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Metamath Proof Explorer

Theorem op1le

Description: If the orthoposet unity is less than or equal to an element, the element equals the unit. ( chle0 analog.) (Contributed by NM, 5-Dec-2011)

Ref Expression
Hypotheses ople1.b 
|- B = ( Base ` K )
ople1.l 
|- .<_ = ( le ` K )
ople1.u 
|- .1. = ( 1. ` K )
Assertion op1le 
|- ( ( K e. OP /\ X e. B ) -> ( .1. .<_ X <-> X = .1. ) )

Proof

Step Hyp Ref Expression
1 ople1.b 
 |-  B = ( Base ` K )
2 ople1.l 
 |-  .<_ = ( le ` K )
3 ople1.u 
 |-  .1. = ( 1. ` K )
4 1 2 3 ople1 
 |-  ( ( K e. OP /\ X e. B ) -> X .<_ .1. )
5 4 biantrurd 
 |-  ( ( K e. OP /\ X e. B ) -> ( .1. .<_ X <-> ( X .<_ .1. /\ .1. .<_ X ) ) )
6 opposet 
 |-  ( K e. OP -> K e. Poset )
7 6 adantr 
 |-  ( ( K e. OP /\ X e. B ) -> K e. Poset )
8 simpr 
 |-  ( ( K e. OP /\ X e. B ) -> X e. B )
9 1 3 op1cl 
 |-  ( K e. OP -> .1. e. B )
10 9 adantr 
 |-  ( ( K e. OP /\ X e. B ) -> .1. e. B )
11 1 2 posasymb 
 |-  ( ( K e. Poset /\ X e. B /\ .1. e. B ) -> ( ( X .<_ .1. /\ .1. .<_ X ) <-> X = .1. ) )
12 7 8 10 11 syl3anc 
 |-  ( ( K e. OP /\ X e. B ) -> ( ( X .<_ .1. /\ .1. .<_ X ) <-> X = .1. ) )
13 5 12 bitrd 
 |-  ( ( K e. OP /\ X e. B ) -> ( .1. .<_ X <-> X = .1. ) )