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Theorem olm11 28696
Description: The meet of an ortholattice element with one equals itself. (chm1i 22031 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 28683 . . . . . . 7  |-  ( K  e.  OL  ->  K  e.  OP )
21adantr 451 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  OP )
3 eqid 2284 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 olm1.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
5 eqid 2284 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
63, 4, 5opoc1 28671 . . . . . 6  |-  ( K  e.  OP  ->  (
( oc `  K
) `  .1.  )  =  ( 0. `  K ) )
72, 6syl 15 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  .1.  )  =  ( 0. `  K ) )
87oveq2d 5836 . . . 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 28663 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
111, 10sylan 457 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
12 eqid 2284 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
139, 12, 3olj01 28694 . . . . 5  |-  ( ( K  e.  OL  /\  ( ( oc `  K ) `  X
)  e.  B )  ->  ( ( ( oc `  K ) `
 X ) (
join `  K )
( 0. `  K
) )  =  ( ( oc `  K
) `  X )
)
1411, 13syldan 456 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( 0.
`  K ) )  =  ( ( oc
`  K ) `  X ) )
158, 14eqtrd 2316 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 .1.  ) )  =  ( ( oc
`  K ) `  X ) )
1615fveq2d 5490 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) )
179, 4op1cl 28654 . . . 4  |-  ( K  e.  OP  ->  .1.  e.  B )
182, 17syl 15 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .1.  e.  B )
19 olm1.m . . . 4  |-  ./\  =  ( meet `  K )
209, 12, 19, 5oldmj4 28693 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .1.  e.  B )  -> 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( X  ./\  .1.  ) )
2118, 20mpd3an3 1278 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( X  ./\  .1.  ) )
229, 5opococ 28664 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
231, 22sylan 457 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
2416, 21, 233eqtr3d 2324 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   ` cfv 5221  (class class class)co 5820   Basecbs 13144   occoc 13212   joincjn 14074   meetcmee 14075   0.cp0 14139   1.cp1 14140   OPcops 28641   OLcol 28643
This theorem is referenced by:  olm12  28697  lhpmcvr3  29493  trljat1  29634  trljat2  29635  cdlemc1  29659  cdlemc6  29664  cdleme0cp  29682  cdleme0cq  29683  cdleme1  29695  cdleme4  29706  cdleme5  29708  cdleme8  29718  cdleme9  29721  cdleme10  29722  cdleme20c  29779  cdleme20j  29786  cdleme22e  29812  cdleme22eALTN  29813  cdleme30a  29846  cdleme35b  29918  cdleme35e  29921  cdleme42a  29939  trlcoabs2N  30190  trlcolem  30194  cdlemi1  30286  cdlemk4  30302  dia2dimlem1  30533  cdlemn10  30675  dihglbcpreN  30769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-oposet 28645  df-ol 28647
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