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Theorem latjidm 14180
Description: Lattice join is idempotent. (chjidm 22099 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 2283 . 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 14156 . . 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 14159 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
91, 2, 4latjle12 14168 . . . 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 14166 . . 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 14163 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151
This theorem is referenced by:  lubsn  14200  latjjdi  14209  latjjdir  14210  cvlsupr2  29533  hlatjidm  29558  cvrat3  29631  snatpsubN  29939  dalawlem7  30066  cdleme11  30459  cdleme23b  30539  cdlemg33a  30895  trljco  30929  doca2N  31316  djajN  31327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-join 14110  df-lat 14152
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