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Theorem olj01 33228
Description: An ortholattice element joined with zero equals itself. (chj0 25072 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 33217 . . . 4  |-  ( K  e.  OL  ->  K  e.  OP )
2 olj0.b . . . . 5  |-  B  =  ( Base `  K
)
3 olj0.z . . . . 5  |-  .0.  =  ( 0. `  K )
42, 3op0cl 33187 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
51, 4syl 16 . . 3  |-  ( K  e.  OL  ->  .0.  e.  B )
65adantr 465 . 2  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  e.  B )
7 eqid 2454 . . 3  |-  ( le
`  K )  =  ( le `  K
)
8 ollat 33216 . . . 4  |-  ( K  e.  OL  ->  K  e.  Lat )
983ad2ant1 1009 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  K  e.  Lat )
10 olj0.j . . . . 5  |-  .\/  =  ( join `  K )
112, 10latjcl 15343 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  e.  B )
128, 11syl3an1 1252 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  e.  B )
13 simp2 989 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X  e.  B )
142, 7latref 15345 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X ( le `  K ) X )
158, 14sylan 471 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  X ( le `  K ) X )
16153adant3 1008 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) X )
172, 7, 3op0le 33189 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
181, 17sylan 471 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
19183adant3 1008 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  ( le `  K
) X )
20 simp3 990 . . . . 5  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  e.  B )
212, 7, 10latjle12 15354 . . . . 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 1221 . . . 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 912 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  ) ( le `  K ) X )
242, 7, 10latlej1 15352 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) ( X 
.\/  .0.  ) )
258, 24syl3an1 1252 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  ->  X ( le `  K ) ( X 
.\/  .0.  ) )
262, 7, 9, 12, 13, 23, 25latasymd 15349 . 2  |-  ( ( K  e.  OL  /\  X  e.  B  /\  .0.  e.  B )  -> 
( X  .\/  .0.  )  =  X )
276, 26mpd3an3 1316 1  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  .0.  )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   0.cp0 15329   Latclat 15337   OPcops 33175   OLcol 33177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-oposet 33179  df-ol 33181
This theorem is referenced by:  olj02  33229  olm11  33230  omllaw3  33248  omlspjN  33264  2at0mat0  33527  lhp2at0nle  34037  lhple  34044  cdlemc6  34198  cdleme3c  34232  cdleme7e  34249  cdlemednpq  34301  cdlemefrs29pre00  34397  cdlemefrs29bpre0  34398  cdlemefrs29cpre1  34400  cdleme32fva  34439  cdleme42ke  34487  cdlemg12e  34649  cdlemg31d  34702  trljco  34742  cdlemkid2  34926  dihvalcqat  35242  dihmeetlem7N  35313  dihjatc1  35314  djh01  35415
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