MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latj13 Structured version   Unicode version

Theorem latj13 14529
Description: Swap 1sd and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)
Hypotheses
Ref Expression
latjass.b  |-  B  =  ( Base `  K
)
latjass.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latj13  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  ( Y  .\/  Z ) )  =  ( Z  .\/  ( Y 
.\/  X ) ) )

Proof of Theorem latj13
StepHypRef Expression
1 simpl 445 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
2 simpr2 965 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
3 simpr3 966 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
4 simpr1 964 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
5 latjass.b . . . 4  |-  B  =  ( Base `  K
)
6 latjass.j . . . 4  |-  .\/  =  ( join `  K )
75, 6latj32 14528 . . 3  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  X  e.  B
) )  ->  (
( Y  .\/  Z
)  .\/  X )  =  ( ( Y 
.\/  X )  .\/  Z ) )
81, 2, 3, 4, 7syl13anc 1187 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .\/  Z
)  .\/  X )  =  ( ( Y 
.\/  X )  .\/  Z ) )
95, 6latjcl 14481 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
1093adant3r1 1163 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
115, 6latjcom 14490 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Y  .\/  Z )  e.  B )  -> 
( X  .\/  ( Y  .\/  Z ) )  =  ( ( Y 
.\/  Z )  .\/  X ) )
121, 4, 10, 11syl3anc 1185 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  ( Y  .\/  Z ) )  =  ( ( Y  .\/  Z
)  .\/  X )
)
135, 6latjcl 14481 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .\/  X
)  e.  B )
141, 2, 4, 13syl3anc 1185 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  X )  e.  B )
155, 6latjcom 14490 . . 3  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  ( Y  .\/  X )  e.  B )  -> 
( Z  .\/  ( Y  .\/  X ) )  =  ( ( Y 
.\/  X )  .\/  Z ) )
161, 3, 14, 15syl3anc 1185 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .\/  ( Y  .\/  X ) )  =  ( ( Y  .\/  X
)  .\/  Z )
)
178, 12, 163eqtr4d 2480 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  ( Y  .\/  Z ) )  =  ( Z  .\/  ( Y 
.\/  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   Basecbs 13471   joincjn 14403   Latclat 14476
This theorem is referenced by:  3atlem1  30342  dalawlem3  30732  dalawlem6  30735  cdleme1  31086  cdleme11g  31124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-lub 14433  df-join 14435  df-lat 14477
  Copyright terms: Public domain W3C validator