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Theorem olm12 30026
Description: The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
olm1.b  |-  B  =  ( Base `  K
)
olm1.m  |-  ./\  =  ( meet `  K )
olm1.u  |-  .1.  =  ( 1. `  K )
Assertion
Ref Expression
olm12  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .1.  ./\  X
)  =  X )

Proof of Theorem olm12
StepHypRef Expression
1 ollat 30011 . . . 4  |-  ( K  e.  OL  ->  K  e.  Lat )
21adantr 452 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  Lat )
3 olop 30012 . . . . 5  |-  ( K  e.  OL  ->  K  e.  OP )
43adantr 452 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  OP )
5 olm1.b . . . . 5  |-  B  =  ( Base `  K
)
6 olm1.u . . . . 5  |-  .1.  =  ( 1. `  K )
75, 6op1cl 29983 . . . 4  |-  ( K  e.  OP  ->  .1.  e.  B )
84, 7syl 16 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .1.  e.  B )
9 simpr 448 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  X  e.  B )
10 olm1.m . . . 4  |-  ./\  =  ( meet `  K )
115, 10latmcom 14504 . . 3  |-  ( ( K  e.  Lat  /\  .1.  e.  B  /\  X  e.  B )  ->  (  .1.  ./\  X )  =  ( X  ./\  .1.  ) )
122, 8, 9, 11syl3anc 1184 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .1.  ./\  X
)  =  ( X 
./\  .1.  ) )
135, 10, 6olm11 30025 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )
1412, 13eqtrd 2468 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .1.  ./\  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   meetcmee 14402   1.cp1 14467   Latclat 14474   OPcops 29970   OLcol 29972
This theorem is referenced by:  dih1  32084  dihjatc  32215
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-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-oposet 29974  df-ol 29976
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