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Metamath Proof Explorer

Theorem doch11

Description: Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014)

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 doch11 
|- ( ph -> ( ( ._|_ ` X ) = ( ._|_ ` Y ) <-> X = Y ) )

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 6 5 dochord 
 |-  ( ph -> ( Y C_ X <-> ( ._|_ ` X ) C_ ( ._|_ ` Y ) ) )
8 1 2 3 4 5 6 dochord 
 |-  ( ph -> ( X C_ Y <-> ( ._|_ ` Y ) C_ ( ._|_ ` X ) ) )
9 7 8 anbi12d 
 |-  ( ph -> ( ( Y C_ X /\ X C_ Y ) <-> ( ( ._|_ ` X ) C_ ( ._|_ ` Y ) /\ ( ._|_ ` Y ) C_ ( ._|_ ` X ) ) ) )
10 eqcom 
 |-  ( X = Y <-> Y = X )
11 eqss 
 |-  ( Y = X <-> ( Y C_ X /\ X C_ Y ) )
12 10 11 bitri 
 |-  ( X = Y <-> ( Y C_ X /\ X C_ Y ) )
13 eqss 
 |-  ( ( ._|_ ` X ) = ( ._|_ ` Y ) <-> ( ( ._|_ ` X ) C_ ( ._|_ ` Y ) /\ ( ._|_ ` Y ) C_ ( ._|_ ` X ) ) )
14 9 12 13 3bitr4g 
 |-  ( ph -> ( X = Y <-> ( ._|_ ` X ) = ( ._|_ ` Y ) ) )
15 14 bicomd 
 |-  ( ph -> ( ( ._|_ ` X ) = ( ._|_ ` Y ) <-> X = Y ) )