MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  latabs1 Unicode version

Theorem latabs1 14037
Description: Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 21925 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
latabs1.b  |-  B  =  ( Base `  K
)
latabs1.j  |-  .\/  =  ( join `  K )
latabs1.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latabs1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  ( X  ./\  Y ) )  =  X )

Proof of Theorem latabs1
StepHypRef Expression
1 latabs1.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2253 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 latabs1.m . . 3  |-  ./\  =  ( meet `  K )
41, 2, 3latmle1 14026 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
) ( le `  K ) X )
51, 3latmcl 14001 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
6 latabs1.j . . . . 5  |-  .\/  =  ( join `  K )
71, 2, 6latleeqj2 14014 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( X  ./\  Y
) ( le `  K ) X  <->  ( X  .\/  ( X  ./\  Y
) )  =  X ) )
873com23 1162 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  ./\  Y )  e.  B )  -> 
( ( X  ./\  Y ) ( le `  K ) X  <->  ( X  .\/  ( X  ./\  Y
) )  =  X ) )
95, 8syld3an3 1232 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) ( le `  K ) X  <->  ( X  .\/  ( X  ./\  Y
) )  =  X ) )
104, 9mpbid 203 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  ( X  ./\  Y ) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Latclat 13995
This theorem is referenced by:  latdisdlem  14127  cvrexchlem  28512
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-lat 13996
  Copyright terms: Public domain W3C validator