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Theorem olm11 29488
Description: The meet of an ortholattice element with one equals itself. (chm1i 22348 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 29475 . . . . . . 7  |-  ( K  e.  OL  ->  K  e.  OP )
21adantr 451 . . . . . 6  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  K  e.  OP )
3 eqid 2366 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 olm1.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
5 eqid 2366 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
63, 4, 5opoc1 29463 . . . . . 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 5997 . . . 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 29455 . . . . . 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 2366 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
139, 12, 3olj01 29486 . . . . 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 2398 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 .1.  ) )  =  ( ( oc
`  K ) `  X ) )
1615fveq2d 5636 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) )
179, 4op1cl 29446 . . . 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 29485 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .1.  e.  B )  -> 
( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( X  ./\  .1.  ) )
2118, 20mpd3an3 1279 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  .1.  ) ) )  =  ( X  ./\  .1.  ) )
229, 5opococ 29456 . . 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 2406 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   ` cfv 5358  (class class class)co 5981   Basecbs 13356   occoc 13424   joincjn 14288   meetcmee 14289   0.cp0 14353   1.cp1 14354   OPcops 29433   OLcol 29435
This theorem is referenced by:  olm12  29489  lhpmcvr3  30285  trljat1  30426  trljat2  30427  cdlemc1  30451  cdlemc6  30456  cdleme0cp  30474  cdleme0cq  30475  cdleme1  30487  cdleme4  30498  cdleme5  30500  cdleme8  30510  cdleme9  30513  cdleme10  30514  cdleme20c  30571  cdleme20j  30578  cdleme22e  30604  cdleme22eALTN  30605  cdleme30a  30638  cdleme35b  30710  cdleme35e  30713  cdleme42a  30731  trlcoabs2N  30982  trlcolem  30986  cdlemi1  31078  cdlemk4  31094  dia2dimlem1  31325  cdlemn10  31467  dihglbcpreN  31561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-oposet 29437  df-ol 29439
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