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Theorem omllaw5N 33211
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 25163 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 988 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OML )
2 simp2 989 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 omllat 33206 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
4 omllaw5.b . . . . 5  |-  B  =  ( Base `  K
)
5 omllaw5.j . . . . 5  |-  .\/  =  ( join `  K )
64, 5latjcl 15335 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
73, 6syl3an1 1252 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
81, 2, 73jca 1168 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( K  e.  OML  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B ) )
9 eqid 2452 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
104, 9, 5latlej1 15344 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
113, 10syl3an1 1252 . 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 33208 . 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 60 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
(  ._|_  `  X )  ./\  ( X  .\/  Y
) ) )  =  ( X  .\/  Y
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   occoc 14360   joincjn 15228   meetcmee 15229   Latclat 15329   OMLcoml 33139
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-lat 15330  df-oposet 33140  df-ol 33142  df-oml 33143
This theorem is referenced by: (None)
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