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Theorem isoml 30037
Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isoml.b  |-  B  =  ( Base `  K
)
isoml.l  |-  .<_  =  ( le `  K )
isoml.j  |-  .\/  =  ( join `  K )
isoml.m  |-  ./\  =  ( meet `  K )
isoml.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
isoml  |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) ) )
Distinct variable groups:    x, y, B    x, K, y
Allowed substitution hints:    .\/ ( x, y)    .<_ ( x, y)    ./\ ( x, y)    ._|_ ( x, y)

Proof of Theorem isoml
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fveq2 5729 . . . 4  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
2 isoml.b . . . 4  |-  B  =  ( Base `  K
)
31, 2syl6eqr 2487 . . 3  |-  ( k  =  K  ->  ( Base `  k )  =  B )
4 fveq2 5729 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
5 isoml.l . . . . . . 7  |-  .<_  =  ( le `  K )
64, 5syl6eqr 2487 . . . . . 6  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
76breqd 4224 . . . . 5  |-  ( k  =  K  ->  (
x ( le `  k ) y  <->  x  .<_  y ) )
8 fveq2 5729 . . . . . . . 8  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
9 isoml.j . . . . . . . 8  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2487 . . . . . . 7  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
11 eqidd 2438 . . . . . . 7  |-  ( k  =  K  ->  x  =  x )
12 fveq2 5729 . . . . . . . . 9  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
13 isoml.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
1412, 13syl6eqr 2487 . . . . . . . 8  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
15 eqidd 2438 . . . . . . . 8  |-  ( k  =  K  ->  y  =  y )
16 fveq2 5729 . . . . . . . . . 10  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
17 isoml.o . . . . . . . . . 10  |-  ._|_  =  ( oc `  K )
1816, 17syl6eqr 2487 . . . . . . . . 9  |-  ( k  =  K  ->  ( oc `  k )  = 
._|_  )
1918fveq1d 5731 . . . . . . . 8  |-  ( k  =  K  ->  (
( oc `  k
) `  x )  =  (  ._|_  `  x
) )
2014, 15, 19oveq123d 6103 . . . . . . 7  |-  ( k  =  K  ->  (
y ( meet `  k
) ( ( oc
`  k ) `  x ) )  =  ( y  ./\  (  ._|_  `  x ) ) )
2110, 11, 20oveq123d 6103 . . . . . 6  |-  ( k  =  K  ->  (
x ( join `  k
) ( y (
meet `  k )
( ( oc `  k ) `  x
) ) )  =  ( x  .\/  (
y  ./\  (  ._|_  `  x ) ) ) )
2221eqeq2d 2448 . . . . 5  |-  ( k  =  K  ->  (
y  =  ( x ( join `  k
) ( y (
meet `  k )
( ( oc `  k ) `  x
) ) )  <->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) )
237, 22imbi12d 313 . . . 4  |-  ( k  =  K  ->  (
( x ( le
`  k ) y  ->  y  =  ( x ( join `  k
) ( y (
meet `  k )
( ( oc `  k ) `  x
) ) ) )  <-> 
( x  .<_  y  -> 
y  =  ( x 
.\/  ( y  ./\  (  ._|_  `  x )
) ) ) ) )
243, 23raleqbidv 2917 . . 3  |-  ( k  =  K  ->  ( A. y  e.  ( Base `  k ) ( x ( le `  k ) y  -> 
y  =  ( x ( join `  k
) ( y (
meet `  k )
( ( oc `  k ) `  x
) ) ) )  <->  A. y  e.  B  ( x  .<_  y  -> 
y  =  ( x 
.\/  ( y  ./\  (  ._|_  `  x )
) ) ) ) )
253, 24raleqbidv 2917 . 2  |-  ( k  =  K  ->  ( A. x  e.  ( Base `  k ) A. y  e.  ( Base `  k ) ( x ( le `  k
) y  ->  y  =  ( x (
join `  k )
( y ( meet `  k ) ( ( oc `  k ) `
 x ) ) ) )  <->  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) ) )
26 df-oml 29978 . 2  |-  OML  =  { k  e.  OL  |  A. x  e.  (
Base `  k ) A. y  e.  ( Base `  k ) ( x ( le `  k ) y  -> 
y  =  ( x ( join `  k
) ( y (
meet `  k )
( ( oc `  k ) `  x
) ) ) ) }
2725, 26elrab2 3095 1  |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   occoc 13538   joincjn 14402   meetcmee 14403   OLcol 29973   OMLcoml 29974
This theorem is referenced by:  isomliN  30038  omlol  30039  omllaw  30042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085  df-oml 29978
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