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Metamath Proof Explorer

Theorem grpinv11

Description: The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015) (Proof shortened by SN, 8-Jul-2025)

Ref Expression
Hypotheses grpinvinv.b 
|- B = ( Base ` G )
grpinvinv.n 
|- N = ( invg ` G )
grpinv11.g 
|- ( ph -> G e. Grp )
grpinv11.x 
|- ( ph -> X e. B )
grpinv11.y 
|- ( ph -> Y e. B )
Assertion grpinv11 
|- ( ph -> ( ( N ` X ) = ( N ` Y ) <-> X = Y ) )

Proof

Step Hyp Ref Expression
1 grpinvinv.b 
 |-  B = ( Base ` G )
2 grpinvinv.n 
 |-  N = ( invg ` G )
3 grpinv11.g 
 |-  ( ph -> G e. Grp )
4 grpinv11.x 
 |-  ( ph -> X e. B )
5 grpinv11.y 
 |-  ( ph -> Y e. B )
6 fveq2 
 |-  ( ( N ` X ) = ( N ` Y ) -> ( N ` ( N ` X ) ) = ( N ` ( N ` Y ) ) )
7 1 2 grpinvinv 
 |-  ( ( G e. Grp /\ X e. B ) -> ( N ` ( N ` X ) ) = X )
8 3 4 7 syl2anc 
 |-  ( ph -> ( N ` ( N ` X ) ) = X )
9 1 2 grpinvinv 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y )
10 3 5 9 syl2anc 
 |-  ( ph -> ( N ` ( N ` Y ) ) = Y )
11 8 10 eqeq12d 
 |-  ( ph -> ( ( N ` ( N ` X ) ) = ( N ` ( N ` Y ) ) <-> X = Y ) )
12 6 11 imbitrid 
 |-  ( ph -> ( ( N ` X ) = ( N ` Y ) -> X = Y ) )
13 fveq2 
 |-  ( X = Y -> ( N ` X ) = ( N ` Y ) )
14 12 13 impbid1 
 |-  ( ph -> ( ( N ` X ) = ( N ` Y ) <-> X = Y ) )