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Metamath Proof Explorer

Theorem omllaw5N

Description: The orthomodular law. Remark in Kalmbach p. 22. ( pjoml5 analog.) (Contributed by NM, 14-Nov-2011) (New usage is discouraged.)

Ref Expression
Hypotheses omllaw5.b 
|- B = ( Base ` K )
omllaw5.j 
|- .\/ = ( join ` K )
omllaw5.m 
|- ./\ = ( meet ` K )
omllaw5.o 
|- ._|_ = ( oc ` K )
Assertion omllaw5N 
|- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .\/ ( ( ._|_ ` X ) ./\ ( X .\/ Y ) ) ) = ( X .\/ Y ) )

Proof

Step Hyp Ref Expression
1 omllaw5.b 
 |-  B = ( Base ` K )
2 omllaw5.j 
 |-  .\/ = ( join ` K )
3 omllaw5.m 
 |-  ./\ = ( meet ` K )
4 omllaw5.o 
 |-  ._|_ = ( oc ` K )
5 simp1 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> K e. OML )
6 simp2 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> X e. B )
7 omllat 
 |-  ( K e. OML -> K e. Lat )
8 1 2 latjcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
9 7 8 syl3an1 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
10 5 6 9 3jca 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( K e. OML /\ X e. B /\ ( X .\/ Y ) e. B ) )
11 eqid 
 |-  ( le ` K ) = ( le ` K )
12 1 11 2 latlej1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X ( le ` K ) ( X .\/ Y ) )
13 7 12 syl3an1 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> X ( le ` K ) ( X .\/ Y ) )
14 1 11 2 3 4 omllaw2N 
 |-  ( ( K e. OML /\ X e. B /\ ( X .\/ Y ) e. B ) -> ( X ( le ` K ) ( X .\/ Y ) -> ( X .\/ ( ( ._|_ ` X ) ./\ ( X .\/ Y ) ) ) = ( X .\/ Y ) ) )
15 10 13 14 sylc 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .\/ ( ( ._|_ ` X ) ./\ ( X .\/ Y ) ) ) = ( X .\/ Y ) )