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Theorem isomliN 30051
Description: Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
isomli.0  |-  K  e.  OL
isomli.b  |-  B  =  ( Base `  K
)
isomli.l  |-  .<_  =  ( le `  K )
isomli.j  |-  .\/  =  ( join `  K )
isomli.m  |-  ./\  =  ( meet `  K )
isomli.o  |-  ._|_  =  ( oc `  K )
isomli.7  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .<_  y  -> 
y  =  ( x 
.\/  ( y  ./\  (  ._|_  `  x )
) ) ) )
Assertion
Ref Expression
isomliN  |-  K  e. 
OML
Distinct variable groups:    x, y, B    x, K, y
Allowed substitution hints:    .\/ ( x, y)    .<_ ( x, y)    ./\ ( x, y)    ._|_ ( x, y)

Proof of Theorem isomliN
StepHypRef Expression
1 isomli.0 . 2  |-  K  e.  OL
2 isomli.7 . . 3  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .<_  y  -> 
y  =  ( x 
.\/  ( y  ./\  (  ._|_  `  x )
) ) ) )
32rgen2a 2622 . 2  |-  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) )
4 isomli.b . . 3  |-  B  =  ( Base `  K
)
5 isomli.l . . 3  |-  .<_  =  ( le `  K )
6 isomli.j . . 3  |-  .\/  =  ( join `  K )
7 isomli.m . . 3  |-  ./\  =  ( meet `  K )
8 isomli.o . . 3  |-  ._|_  =  ( oc `  K )
94, 5, 6, 7, 8isoml 30050 . 2  |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) ) )
101, 3, 9mpbir2an 886 1  |-  K  e. 
OML
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   joincjn 14094   meetcmee 14095   OLcol 29986   OMLcoml 29987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-oml 29991
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