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Metamath Proof Explorer

Theorem opoc1

Description: Orthocomplement of orthoposet unity. (Contributed by NM, 24-Jan-2012)

Ref Expression
Hypotheses opoc1.z 
|- .0. = ( 0. ` K )
opoc1.u 
|- .1. = ( 1. ` K )
opoc1.o 
|- ._|_ = ( oc ` K )
Assertion opoc1 
|- ( K e. OP -> ( ._|_ ` .1. ) = .0. )

Proof

Step Hyp Ref Expression
1 opoc1.z 
 |-  .0. = ( 0. ` K )
2 opoc1.u 
 |-  .1. = ( 1. ` K )
3 opoc1.o 
 |-  ._|_ = ( oc ` K )
4 eqid 
 |-  ( Base ` K ) = ( Base ` K )
5 4 1 op0cl 
 |-  ( K e. OP -> .0. e. ( Base ` K ) )
6 4 3 opoccl 
 |-  ( ( K e. OP /\ .0. e. ( Base ` K ) ) -> ( ._|_ ` .0. ) e. ( Base ` K ) )
7 5 6 mpdan 
 |-  ( K e. OP -> ( ._|_ ` .0. ) e. ( Base ` K ) )
8 eqid 
 |-  ( le ` K ) = ( le ` K )
9 4 8 2 ople1 
 |-  ( ( K e. OP /\ ( ._|_ ` .0. ) e. ( Base ` K ) ) -> ( ._|_ ` .0. ) ( le ` K ) .1. )
10 7 9 mpdan 
 |-  ( K e. OP -> ( ._|_ ` .0. ) ( le ` K ) .1. )
11 4 2 op1cl 
 |-  ( K e. OP -> .1. e. ( Base ` K ) )
12 4 8 3 oplecon1b 
 |-  ( ( K e. OP /\ .1. e. ( Base ` K ) /\ .0. e. ( Base ` K ) ) -> ( ( ._|_ ` .1. ) ( le ` K ) .0. <-> ( ._|_ ` .0. ) ( le ` K ) .1. ) )
13 11 5 12 mpd3an23 
 |-  ( K e. OP -> ( ( ._|_ ` .1. ) ( le ` K ) .0. <-> ( ._|_ ` .0. ) ( le ` K ) .1. ) )
14 10 13 mpbird 
 |-  ( K e. OP -> ( ._|_ ` .1. ) ( le ` K ) .0. )
15 4 3 opoccl 
 |-  ( ( K e. OP /\ .1. e. ( Base ` K ) ) -> ( ._|_ ` .1. ) e. ( Base ` K ) )
16 11 15 mpdan 
 |-  ( K e. OP -> ( ._|_ ` .1. ) e. ( Base ` K ) )
17 4 8 1 ople0 
 |-  ( ( K e. OP /\ ( ._|_ ` .1. ) e. ( Base ` K ) ) -> ( ( ._|_ ` .1. ) ( le ` K ) .0. <-> ( ._|_ ` .1. ) = .0. ) )
18 16 17 mpdan 
 |-  ( K e. OP -> ( ( ._|_ ` .1. ) ( le ` K ) .0. <-> ( ._|_ ` .1. ) = .0. ) )
19 14 18 mpbid 
 |-  ( K e. OP -> ( ._|_ ` .1. ) = .0. )