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Theorem isomliN 29974
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 2764 . 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 29973 . 2  |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  ->  y  =  ( x  .\/  ( y 
./\  (  ._|_  `  x
) ) ) ) ) )
101, 3, 9mpbir2an 887 1  |-  K  e. 
OML
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   occoc 13529   joincjn 14393   meetcmee 14394   OLcol 29909   OMLcoml 29910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-oml 29914
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