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Theorem latj12 14488
Description: Swap 1st and 2nd members of lattice join. (chj12 22997 analog.) (Contributed by NM, 4-Jun-2012.)
Hypotheses
Ref Expression
latjass.b  |-  B  =  ( Base `  K
)
latjass.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latj12  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  ( Y  .\/  Z ) )  =  ( Y  .\/  ( X 
.\/  Z ) ) )

Proof of Theorem latj12
StepHypRef Expression
1 latjass.b . . . . 5  |-  B  =  ( Base `  K
)
2 latjass.j . . . . 5  |-  .\/  =  ( join `  K )
31, 2latjcom 14451 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  =  ( Y 
.\/  X ) )
433adant3r3 1164 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  Y )  =  ( Y  .\/  X
) )
54oveq1d 6063 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( ( Y 
.\/  X )  .\/  Z ) )
61, 2latjass 14487 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( X  .\/  ( Y  .\/  Z ) ) )
7 simpl 444 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
8 simpr2 964 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
9 simpr1 963 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
10 simpr3 965 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
111, 2latjass 14487 . . 3  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  X  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .\/  X
)  .\/  Z )  =  ( Y  .\/  ( X  .\/  Z ) ) )
127, 8, 9, 10, 11syl13anc 1186 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .\/  X
)  .\/  Z )  =  ( Y  .\/  ( X  .\/  Z ) ) )
135, 6, 123eqtr3d 2452 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  ( Y  .\/  Z ) )  =  ( Y  .\/  ( X 
.\/  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   Basecbs 13432   joincjn 14364   Latclat 14437
This theorem is referenced by:  latj31  14491  latj4  14493  4atlem4b  30094  4atlem4c  30095  dalawlem3  30367  cdleme1  30721  cdleme5  30734  cdleme11g  30759
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-lub 14394  df-join 14396  df-lat 14438
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