Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hlatj12 Structured version   Unicode version

Theorem hlatj12 33020
Description: Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 15272 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 33017 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
433adant3r3 1198 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
54oveq1d 6111 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( Q 
.\/  P )  .\/  R ) )
61, 2hlatjass 33019 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( P  .\/  ( Q  .\/  R ) ) )
7 simpl 457 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
8 simpr2 995 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
9 simpr1 994 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
10 simpr3 996 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
111, 2hlatjass 33019 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  P  e.  A  /\  R  e.  A
) )  ->  (
( Q  .\/  P
)  .\/  R )  =  ( Q  .\/  ( P  .\/  R ) ) )
127, 8, 9, 10, 11syl13anc 1220 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  .\/  P
)  .\/  R )  =  ( Q  .\/  ( P  .\/  R ) ) )
135, 6, 123eqtr3d 2483 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   joincjn 15119   Atomscatm 32913   HLchlt 33000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-poset 15121  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-lat 15221  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001
This theorem is referenced by:  3atlem1  33132  3atlem2  33133  dalawlem12  33531  cdleme35b  34099
  Copyright terms: Public domain W3C validator