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Theorem hlatj12 32854
Description: Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 16330 for atoms. (Contributed by NM, 4-Jun-2012.)
Hypotheses
Ref Expression
hlatjcom.j  |-  .\/  =  ( join `  K )
hlatjcom.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlatj12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  ( Q  .\/  R ) )  =  ( Q  .\/  ( P 
.\/  R ) ) )

Proof of Theorem hlatj12
StepHypRef Expression
1 hlatjcom.j . . . . 5  |-  .\/  =  ( join `  K )
2 hlatjcom.a . . . . 5  |-  A  =  ( Atoms `  K )
31, 2hlatjcom 32851 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
433adant3r3 1216 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
54oveq1d 6316 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( Q 
.\/  P )  .\/  R ) )
61, 2hlatjass 32853 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( P  .\/  ( Q  .\/  R ) ) )
7 simpl 458 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
8 simpr2 1012 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
9 simpr1 1011 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
10 simpr3 1013 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
111, 2hlatjass 32853 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  P  e.  A  /\  R  e.  A
) )  ->  (
( Q  .\/  P
)  .\/  R )  =  ( Q  .\/  ( P  .\/  R ) ) )
127, 8, 9, 10, 11syl13anc 1266 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  .\/  P
)  .\/  R )  =  ( Q  .\/  ( P  .\/  R ) ) )
135, 6, 123eqtr3d 2471 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  ( Q  .\/  R ) )  =  ( Q  .\/  ( P 
.\/  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   ` cfv 5597  (class class class)co 6301   joincjn 16176   Atomscatm 32747   HLchlt 32834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-preset 16160  df-poset 16178  df-lub 16207  df-glb 16208  df-join 16209  df-meet 16210  df-lat 16279  df-ats 32751  df-atl 32782  df-cvlat 32806  df-hlat 32835
This theorem is referenced by:  3atlem1  32966  3atlem2  32967  dalawlem12  33365  cdleme35b  33935
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