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Theorem latj31 16053
Description: Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)
Hypotheses
Ref Expression
latjass.b  |-  B  =  ( Base `  K
)
latjass.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latj31  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( ( Z 
.\/  Y )  .\/  X ) )

Proof of Theorem latj31
StepHypRef Expression
1 simpl 455 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
2 simpr3 1005 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
3 simpr1 1003 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
4 simpr2 1004 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
5 latjass.b . . . 4  |-  B  =  ( Base `  K
)
6 latjass.j . . . 4  |-  .\/  =  ( join `  K )
75, 6latj12 16050 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( Z  .\/  ( X  .\/  Y ) )  =  ( X  .\/  ( Z 
.\/  Y ) ) )
81, 2, 3, 4, 7syl13anc 1232 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .\/  ( X  .\/  Y ) )  =  ( X  .\/  ( Z 
.\/  Y ) ) )
95, 6latjcl 16005 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
1093adant3r3 1208 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  Y )  e.  B )
115, 6latjcom 16013 . . 3  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( Z  .\/  ( X  .\/  Y ) ) )
121, 10, 2, 11syl3anc 1230 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( Z  .\/  ( X  .\/  Y ) ) )
135, 6latjcl 16005 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .\/  Y
)  e.  B )
141, 2, 4, 13syl3anc 1230 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .\/  Y )  e.  B )
155, 6latjcom 16013 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  .\/  Y )  e.  B  /\  X  e.  B )  ->  (
( Z  .\/  Y
)  .\/  X )  =  ( X  .\/  ( Z  .\/  Y ) ) )
161, 14, 3, 15syl3anc 1230 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( Z  .\/  Y
)  .\/  X )  =  ( X  .\/  ( Z  .\/  Y ) ) )
178, 12, 163eqtr4d 2453 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( ( Z 
.\/  Y )  .\/  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   Basecbs 14841   joincjn 15897   Latclat 15999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-lat 16000
This theorem is referenced by:  latjrot  16054  4noncolr3  32470  3atlem5  32504  lplnexllnN  32581  dalawlem11  32898  cdleme20bN  33329
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