712, 7atbase 29404 . . . 4  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1817ad2antll 710 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  S  e.  ( Base `  K
) )
1912, 6latj12 14452 . . 3  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  R )  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( P  .\/  R
)  .\/  ( Q  .\/  S ) )  =  ( Q  .\/  (
( P  .\/  R
)  .\/  S )
) )
2011, 14, 16, 18, 19syl13anc 1186 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  R
)  .\/  ( Q  .\/  S ) )  =  ( Q  .\/  (
( P  .\/  R
)  .\/  S )
) )
219, 20eqtrd 2419 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( Q  .\/  (
( P  .\/  R
)  .\/  S )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   Latclat 14401   Atomscatm 29378   HLchlt 29465
This theorem is referenced by:  4atlem11a  29721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-lub 14358  df-join 14360  df-lat 14402  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466
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