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Theorem olj01 29924
Description: An ortholattice element joined with zero equals itself. (chj0 22989 analog.) (Contributed by NM, 19-Oct-2011.)
Hypotheses
Ref Expression
olj0.b  |-  B  =  ( Base `  K
)
olj0.j  |-  .\/  =  ( join `  K )
olj0.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
olj01  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  .0.  )  =  X )

Proof of Theorem olj01
StepHypRef Expression
1 olop 29913 . . . 4  |-  ( K  e.  OL  ->  K  e.  OP )
2 olj0.b . . . . 5  |-  B  =  ( Base `  K
)
3 olj0.z . . . . 5  |-  .0.  =  ( 0. `  K )
42, 3op0cl 29883 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
51, 4syl 16 . . 3  |-  ( K  e.  OL  ->  .0.  e.  B )
65adantr 452 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  e.  B )
7 eqid 2435 . . 3  |-  ( le
`  K )  =  ( le `  K
)
8 ollat 29912 . . . 4  |-  ( K  e.  OL  ->  K  e.  Lat )
983ad2ant1 978 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  K  e.  Lat )
10 olj0.j . . . . 5  |-  .\/  =  ( join `  K )
112, 10latjcl 14469 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  e.  B )
128, 11syl3an1 1217 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  e.  B )
13 simp2 958 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X  e.  B )
142, 7latref 14472 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
158, 14sylan 458 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  X ( le `  K ) X )
16153adant3 977 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) X )
172, 7, 3op0le 29885 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
181, 17sylan 458 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
19183adant3 977 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  ( le `  K
) X )
20 simp3 959 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  e.  B )
212, 7, 10latjle12 14481 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( X ( le `  K ) X  /\  .0.  ( le `  K
) X )  <->  ( X  .\/  .0.  ) ( le
`  K ) X ) )
229, 13, 20, 13, 21syl13anc 1186 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( ( X ( le `  K ) X  /\  .0.  ( le `  K ) X )  <->  ( X  .\/  .0.  ) ( le `  K ) X ) )
2316, 19, 22mpbi2and 888 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  ) ( le `  K ) X )
242, 7, 10latlej1 14479 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) ( X 
.\/  .0.  ) )
258, 24syl3an1 1217 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) ( X 
.\/  .0.  ) )
262, 7, 9, 12, 13, 23, 25latasymd 14476 . 2  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  =  X )
276, 26mpd3an3 1280 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  .0.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lecple 13526   joincjn 14391   0.cp0 14456   Latclat 14464   OPcops 29871   OLcol 29873
This theorem is referenced by:  olj02  29925  olm11  29926  omllaw3  29944  omlspjN  29960  2at0mat0  30223  lhp2at0nle  30733  lhple  30740  cdlemc6  30894  cdleme3c  30928  cdleme7e  30945  cdlemednpq  30997  cdlemefrs29pre00  31093  cdlemefrs29bpre0  31094  cdlemefrs29cpre1  31096  cdleme32fva  31135  cdleme42ke  31183  cdlemg12e  31345  cdlemg31d  31398  trljco  31438  cdlemkid2  31622  dihvalcqat  31938  dihmeetlem7N  32009  dihjatc1  32010  djh01  32111
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-p0 14458  df-lat 14465  df-oposet 29875  df-ol 29877
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