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Theorem isomliN 29429
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 2609 . 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 29428 . 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 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 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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