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Theorem latj31 15368
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 457 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
2 simpr3 996 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
3 simpr1 994 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
4 simpr2 995 . . 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 15365 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( Z  .\/  ( X  .\/  Y ) )  =  ( X  .\/  ( Z 
.\/  Y ) ) )
81, 2, 3, 4, 7syl13anc 1221 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .\/  ( X  .\/  Y ) )  =  ( X  .\/  ( Z 
.\/  Y ) ) )
95, 6latjcl 15320 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
1093adant3r3 1199 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  Y )  e.  B )
115, 6latjcom 15328 . . 3  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( Z  .\/  ( X  .\/  Y ) ) )
121, 10, 2, 11syl3anc 1219 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( Z  .\/  ( X  .\/  Y ) ) )
135, 6latjcl 15320 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .\/  Y
)  e.  B )
141, 2, 4, 13syl3anc 1219 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .\/  Y )  e.  B )
155, 6latjcom 15328 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  .\/  Y )  e.  B  /\  X  e.  B )  ->  (
( Z  .\/  Y
)  .\/  X )  =  ( X  .\/  ( Z  .\/  Y ) ) )
161, 14, 3, 15syl3anc 1219 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( Z  .\/  Y
)  .\/  X )  =  ( X  .\/  ( Z  .\/  Y ) ) )
178, 12, 163eqtr4d 2501 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5513  (class class class)co 6187   Basecbs 14273   joincjn 15213   Latclat 15314
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
This theorem is referenced by:  latjrot  15369  4noncolr3  33400  3atlem5  33434  lplnexllnN  33511  dalawlem11  33828  cdleme20bN  34257
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