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Theorem latjidm 14503
Description: Lattice join is idempotent. (chjidm 23022 analog.) (Contributed by NM, 8-Oct-2011.)
Hypotheses
Ref Expression
latidm.b  |-  B  =  ( Base `  K
)
latidm.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latjidm  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  =  X )

Proof of Theorem latjidm
StepHypRef Expression
1 latidm.b . 2  |-  B  =  ( Base `  K
)
2 eqid 2436 . 2  |-  ( le
`  K )  =  ( le `  K
)
3 simpl 444 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  K  e.  Lat )
4 latidm.j . . . 4  |-  .\/  =  ( join `  K )
51, 4latjcl 14479 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  X  e.  B )  ->  ( X  .\/  X
)  e.  B )
653anidm23 1243 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  e.  B )
7 simpr 448 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  e.  B )
81, 2latref 14482 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
91, 2, 4latjle12 14491 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  X  e.  B  /\  X  e.  B
) )  ->  (
( X ( le
`  K ) X  /\  X ( le
`  K ) X )  <->  ( X  .\/  X ) ( le `  K ) X ) )
103, 7, 7, 7, 9syl13anc 1186 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( ( X ( le `  K ) X  /\  X ( le `  K ) X )  <->  ( X  .\/  X ) ( le
`  K ) X ) )
118, 8, 10mpbi2and 888 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
) ( le `  K ) X )
121, 2, 4latlej1 14489 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  X  e.  B )  ->  X ( le `  K ) ( X 
.\/  X ) )
13123anidm23 1243 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) ( X 
.\/  X ) )
141, 2, 3, 6, 7, 11, 13latasymd 14486 1  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   Latclat 14474
This theorem is referenced by:  lubsn  14523  latjjdi  14532  latjjdir  14533  cvlsupr2  30141  hlatjidm  30166  cvrat3  30239  snatpsubN  30547  dalawlem7  30674  cdleme11  31067  cdleme23b  31147  cdlemg33a  31503  trljco  31537  doca2N  31924  djajN  31935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-lub 14431  df-join 14433  df-lat 14475
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