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Theorem hlatj4 29563
Description: Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 14207 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 29553 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
3 simp2l 981 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  A )
4 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
5 hlatjcom.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atbase 29479 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
73, 6syl 15 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  ( Base `  K ) )
8 simp2r 982 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  A )
94, 5atbase 29479 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
108, 9syl 15 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  ( Base `  K ) )
11 simp3l 983 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  A )
124, 5atbase 29479 . . 3  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1311, 12syl 15 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  ( Base `  K ) )
14 simp3r 984 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  S  e.  A )
154, 5atbase 29479 . . 3  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1614, 15syl 15 . 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 14207 . 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 1191 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  4atlem4b  29789  4atlem11  29798  dalem2  29850  dalem23  29885  cdleme16c  30469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-join 14110  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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