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Theorem olj01 30085
Description: An ortholattice element joined with zero equals itself. (chj0 23001 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 30074 . . . 4  |-  ( K  e.  OL  ->  K  e.  OP )
2 olj0.b . . . . 5  |-  B  =  ( Base `  K
)
3 olj0.z . . . . 5  |-  .0.  =  ( 0. `  K )
42, 3op0cl 30044 . . . 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 2438 . . 3  |-  ( le
`  K )  =  ( le `  K
)
8 ollat 30073 . . . 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 14481 . . . 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 14484 . . . . . 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 30046 . . . . . 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 14493 . . . . 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 14491 . . . 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 14488 . 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 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   0.cp0 14468   Latclat 14476   OPcops 30032   OLcol 30034
This theorem is referenced by:  olj02  30086  olm11  30087  omllaw3  30105  omlspjN  30121  2at0mat0  30384  lhp2at0nle  30894  lhple  30901  cdlemc6  31055  cdleme3c  31089  cdleme7e  31106  cdlemednpq  31158  cdlemefrs29pre00  31254  cdlemefrs29bpre0  31255  cdlemefrs29cpre1  31257  cdleme32fva  31296  cdleme42ke  31344  cdlemg12e  31506  cdlemg31d  31559  trljco  31599  cdlemkid2  31783  dihvalcqat  32099  dihmeetlem7N  32170  dihjatc1  32171  djh01  32272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-lub 14433  df-glb 14434  df-join 14435  df-p0 14470  df-lat 14477  df-oposet 30036  df-ol 30038
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