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Theorem isoml 29721
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 5687 . . . 4  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
2 isoml.b . . . 4  |-  B  =  ( Base `  K
)
31, 2syl6eqr 2454 . . 3  |-  ( k  =  K  ->  ( Base `  k )  =  B )
4 fveq2 5687 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
5 isoml.l . . . . . . 7  |-  .<_  =  ( le `  K )
64, 5syl6eqr 2454 . . . . . 6  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
76breqd 4183 . . . . 5  |-  ( k  =  K  ->  (
x ( le `  k ) y  <->  x  .<_  y ) )
8 fveq2 5687 . . . . . . . 8  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
9 isoml.j . . . . . . . 8  |-  .\/  =  ( join `  K )
108, 9syl6eqr 2454 . . . . . . 7  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
11 eqidd 2405 . . . . . . 7  |-  ( k  =  K  ->  x  =  x )
12 fveq2 5687 . . . . . . . . 9  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
13 isoml.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
1412, 13syl6eqr 2454 . . . . . . . 8  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
15 eqidd 2405 . . . . . . . 8  |-  ( k  =  K  ->  y  =  y )
16 fveq2 5687 . . . . . . . . . 10  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
17 isoml.o . . . . . . . . . 10  |-  ._|_  =  ( oc `  K )
1816, 17syl6eqr 2454 . . . . . . . . 9  |-  ( k  =  K  ->  ( oc `  k )  = 
._|_  )
1918fveq1d 5689 . . . . . . . 8  |-  ( k  =  K  ->  (
( oc `  k
) `  x )  =  (  ._|_  `  x
) )
2014, 15, 19oveq123d 6061 . . . . . . 7  |-  ( k  =  K  ->  (
y ( meet `  k
) ( ( oc
`  k ) `  x ) )  =  ( y  ./\  (  ._|_  `  x ) ) )
2110, 11, 20oveq123d 6061 . . . . . 6  |-  ( k  =  K  ->  (
x ( join `  k
) ( y (
meet `  k )
( ( oc `  k ) `  x
) ) )  =  ( x  .\/  (
y  ./\  (  ._|_  `  x ) ) ) )
2221eqeq2d 2415 . . . . 5  |-  ( k  =  K  ->  (
y  =  ( x ( join `  k
) ( y (
meet `  k )
( ( oc `  k ) `  x
) ) )  <->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) )
237, 22imbi12d 312 . . . 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 2876 . . 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 2876 . 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 29662 . 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 3054 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   occoc 13492   joincjn 14356   meetcmee 14357   OLcol 29657   OMLcoml 29658
This theorem is referenced by:  isomliN  29722  omlol  29723  omllaw  29726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-oml 29662
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