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Theorem olm11 29344
Description: The meet of an ortholattice element with one equals itself. (chm1i 22808 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 29331 . . . . . . 7  |-  ( K  e.  OL  ->  K  e.  OP )
21adantr 452 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  OP )
3 eqid 2389 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 olm1.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
5 eqid 2389 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
63, 4, 5opoc1 29319 . . . . . 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 6038 . . . 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 29311 . . . . . 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 2389 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
139, 12, 3olj01 29342 . . . . 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 2421 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 .1.  ) )  =  ( ( oc
`  K ) `  X ) )
1615fveq2d 5674 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) )
179, 4op1cl 29302 . . . 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 29341 . . 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 29312 . . 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 2429 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   Basecbs 13398   occoc 13466   joincjn 14330   meetcmee 14331   0.cp0 14395   1.cp1 14396   OPcops 29289   OLcol 29291
This theorem is referenced by:  olm12  29345  lhpmcvr3  30141  trljat1  30282  trljat2  30283  cdlemc1  30307  cdlemc6  30312  cdleme0cp  30330  cdleme0cq  30331  cdleme1  30343  cdleme4  30354  cdleme5  30356  cdleme8  30366  cdleme9  30369  cdleme10  30370  cdleme20c  30427  cdleme20j  30434  cdleme22e  30460  cdleme22eALTN  30461  cdleme30a  30494  cdleme35b  30566  cdleme35e  30569  cdleme42a  30587  trlcoabs2N  30838  trlcolem  30842  cdlemi1  30934  cdlemk4  30950  dia2dimlem1  31181  cdlemn10  31323  dihglbcpreN  31417
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-oposet 29293  df-ol 29295
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