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Metamath Proof Explorer

Theorem oplecon3

Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011)

Ref Expression
Hypotheses opcon3.b 
|- B = ( Base ` K )
opcon3.l 
|- .<_ = ( le ` K )
opcon3.o 
|- ._|_ = ( oc ` K )
Assertion oplecon3 
|- ( ( K e. OP /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> ( ._|_ ` Y ) .<_ ( ._|_ ` X ) ) )

Proof

Step Hyp Ref Expression
1 opcon3.b 
 |-  B = ( Base ` K )
2 opcon3.l 
 |-  .<_ = ( le ` K )
3 opcon3.o 
 |-  ._|_ = ( oc ` K )
4 eqid 
 |-  ( join ` K ) = ( join ` K )
5 eqid 
 |-  ( meet ` K ) = ( meet ` K )
6 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
7 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
8 1 2 3 4 5 6 7 oposlem 
 |-  ( ( K e. OP /\ X e. B /\ Y e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X .<_ Y -> ( ._|_ ` Y ) .<_ ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ( meet ` K ) ( ._|_ ` X ) ) = ( 0. ` K ) ) )
9 8 simp1d 
 |-  ( ( K e. OP /\ X e. B /\ Y e. B ) -> ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X .<_ Y -> ( ._|_ ` Y ) .<_ ( ._|_ ` X ) ) ) )
10 9 simp3d 
 |-  ( ( K e. OP /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> ( ._|_ ` Y ) .<_ ( ._|_ ` X ) ) )