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Metamath Proof Explorer

Theorem opnoncon

Description: Law of contradiction for orthoposets. ( chocin analog.) (Contributed by NM, 13-Sep-2011)

Ref Expression
Hypotheses opnoncon.b 
|- B = ( Base ` K )
opnoncon.o 
|- ._|_ = ( oc ` K )
opnoncon.m 
|- ./\ = ( meet ` K )
opnoncon.z 
|- .0. = ( 0. ` K )
Assertion opnoncon 
|- ( ( K e. OP /\ X e. B ) -> ( X ./\ ( ._|_ ` X ) ) = .0. )

Proof

Step Hyp Ref Expression
1 opnoncon.b 
 |-  B = ( Base ` K )
2 opnoncon.o 
 |-  ._|_ = ( oc ` K )
3 opnoncon.m 
 |-  ./\ = ( meet ` K )
4 opnoncon.z 
 |-  .0. = ( 0. ` K )
5 eqid 
 |-  ( le ` K ) = ( le ` K )
6 eqid 
 |-  ( join ` K ) = ( join ` K )
7 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
8 1 5 2 6 3 4 7 oposlem 
 |-  ( ( K e. OP /\ X e. B /\ X e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ./\ ( ._|_ ` X ) ) = .0. ) )
9 8 3anidm23 
 |-  ( ( K e. OP /\ X e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ./\ ( ._|_ ` X ) ) = .0. ) )
10 9 simp3d 
 |-  ( ( K e. OP /\ X e. B ) -> ( X ./\ ( ._|_ ` X ) ) = .0. )