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Theorem omllaw 34446
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 34441 . . . 4  |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) ) )
76simprbi 464 . . 3  |-  ( K  e.  OML  ->  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) )
8 breq1 4456 . . . . 5  |-  ( x  =  X  ->  (
x  .<_  y  <->  X  .<_  y ) )
9 id 22 . . . . . . 7  |-  ( x  =  X  ->  x  =  X )
10 fveq2 5872 . . . . . . . 8  |-  ( x  =  X  ->  (  ._|_  `  x )  =  (  ._|_  `  X ) )
1110oveq2d 6311 . . . . . . 7  |-  ( x  =  X  ->  (
y  ./\  (  ._|_  `  x ) )  =  ( y  ./\  (  ._|_  `  X ) ) )
129, 11oveq12d 6313 . . . . . 6  |-  ( x  =  X  ->  (
x  .\/  ( y  ./\  (  ._|_  `  x
) ) )  =  ( X  .\/  (
y  ./\  (  ._|_  `  X ) ) ) )
1312eqeq2d 2481 . . . . 5  |-  ( x  =  X  ->  (
y  =  ( x 
.\/  ( y  ./\  (  ._|_  `  x )
) )  <->  y  =  ( X  .\/  ( y 
./\  (  ._|_  `  X
) ) ) ) )
148, 13imbi12d 320 . . . 4  |-  ( x  =  X  ->  (
( x  .<_  y  -> 
y  =  ( x 
.\/  ( y  ./\  (  ._|_  `  x )
) ) )  <->  ( X  .<_  y  ->  y  =  ( X  .\/  ( y 
./\  (  ._|_  `  X
) ) ) ) ) )
15 breq2 4457 . . . . 5  |-  ( y  =  Y  ->  ( X  .<_  y  <->  X  .<_  Y ) )
16 id 22 . . . . . 6  |-  ( y  =  Y  ->  y  =  Y )
17 oveq1 6302 . . . . . . 7  |-  ( y  =  Y  ->  (
y  ./\  (  ._|_  `  X ) )  =  ( Y  ./\  (  ._|_  `  X ) ) )
1817oveq2d 6311 . . . . . 6  |-  ( y  =  Y  ->  ( X  .\/  ( y  ./\  (  ._|_  `  X )
) )  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) )
1916, 18eqeq12d 2489 . . . . 5  |-  ( y  =  Y  ->  (
y  =  ( X 
.\/  ( y  ./\  (  ._|_  `  X )
) )  <->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) )
2015, 19imbi12d 320 . . . 4  |-  ( y  =  Y  ->  (
( X  .<_  y  -> 
y  =  ( X 
.\/  ( y  ./\  (  ._|_  `  X )
) ) )  <->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) ) )
2114, 20rspc2v 3228 . . 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 30 . 2  |-  ( K  e.  OML  ->  (
( X  e.  B  /\  Y  e.  B
)  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y 
./\  (  ._|_  `  X
) ) ) ) ) )
23223impib 1194 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14506   lecple 14578   occoc 14579   joincjn 15447   meetcmee 15448   OLcol 34377   OMLcoml 34378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-oml 34382
This theorem is referenced by:  omllaw2N  34447  omllaw3  34448  omllaw4  34449
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