Metamath Proof Explorer


Theorem grpinv11

Description: The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015)

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 6 adantl
 |-  ( ( ph /\ ( N ` X ) = ( N ` Y ) ) -> ( N ` ( N ` X ) ) = ( N ` ( N ` Y ) ) )
8 1 2 grpinvinv
 |-  ( ( G e. Grp /\ X e. B ) -> ( N ` ( N ` X ) ) = X )
9 3 4 8 syl2anc
 |-  ( ph -> ( N ` ( N ` X ) ) = X )
10 9 adantr
 |-  ( ( ph /\ ( N ` X ) = ( N ` Y ) ) -> ( N ` ( N ` X ) ) = X )
11 1 2 grpinvinv
 |-  ( ( G e. Grp /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y )
12 3 5 11 syl2anc
 |-  ( ph -> ( N ` ( N ` Y ) ) = Y )
13 12 adantr
 |-  ( ( ph /\ ( N ` X ) = ( N ` Y ) ) -> ( N ` ( N ` Y ) ) = Y )
14 7 10 13 3eqtr3d
 |-  ( ( ph /\ ( N ` X ) = ( N ` Y ) ) -> X = Y )
15 14 ex
 |-  ( ph -> ( ( N ` X ) = ( N ` Y ) -> X = Y ) )
16 fveq2
 |-  ( X = Y -> ( N ` X ) = ( N ` Y ) )
17 15 16 impbid1
 |-  ( ph -> ( ( N ` X ) = ( N ` Y ) <-> X = Y ) )