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Theorem olm11 29926
Description: The meet of an ortholattice element with one equals itself. (chm1i 22948 analog.) (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
olm11  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )

Proof of Theorem olm11
StepHypRef Expression
1 olop 29913 . . . . . . 7  |-  ( K  e.  OL  ->  K  e.  OP )
21adantr 452 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  OP )
3 eqid 2435 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 olm1.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
5 eqid 2435 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
63, 4, 5opoc1 29901 . . . . . 6  |-  ( K  e.  OP  ->  (
( oc `  K
) `  .1.  )  =  ( 0. `  K ) )
72, 6syl 16 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  .1.  )  =  ( 0. `  K ) )
87oveq2d 6089 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 .1.  ) )  =  ( ( ( oc `  K ) `
 X ) (
join `  K )
( 0. `  K
) ) )
9 olm1.b . . . . . . 7  |-  B  =  ( Base `  K
)
109, 5opoccl 29893 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
111, 10sylan 458 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
12 eqid 2435 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
139, 12, 3olj01 29924 . . . . 5  |-  ( ( K  e.  OL  /\  ( ( oc `  K ) `  X
)  e.  B )  ->  ( ( ( oc `  K ) `
 X ) (
join `  K )
( 0. `  K
) )  =  ( ( oc `  K
) `  X )
)
1411, 13syldan 457 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( 0.
`  K ) )  =  ( ( oc
`  K ) `  X ) )
158, 14eqtrd 2467 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 .1.  ) )  =  ( ( oc
`  K ) `  X ) )
1615fveq2d 5724 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) )
179, 4op1cl 29884 . . . 4  |-  ( K  e.  OP  ->  .1.  e.  B )
182, 17syl 16 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .1.  e.  B )
19 olm1.m . . . 4  |-  ./\  =  ( meet `  K )
209, 12, 19, 5oldmj4 29923 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .1.  e.  B )  -> 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( X  ./\  .1.  ) )
2118, 20mpd3an3 1280 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( X  ./\  .1.  ) )
229, 5opococ 29894 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
231, 22sylan 458 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
2416, 21, 233eqtr3d 2475 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13459   occoc 13527   joincjn 14391   meetcmee 14392   0.cp0 14456   1.cp1 14457   OPcops 29871   OLcol 29873
This theorem is referenced by:  olm12  29927  lhpmcvr3  30723  trljat1  30864  trljat2  30865  cdlemc1  30889  cdlemc6  30894  cdleme0cp  30912  cdleme0cq  30913  cdleme1  30925  cdleme4  30936  cdleme5  30938  cdleme8  30948  cdleme9  30951  cdleme10  30952  cdleme20c  31009  cdleme20j  31016  cdleme22e  31042  cdleme22eALTN  31043  cdleme30a  31076  cdleme35b  31148  cdleme35e  31151  cdleme42a  31169  trlcoabs2N  31420  trlcolem  31424  cdlemi1  31516  cdlemk4  31532  dia2dimlem1  31763  cdlemn10  31905  dihglbcpreN  31999
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-oposet 29875  df-ol 29877
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