Metamath Proof Explorer

Theorem grpinveu

Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of Herstein p. 55. (Contributed by NM, 24-Aug-2011)

Ref Expression
Hypotheses grpinveu.b 
|- B = ( Base ` G )
grpinveu.p 
|- .+ = ( +g ` G )
grpinveu.o 
|- .0. = ( 0g ` G )
Assertion grpinveu 
|- ( ( G e. Grp /\ X e. B ) -> E! y e. B ( y .+ X ) = .0. )

Proof

Step Hyp Ref Expression
1 grpinveu.b 
 |-  B = ( Base ` G )
2 grpinveu.p 
 |-  .+ = ( +g ` G )
3 grpinveu.o 
 |-  .0. = ( 0g ` G )
4 1 2 3 grpinvex 
 |-  ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
5 eqtr3 
 |-  ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> ( y .+ X ) = ( z .+ X ) )
6 1 2 grprcan 
 |-  ( ( G e. Grp /\ ( y e. B /\ z e. B /\ X e. B ) ) -> ( ( y .+ X ) = ( z .+ X ) <-> y = z ) )
7 5 6 syl5ib 
 |-  ( ( G e. Grp /\ ( y e. B /\ z e. B /\ X e. B ) ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
8 7 3exp2 
 |-  ( G e. Grp -> ( y e. B -> ( z e. B -> ( X e. B -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) ) ) ) )
9 8 com24 
 |-  ( G e. Grp -> ( X e. B -> ( z e. B -> ( y e. B -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) ) ) ) )
10 9 imp41 
 |-  ( ( ( ( G e. Grp /\ X e. B ) /\ z e. B ) /\ y e. B ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
11 10 an32s 
 |-  ( ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) /\ z e. B ) -> ( ( ( y .+ X ) = .0. /\ ( z .+ X ) = .0. ) -> y = z ) )
12 11 expd 
 |-  ( ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) /\ z e. B ) -> ( ( y .+ X ) = .0. -> ( ( z .+ X ) = .0. -> y = z ) ) )
13 12 ralrimdva 
 |-  ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) -> ( ( y .+ X ) = .0. -> A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) )
14 13 ancld 
 |-  ( ( ( G e. Grp /\ X e. B ) /\ y e. B ) -> ( ( y .+ X ) = .0. -> ( ( y .+ X ) = .0. /\ A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) ) )
15 14 reximdva 
 |-  ( ( G e. Grp /\ X e. B ) -> ( E. y e. B ( y .+ X ) = .0. -> E. y e. B ( ( y .+ X ) = .0. /\ A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) ) )
16 4 15 mpd 
 |-  ( ( G e. Grp /\ X e. B ) -> E. y e. B ( ( y .+ X ) = .0. /\ A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) )
17 oveq1 
 |-  ( y = z -> ( y .+ X ) = ( z .+ X ) )
18 17 eqeq1d 
 |-  ( y = z -> ( ( y .+ X ) = .0. <-> ( z .+ X ) = .0. ) )
19 18 reu8 
 |-  ( E! y e. B ( y .+ X ) = .0. <-> E. y e. B ( ( y .+ X ) = .0. /\ A. z e. B ( ( z .+ X ) = .0. -> y = z ) ) )
20 16 19 sylibr 
 |-  ( ( G e. Grp /\ X e. B ) -> E! y e. B ( y .+ X ) = .0. )