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Theorem olj01 30037
Description: An ortholattice element joined with zero equals itself. (chj0 22092 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 30026 . . . 4  |-  ( K  e.  OL  ->  K  e.  OP )
2 olj0.b . . . . 5  |-  B  =  ( Base `  K
)
3 olj0.z . . . . 5  |-  .0.  =  ( 0. `  K )
42, 3op0cl 29996 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
51, 4syl 15 . . 3  |-  ( K  e.  OL  ->  .0.  e.  B )
65adantr 451 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  e.  B )
7 eqid 2296 . . 3  |-  ( le
`  K )  =  ( le `  K
)
8 ollat 30025 . . . 4  |-  ( K  e.  OL  ->  K  e.  Lat )
983ad2ant1 976 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  K  e.  Lat )
10 olj0.j . . . . 5  |-  .\/  =  ( join `  K )
112, 10latjcl 14172 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  e.  B )
128, 11syl3an1 1215 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  e.  B )
13 simp2 956 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X  e.  B )
142, 7latref 14175 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
158, 14sylan 457 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  X ( le `  K ) X )
16153adant3 975 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) X )
172, 7, 3op0le 29998 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
181, 17sylan 457 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
19183adant3 975 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  ( le `  K
) X )
20 simp3 957 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  e.  B )
212, 7, 10latjle12 14184 . . . . 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 1184 . . . 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 887 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  ) ( le `  K ) X )
242, 7, 10latlej1 14182 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) ( X 
.\/  .0.  ) )
258, 24syl3an1 1215 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) ( X 
.\/  .0.  ) )
262, 7, 9, 12, 13, 23, 25latasymd 14179 . 2  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  =  X )
276, 26mpd3an3 1278 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  .0.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   0.cp0 14159   Latclat 14167   OPcops 29984   OLcol 29986
This theorem is referenced by:  olj02  30038  olm11  30039  omllaw3  30057  omlspjN  30073  2at0mat0  30336  lhp2at0nle  30846  lhple  30853  cdlemc6  31007  cdleme3c  31041  cdleme7e  31058  cdlemednpq  31110  cdlemefrs29pre00  31206  cdlemefrs29bpre0  31207  cdlemefrs29cpre1  31209  cdleme32fva  31248  cdleme42ke  31296  cdlemg12e  31458  cdlemg31d  31511  trljco  31551  cdlemkid2  31735  dihvalcqat  32051  dihmeetlem7N  32122  dihjatc1  32123  djh01  32224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-glb 14125  df-join 14126  df-p0 14161  df-lat 14168  df-oposet 29988  df-ol 29990
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