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Theorem latjidm 14196
Description: Lattice join is idempotent. (chjidm 22115 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 2296 . 2  |-  ( le
`  K )  =  ( le `  K
)
3 simpl 443 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  K  e.  Lat )
4 latidm.j . . . 4  |-  .\/  =  ( join `  K )
51, 4latjcl 14172 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  X  e.  B )  ->  ( X  .\/  X
)  e.  B )
653anidm23 1241 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  e.  B )
7 simpr 447 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  e.  B )
81, 2latref 14175 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
91, 2, 4latjle12 14184 . . . 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 1184 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( ( X ( le `  K ) X  /\  X ( le `  K ) X )  <->  ( X  .\/  X ) ( le
`  K ) X ) )
118, 8, 10mpbi2and 887 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
) ( le `  K ) X )
121, 2, 4latlej1 14182 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  X  e.  B )  ->  X ( le `  K ) ( X 
.\/  X ) )
13123anidm23 1241 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) ( X 
.\/  X ) )
141, 2, 3, 6, 7, 11, 13latasymd 14179 1  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167
This theorem is referenced by:  lubsn  14216  latjjdi  14225  latjjdir  14226  cvlsupr2  30155  hlatjidm  30180  cvrat3  30253  snatpsubN  30561  dalawlem7  30688  cdleme11  31081  cdleme23b  31161  cdlemg33a  31517  trljco  31551  doca2N  31938  djajN  31949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-join 14126  df-lat 14168
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