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Theorem latj31 15575
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 999 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
3 simpr1 997 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
4 simpr2 998 . . 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 15572 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( Z  .\/  ( X  .\/  Y ) )  =  ( X  .\/  ( Z 
.\/  Y ) ) )
81, 2, 3, 4, 7syl13anc 1225 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .\/  ( X  .\/  Y ) )  =  ( X  .\/  ( Z 
.\/  Y ) ) )
95, 6latjcl 15527 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
1093adant3r3 1202 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .\/  Y )  e.  B )
115, 6latjcom 15535 . . 3  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( Z  .\/  ( X  .\/  Y ) ) )
121, 10, 2, 11syl3anc 1223 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .\/  Y
)  .\/  Z )  =  ( Z  .\/  ( X  .\/  Y ) ) )
135, 6latjcl 15527 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  .\/  Y
)  e.  B )
141, 2, 4, 13syl3anc 1223 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Z  .\/  Y )  e.  B )
155, 6latjcom 15535 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  .\/  Y )  e.  B  /\  X  e.  B )  ->  (
( Z  .\/  Y
)  .\/  X )  =  ( X  .\/  ( Z  .\/  Y ) ) )
161, 14, 3, 15syl3anc 1223 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( Z  .\/  Y
)  .\/  X )  =  ( X  .\/  ( Z  .\/  Y ) ) )
178, 12, 163eqtr4d 2511 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 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   joincjn 15420   Latclat 15521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-lat 15522
This theorem is referenced by:  latjrot  15576  4noncolr3  34124  3atlem5  34158  lplnexllnN  34235  dalawlem11  34552  cdleme20bN  34981
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