Metamath Proof Explorer

Theorem omllaw3

Description: Orthomodular law equivalent. Theorem 2(ii) of Kalmbach p. 22. ( pjoml analog.) (Contributed by NM, 19-Oct-2011)

Ref Expression
Hypotheses omllaw3.b 
|- B = ( Base ` K )
omllaw3.l 
|- .<_ = ( le ` K )
omllaw3.m 
|- ./\ = ( meet ` K )
omllaw3.o 
|- ._|_ = ( oc ` K )
omllaw3.z 
|- .0. = ( 0. ` K )
Assertion omllaw3 
|- ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ ( Y ./\ ( ._|_ ` X ) ) = .0. ) -> X = Y ) )

Proof

Step Hyp Ref Expression
1 omllaw3.b 
 |-  B = ( Base ` K )
2 omllaw3.l 
 |-  .<_ = ( le ` K )
3 omllaw3.m 
 |-  ./\ = ( meet ` K )
4 omllaw3.o 
 |-  ._|_ = ( oc ` K )
5 omllaw3.z 
 |-  .0. = ( 0. ` K )
6 oveq2 
 |-  ( ( Y ./\ ( ._|_ ` X ) ) = .0. -> ( X ( join ` K ) ( Y ./\ ( ._|_ ` X ) ) ) = ( X ( join ` K ) .0. ) )
7 6 adantl 
 |-  ( ( ( K e. OML /\ X e. B /\ Y e. B ) /\ ( Y ./\ ( ._|_ ` X ) ) = .0. ) -> ( X ( join ` K ) ( Y ./\ ( ._|_ ` X ) ) ) = ( X ( join ` K ) .0. ) )
8 omlol 
 |-  ( K e. OML -> K e. OL )
9 eqid 
 |-  ( join ` K ) = ( join ` K )
10 1 9 5 olj01 
 |-  ( ( K e. OL /\ X e. B ) -> ( X ( join ` K ) .0. ) = X )
11 8 10 sylan 
 |-  ( ( K e. OML /\ X e. B ) -> ( X ( join ` K ) .0. ) = X )
12 11 3adant3 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X ( join ` K ) .0. ) = X )
13 12 adantr 
 |-  ( ( ( K e. OML /\ X e. B /\ Y e. B ) /\ ( Y ./\ ( ._|_ ` X ) ) = .0. ) -> ( X ( join ` K ) .0. ) = X )
14 7 13 eqtr2d 
 |-  ( ( ( K e. OML /\ X e. B /\ Y e. B ) /\ ( Y ./\ ( ._|_ ` X ) ) = .0. ) -> X = ( X ( join ` K ) ( Y ./\ ( ._|_ ` X ) ) ) )
15 14 adantrl 
 |-  ( ( ( K e. OML /\ X e. B /\ Y e. B ) /\ ( X .<_ Y /\ ( Y ./\ ( ._|_ ` X ) ) = .0. ) ) -> X = ( X ( join ` K ) ( Y ./\ ( ._|_ ` X ) ) ) )
16 1 2 9 3 4 omllaw 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> Y = ( X ( join ` K ) ( Y ./\ ( ._|_ ` X ) ) ) ) )
17 16 imp 
 |-  ( ( ( K e. OML /\ X e. B /\ Y e. B ) /\ X .<_ Y ) -> Y = ( X ( join ` K ) ( Y ./\ ( ._|_ ` X ) ) ) )
18 17 adantrr 
 |-  ( ( ( K e. OML /\ X e. B /\ Y e. B ) /\ ( X .<_ Y /\ ( Y ./\ ( ._|_ ` X ) ) = .0. ) ) -> Y = ( X ( join ` K ) ( Y ./\ ( ._|_ ` X ) ) ) )
19 15 18 eqtr4d 
 |-  ( ( ( K e. OML /\ X e. B /\ Y e. B ) /\ ( X .<_ Y /\ ( Y ./\ ( ._|_ ` X ) ) = .0. ) ) -> X = Y )
20 19 ex 
 |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ ( Y ./\ ( ._|_ ` X ) ) = .0. ) -> X = Y ) )