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Theorem latabs1 14187
Description: Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 22087 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 2284 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 latabs1.m . . 3  |-  ./\  =  ( meet `  K )
41, 2, 3latmle1 14176 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
) ( le `  K ) X )
51, 3latmcl 14151 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
6 latabs1.j . . . . 5  |-  .\/  =  ( join `  K )
71, 2, 6latleeqj2 14164 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( X  ./\  Y
) ( le `  K ) X  <->  ( X  .\/  ( X  ./\  Y
) )  =  X ) )
873com23 1159 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  ./\  Y )  e.  B )  -> 
( ( X  ./\  Y ) ( le `  K ) X  <->  ( X  .\/  ( X  ./\  Y
) )  =  X ) )
95, 8syld3an3 1229 . 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 936    = wceq 1624    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Latclat 14145
This theorem is referenced by:  latdisdlem  14286  cvrexchlem  28875
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-lat 14146
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