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Metamath Proof Explorer

Theorem dochsordN

Description: Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014) (New usage is discouraged.)

Ref Expression
Hypotheses doch11.h 
|- H = ( LHyp ` K )
doch11.i 
|- I = ( ( DIsoH ` K ) ` W )
doch11.o 
|- ._|_ = ( ( ocH ` K ) ` W )
doch11.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
doch11.x 
|- ( ph -> X e. ran I )
doch11.y 
|- ( ph -> Y e. ran I )
Assertion dochsordN 
|- ( ph -> ( X C. Y <-> ( ._|_ ` Y ) C. ( ._|_ ` X ) ) )

Proof

Step Hyp Ref Expression
1 doch11.h 
 |-  H = ( LHyp ` K )
2 doch11.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 doch11.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
4 doch11.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
5 doch11.x 
 |-  ( ph -> X e. ran I )
6 doch11.y 
 |-  ( ph -> Y e. ran I )
7 1 2 3 4 5 6 dochord 
 |-  ( ph -> ( X C_ Y <-> ( ._|_ ` Y ) C_ ( ._|_ ` X ) ) )
8 1 2 3 4 6 5 doch11 
 |-  ( ph -> ( ( ._|_ ` Y ) = ( ._|_ ` X ) <-> Y = X ) )
9 eqcom 
 |-  ( Y = X <-> X = Y )
10 8 9 bitr2di 
 |-  ( ph -> ( X = Y <-> ( ._|_ ` Y ) = ( ._|_ ` X ) ) )
11 10 necon3bid 
 |-  ( ph -> ( X =/= Y <-> ( ._|_ ` Y ) =/= ( ._|_ ` X ) ) )
12 7 11 anbi12d 
 |-  ( ph -> ( ( X C_ Y /\ X =/= Y ) <-> ( ( ._|_ ` Y ) C_ ( ._|_ ` X ) /\ ( ._|_ ` Y ) =/= ( ._|_ ` X ) ) ) )
13 df-pss 
 |-  ( X C. Y <-> ( X C_ Y /\ X =/= Y ) )
14 df-pss 
 |-  ( ( ._|_ ` Y ) C. ( ._|_ ` X ) <-> ( ( ._|_ ` Y ) C_ ( ._|_ ` X ) /\ ( ._|_ ` Y ) =/= ( ._|_ ` X ) ) )
15 12 13 14 3bitr4g 
 |-  ( ph -> ( X C. Y <-> ( ._|_ ` Y ) C. ( ._|_ ` X ) ) )