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Theorem 4atlem4d 30088
Description: Lemma for 4at 30099. 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
4atlem4d  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( S  .\/  (
( P  .\/  Q
)  .\/  R )
) )

Proof of Theorem 4atlem4d
StepHypRef Expression
1 simpl1 960 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  K  e.  HL )
2 hllat 29850 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  K  e.  Lat )
4 eqid 2408 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
5 4at.j . . . . 5  |-  .\/  =  ( join `  K )
6 4at.a . . . . 5  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 29853 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
87adantr 452 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
94, 6atbase 29776 . . . 4  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
109ad2antrl 709 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  R  e.  ( Base `  K
) )
114, 6atbase 29776 . . . 4  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1211ad2antll 710 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  S  e.  ( Base `  K
) )
134, 5latjass 14483 . . 3  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  R  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  ( R 
.\/  S ) ) )
143, 8, 10, 12, 13syl13anc 1186 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  ( R 
.\/  S ) ) )
154, 5latjcl 14438 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
163, 8, 10, 15syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
174, 5latjcom 14447 . . 3  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( (
( P  .\/  Q
)  .\/  R )  .\/  S )  =  ( S  .\/  ( ( P  .\/  Q ) 
.\/  R ) ) )
183, 16, 12, 17syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( S  .\/  (
( P  .\/  Q
)  .\/  R )
) )
1914, 18eqtr3d 2442 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( S  .\/  (
( P  .\/  Q
)  .\/  R )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   Latclat 14433   Atomscatm 29750   HLchlt 29837
This theorem is referenced by:  4atlem9  30089
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-lub 14390  df-join 14392  df-lat 14434  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838
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