Metamath Proof Explorer

Theorem dih11

Description: The isomorphism H is one-to-one. Part of proof after Lemma N of Crawley p. 122 line 6. (Contributed by NM, 7-Mar-2014)

Ref Expression
Hypotheses dih11.b 
|- B = ( Base ` K )
dih11.h 
|- H = ( LHyp ` K )
dih11.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dih11 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( ( I ` X ) = ( I ` Y ) <-> X = Y ) )

Proof

Step Hyp Ref Expression
1 dih11.b 
 |-  B = ( Base ` K )
2 dih11.h 
 |-  H = ( LHyp ` K )
3 dih11.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 eqss 
 |-  ( ( I ` X ) = ( I ` Y ) <-> ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) )
5 eqid 
 |-  ( le ` K ) = ( le ` K )
6 1 5 2 3 dihord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( ( I ` X ) C_ ( I ` Y ) <-> X ( le ` K ) Y ) )
7 1 5 2 3 dihord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ X e. B ) -> ( ( I ` Y ) C_ ( I ` X ) <-> Y ( le ` K ) X ) )
8 7 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( ( I ` Y ) C_ ( I ` X ) <-> Y ( le ` K ) X ) )
9 6 8 anbi12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) <-> ( X ( le ` K ) Y /\ Y ( le ` K ) X ) ) )
10 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> K e. HL )
11 10 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> K e. Lat )
12 1 5 latasymb 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X ( le ` K ) Y /\ Y ( le ` K ) X ) <-> X = Y ) )
13 11 12 syld3an1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( ( X ( le ` K ) Y /\ Y ( le ` K ) X ) <-> X = Y ) )
14 9 13 bitrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) <-> X = Y ) )
15 4 14 bitrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( ( I ` X ) = ( I ` Y ) <-> X = Y ) )