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Theorem olm11 29864
Description: The meet of an ortholattice element with one equals itself. (chm1i 22946 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 29851 . . . . . . 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 29839 . . . . . 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 6088 . . . 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 29831 . . . . . 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 29862 . . . . 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 5723 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) )
179, 4op1cl 29822 . . . 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 29861 . . 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 29832 . . 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 5445  (class class class)co 6072   Basecbs 13457   occoc 13525   joincjn 14389   meetcmee 14390   0.cp0 14454   1.cp1 14455   OPcops 29809   OLcol 29811
This theorem is referenced by:  olm12  29865  lhpmcvr3  30661  trljat1  30802  trljat2  30803  cdlemc1  30827  cdlemc6  30832  cdleme0cp  30850  cdleme0cq  30851  cdleme1  30863  cdleme4  30874  cdleme5  30876  cdleme8  30886  cdleme9  30889  cdleme10  30890  cdleme20c  30947  cdleme20j  30954  cdleme22e  30980  cdleme22eALTN  30981  cdleme30a  31014  cdleme35b  31086  cdleme35e  31089  cdleme42a  31107  trlcoabs2N  31358  trlcolem  31362  cdlemi1  31454  cdlemk4  31470  dia2dimlem1  31701  cdlemn10  31843  dihglbcpreN  31937
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 4692
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-oposet 29813  df-ol 29815
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