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Theorem latmidm 15367
Description: Lattice join is idempotent. (inidm 3660 analog.) (Contributed by NM, 8-Nov-2011.)
Hypotheses
Ref Expression
latmidm.b  |-  B  =  ( Base `  K
)
latmidm.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latmidm  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  ./\  X
)  =  X )

Proof of Theorem latmidm
StepHypRef Expression
1 latmidm.b . 2  |-  B  =  ( Base `  K
)
2 eqid 2451 . 2  |-  ( le
`  K )  =  ( le `  K
)
3 simpl 457 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  K  e.  Lat )
4 latmidm.m . . . 4  |-  ./\  =  ( meet `  K )
51, 4latmcl 15333 . . 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, 2, 4latmle1 15357 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  X  e.  B )  ->  ( X  ./\  X
) ( le `  K ) X )
983anidm23 1278 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  ./\  X
) ( le `  K ) X )
101, 2latref 15334 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
111, 2, 4latlem12 15359 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  X  e.  B  /\  X  e.  B
) )  ->  (
( X ( le
`  K ) X  /\  X ( le
`  K ) X )  <->  X ( le `  K ) ( X 
./\  X ) ) )
123, 7, 7, 7, 11syl13anc 1221 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( ( X ( le `  K ) X  /\  X ( le `  K ) X )  <->  X ( le `  K ) ( X  ./\  X )
) )
1310, 10, 12mpbi2and 912 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) ( X 
./\  X ) )
141, 2, 3, 6, 7, 9, 13latasymd 15338 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 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   meetcmee 15226   Latclat 15326
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-lat 15327
This theorem is referenced by:  latmmdiN  33188  latmmdir  33189  2llnm3N  33522
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