Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omllaw5N Unicode version

Theorem omllaw5N 28716
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 22188 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omllaw5.b  |-  B  =  ( Base `  K
)
omllaw5.j  |-  .\/  =  ( join `  K )
omllaw5.m  |-  ./\  =  ( meet `  K )
omllaw5.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
omllaw5N  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  ( X  .\/  Y
) ) )  =  ( X  .\/  Y
) )

Proof of Theorem omllaw5N
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OML )
2 simp2 956 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 omllat 28711 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
4 omllaw5.b . . . . 5  |-  B  =  ( Base `  K
)
5 omllaw5.j . . . . 5  |-  .\/  =  ( join `  K )
64, 5latjcl 14152 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
73, 6syl3an1 1215 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
81, 2, 73jca 1132 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( K  e.  OML  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B ) )
9 eqid 2284 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
104, 9, 5latlej1 14162 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
113, 10syl3an1 1215 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
12 omllaw5.m . . 3  |-  ./\  =  ( meet `  K )
13 omllaw5.o . . 3  |-  ._|_  =  ( oc `  K )
144, 9, 5, 12, 13omllaw2N 28713 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X ( le
`  K ) ( X  .\/  Y )  ->  ( X  .\/  ( (  ._|_  `  X
)  ./\  ( X  .\/  Y ) ) )  =  ( X  .\/  Y ) ) )
158, 11, 14sylc 56 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  ( X  .\/  Y
) ) )  =  ( X  .\/  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   occoc 13212   joincjn 14074   meetcmee 14075   Latclat 14147   OMLcoml 28644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-lub 14104  df-join 14106  df-meet 14107  df-lat 14148  df-oposet 28645  df-ol 28647  df-oml 28648
  Copyright terms: Public domain W3C validator