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Theorem latj4 14485
Description: Rearrangement of lattice join of 4 classes. (chj4 22990 analog.) (Contributed by NM, 14-Jun-2012.)
Hypotheses
Ref Expression
latjass.b  |-  B  =  ( Base `  K
)
latjass.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latj4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .\/  Y )  .\/  ( Z 
.\/  W ) )  =  ( ( X 
.\/  Z )  .\/  ( Y  .\/  W ) ) )

Proof of Theorem latj4
StepHypRef Expression
1 simp1 957 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  K  e.  Lat )
2 simp2r 984 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Y  e.  B )
3 simp3l 985 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Z  e.  B )
4 simp3r 986 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  W  e.  B )
5 latjass.b . . . . 5  |-  B  =  ( Base `  K
)
6 latjass.j . . . . 5  |-  .\/  =  ( join `  K )
75, 6latj12 14480 . . . 4  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  W  e.  B
) )  ->  ( Y  .\/  ( Z  .\/  W ) )  =  ( Z  .\/  ( Y 
.\/  W ) ) )
81, 2, 3, 4, 7syl13anc 1186 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  ( Z  .\/  W ) )  =  ( Z  .\/  ( Y  .\/  W ) ) )
98oveq2d 6056 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .\/  ( Y  .\/  ( Z  .\/  W ) ) )  =  ( X  .\/  ( Z  .\/  ( Y  .\/  W ) ) ) )
10 simp2l 983 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  X  e.  B )
115, 6latjcl 14434 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  W  e.  B )  ->  ( Z  .\/  W
)  e.  B )
121, 3, 4, 11syl3anc 1184 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Z  .\/  W
)  e.  B )
135, 6latjass 14479 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  ( Z  .\/  W
)  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  ( Z  .\/  W ) )  =  ( X 
.\/  ( Y  .\/  ( Z  .\/  W ) ) ) )
141, 10, 2, 12, 13syl13anc 1186 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .\/  Y )  .\/  ( Z 
.\/  W ) )  =  ( X  .\/  ( Y  .\/  ( Z 
.\/  W ) ) ) )
155, 6latjcl 14434 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .\/  W
)  e.  B )
161, 2, 4, 15syl3anc 1184 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  W
)  e.  B )
175, 6latjass 14479 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .\/  W
)  e.  B ) )  ->  ( ( X  .\/  Z )  .\/  ( Y  .\/  W ) )  =  ( X 
.\/  ( Z  .\/  ( Y  .\/  W ) ) ) )
181, 10, 3, 16, 17syl13anc 1186 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .\/  Z )  .\/  ( Y 
.\/  W ) )  =  ( X  .\/  ( Z  .\/  ( Y 
.\/  W ) ) ) )
199, 14, 183eqtr4d 2446 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .\/  Y )  .\/  ( Z 
.\/  W ) )  =  ( ( X 
.\/  Z )  .\/  ( Y  .\/  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   joincjn 14356   Latclat 14429
This theorem is referenced by:  latj4rot  14486  latjjdi  14487  latjjdir  14488  hlatj4  29856  arglem1N  30672  cdleme11  30752  cdleme20l2  30803
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-join 14388  df-lat 14430
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