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Theorem omllaw5N 34674
Description: The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 26396 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 995 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OML )
2 simp2 996 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
3 omllat 34669 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
4 omllaw5.b . . . . 5  |-  B  =  ( Base `  K
)
5 omllaw5.j . . . . 5  |-  .\/  =  ( join `  K )
64, 5latjcl 15550 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
73, 6syl3an1 1260 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
81, 2, 73jca 1175 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( K  e.  OML  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B ) )
9 eqid 2441 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
104, 9, 5latlej1 15559 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X ( le `  K ) ( X 
.\/  Y ) )
113, 10syl3an1 1260 . 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 34671 . 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 972    = wceq 1381    e. wcel 1802   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   lecple 14576   occoc 14577   joincjn 15442   meetcmee 15443   Latclat 15544   OMLcoml 34602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-lat 15545  df-oposet 34603  df-ol 34605  df-oml 34606
This theorem is referenced by: (None)
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