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Theorem latabs1 14141
Description: Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 22041 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 2256 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 latabs1.m . . 3  |-  ./\  =  ( meet `  K )
41, 2, 3latmle1 14130 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
) ( le `  K ) X )
51, 3latmcl 14105 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
6 latabs1.j . . . . 5  |-  .\/  =  ( join `  K )
71, 2, 6latleeqj2 14118 . . . 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 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   joincjn 14026   meetcmee 14027   Latclat 14099
This theorem is referenced by:  latdisdlem  14240  cvrexchlem  28759
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 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-lat 14100
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