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Theorem latjidm 15358
Description: Lattice join is idempotent. (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 2452 . 2  |-  ( le
`  K )  =  ( le `  K
)
3 simpl 457 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  K  e.  Lat )
4 latidm.j . . . 4  |-  .\/  =  ( join `  K )
51, 4latjcl 15335 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  X  e.  B )  ->  ( X  .\/  X
)  e.  B )
653anidm23 1278 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  e.  B )
7 simpr 461 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  e.  B )
81, 2latref 15337 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
91, 2, 4latjle12 15346 . . . 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 1221 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( ( X ( le `  K ) X  /\  X ( le `  K ) X )  <->  ( X  .\/  X ) ( le
`  K ) X ) )
118, 8, 10mpbi2and 912 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
) ( le `  K ) X )
121, 2, 4latlej1 15344 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  X  e.  B )  ->  X ( le `  K ) ( X 
.\/  X ) )
13123anidm23 1278 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) ( X 
.\/  X ) )
141, 2, 3, 6, 7, 11, 13latasymd 15341 1  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   Latclat 15329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-lat 15330
This theorem is referenced by:  lubsn  15378  latjjdi  15387  latjjdir  15388  cvlsupr2  33307  hlatjidm  33332  cvrat3  33405  snatpsubN  33713  dalawlem7  33840  cdleme11  34233  cdleme23b  34313  cdlemg33a  34669  trljco  34703  doca2N  35090  djajN  35101
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