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Metamath Proof Explorer

Theorem dihcnv11

Description: The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014)

Ref Expression
Hypotheses dihcnv11.h 
|- H = ( LHyp ` K )
dihcnv11.i 
|- I = ( ( DIsoH ` K ) ` W )
dihcnv11.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dihcnv11.x 
|- ( ph -> X e. ran I )
dihcnv11.y 
|- ( ph -> Y e. ran I )
Assertion dihcnv11 
|- ( ph -> ( ( `' I ` X ) = ( `' I ` Y ) <-> X = Y ) )

Proof

Step Hyp Ref Expression
1 dihcnv11.h 
 |-  H = ( LHyp ` K )
2 dihcnv11.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 dihcnv11.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
4 dihcnv11.x 
 |-  ( ph -> X e. ran I )
5 dihcnv11.y 
 |-  ( ph -> Y e. ran I )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 1 2 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. ( Base ` K ) )
8 3 4 7 syl2anc 
 |-  ( ph -> ( `' I ` X ) e. ( Base ` K ) )
9 6 1 2 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( `' I ` Y ) e. ( Base ` K ) )
10 3 5 9 syl2anc 
 |-  ( ph -> ( `' I ` Y ) e. ( Base ` K ) )
11 6 1 2 dih11 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( `' I ` X ) e. ( Base ` K ) /\ ( `' I ` Y ) e. ( Base ` K ) ) -> ( ( I ` ( `' I ` X ) ) = ( I ` ( `' I ` Y ) ) <-> ( `' I ` X ) = ( `' I ` Y ) ) )
12 3 8 10 11 syl3anc 
 |-  ( ph -> ( ( I ` ( `' I ` X ) ) = ( I ` ( `' I ` Y ) ) <-> ( `' I ` X ) = ( `' I ` Y ) ) )
13 1 2 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )
14 3 4 13 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` X ) ) = X )
15 1 2 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( I ` ( `' I ` Y ) ) = Y )
16 3 5 15 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` Y ) ) = Y )
17 14 16 eqeq12d 
 |-  ( ph -> ( ( I ` ( `' I ` X ) ) = ( I ` ( `' I ` Y ) ) <-> X = Y ) )
18 12 17 bitr3d 
 |-  ( ph -> ( ( `' I ` X ) = ( `' I ` Y ) <-> X = Y ) )