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Theorem olj01 30121
Description: An ortholattice element joined with zero equals itself. (chj0 23030 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 30110 . . . 4  |-  ( K  e.  OL  ->  K  e.  OP )
2 olj0.b . . . . 5  |-  B  =  ( Base `  K
)
3 olj0.z . . . . 5  |-  .0.  =  ( 0. `  K )
42, 3op0cl 30080 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
51, 4syl 16 . . 3  |-  ( K  e.  OL  ->  .0.  e.  B )
65adantr 453 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  e.  B )
7 eqid 2442 . . 3  |-  ( le
`  K )  =  ( le `  K
)
8 ollat 30109 . . . 4  |-  ( K  e.  OL  ->  K  e.  Lat )
983ad2ant1 979 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  K  e.  Lat )
10 olj0.j . . . . 5  |-  .\/  =  ( join `  K )
112, 10latjcl 14510 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  e.  B )
128, 11syl3an1 1218 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  e.  B )
13 simp2 959 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X  e.  B )
142, 7latref 14513 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
158, 14sylan 459 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  X ( le `  K ) X )
16153adant3 978 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) X )
172, 7, 3op0le 30082 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
181, 17sylan 459 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
19183adant3 978 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  ( le `  K
) X )
20 simp3 960 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  e.  B )
212, 7, 10latjle12 14522 . . . . 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 1187 . . . 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 889 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  ) ( le `  K ) X )
242, 7, 10latlej1 14520 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) ( X 
.\/  .0.  ) )
258, 24syl3an1 1218 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) ( X 
.\/  .0.  ) )
262, 7, 9, 12, 13, 23, 25latasymd 14517 . 2  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  =  X )
276, 26mpd3an3 1281 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  .0.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   joincjn 14432   0.cp0 14497   Latclat 14505   OPcops 30068   OLcol 30070
This theorem is referenced by:  olj02  30122  olm11  30123  omllaw3  30141  omlspjN  30157  2at0mat0  30420  lhp2at0nle  30930  lhple  30937  cdlemc6  31091  cdleme3c  31125  cdleme7e  31142  cdlemednpq  31194  cdlemefrs29pre00  31290  cdlemefrs29bpre0  31291  cdlemefrs29cpre1  31293  cdleme32fva  31332  cdleme42ke  31380  cdlemg12e  31542  cdlemg31d  31595  trljco  31635  cdlemkid2  31819  dihvalcqat  32135  dihmeetlem7N  32206  dihjatc1  32207  djh01  32308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-lub 14462  df-glb 14463  df-join 14464  df-p0 14499  df-lat 14506  df-oposet 30072  df-ol 30074
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