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Theorem olm12 29711
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 29696 . . . 4  |-  ( K  e.  OL  ->  K  e.  Lat )
21adantr 452 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  Lat )
3 olop 29697 . . . . 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 29668 . . . 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 14459 . . 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 29710 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )
1412, 13eqtrd 2436 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .1.  ./\  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   meetcmee 14357   1.cp1 14422   Latclat 14429   OPcops 29655   OLcol 29657
This theorem is referenced by:  dih1  31769  dihjatc  31900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-oposet 29659  df-ol 29661
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