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Theorem omllaw 29433
Description: The orthomodular law. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
omllaw.b  |-  B  =  ( Base `  K
)
omllaw.l  |-  .<_  =  ( le `  K )
omllaw.j  |-  .\/  =  ( join `  K )
omllaw.m  |-  ./\  =  ( meet `  K )
omllaw.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
omllaw  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )

Proof of Theorem omllaw
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omllaw.b . . . . 5  |-  B  =  ( Base `  K
)
2 omllaw.l . . . . 5  |-  .<_  =  ( le `  K )
3 omllaw.j . . . . 5  |-  .\/  =  ( join `  K )
4 omllaw.m . . . . 5  |-  ./\  =  ( meet `  K )
5 omllaw.o . . . . 5  |-  ._|_  =  ( oc `  K )
61, 2, 3, 4, 5isoml 29428 . . . 4  |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) ) )
76simprbi 450 . . 3  |-  ( K  e.  OML  ->  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) )
8 breq1 4026 . . . . 5  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
9 id 19 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
10 fveq2 5525 . . . . . . . 8  |-  ( x  =  X  ->  (  ._|_  `  x )  =  (  ._|_  `  X ) )
1110oveq2d 5874 . . . . . . 7  |-  ( x  =  X  ->  (
y  ./\  (  ._|_  `  x ) )  =  ( y  ./\  (  ._|_  `  X ) ) )
129, 11oveq12d 5876 . . . . . 6  |-  ( x  =  X  ->  (
x  .\/  ( y  ./\  (  ._|_  `  x
) ) )  =  ( X  .\/  (
y  ./\  (  ._|_  `  X ) ) ) )
1312eqeq2d 2294 . . . . 5  |-  ( x  =  X  ->  (
y  =  ( x 
.\/  ( y  ./\  (  ._|_  `  x )
) )  <->  y  =  ( X  .\/  ( y 
./\  (  ._|_  `  X
) ) ) ) )
148, 13imbi12d 311 . . . 4  |-  ( x  =  X  ->  (
( x  .<_  y  -> 
y  =  ( x 
.\/  ( y  ./\  (  ._|_  `  x )
) ) )  <->  ( X  .<_  y  ->  y  =  ( X  .\/  ( y 
./\  (  ._|_  `  X
) ) ) ) ) )
15 breq2 4027 . . . . 5  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
16 id 19 . . . . . 6  |-  ( y  =  Y  ->  y  =  Y )
17 oveq1 5865 . . . . . . 7  |-  ( y  =  Y  ->  (
y  ./\  (  ._|_  `  X ) )  =  ( Y  ./\  (  ._|_  `  X ) ) )
1817oveq2d 5874 . . . . . 6  |-  ( y  =  Y  ->  ( X  .\/  ( y  ./\  (  ._|_  `  X )
) )  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) )
1916, 18eqeq12d 2297 . . . . 5  |-  ( y  =  Y  ->  (
y  =  ( X 
.\/  ( y  ./\  (  ._|_  `  X )
) )  <->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) )
2015, 19imbi12d 311 . . . 4  |-  ( y  =  Y  ->  (
( X  .<_  y  -> 
y  =  ( X 
.\/  ( y  ./\  (  ._|_  `  X )
) ) )  <->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) ) )
2114, 20rspc2v 2890 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) ) )
227, 21syl5com 26 . 2  |-  ( K  e.  OML  ->  (
( X  e.  B  /\  Y  e.  B
)  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) ) )
23223impib 1149 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   OLcol 29364   OMLcoml 29365
This theorem is referenced by:  omllaw2N  29434  omllaw3  29435  omllaw4  29436
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 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-oml 29369
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