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Metamath Proof Explorer

Theorem grppncan

Description: Cancellation law for subtraction ( pncan analog). (Contributed by NM, 16-Apr-2014)

Ref Expression
Hypotheses grpsubadd.b 
|- B = ( Base ` G )
grpsubadd.p 
|- .+ = ( +g ` G )
grpsubadd.m 
|- .- = ( -g ` G )
Assertion grppncan 
|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- Y ) = X )

Proof

Step Hyp Ref Expression
1 grpsubadd.b 
 |-  B = ( Base ` G )
2 grpsubadd.p 
 |-  .+ = ( +g ` G )
3 grpsubadd.m 
 |-  .- = ( -g ` G )
4 simp1 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> G e. Grp )
5 simp2 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> X e. B )
6 simp3 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> Y e. B )
7 1 2 3 grpaddsubass 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Y e. B ) ) -> ( ( X .+ Y ) .- Y ) = ( X .+ ( Y .- Y ) ) )
8 4 5 6 6 7 syl13anc 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- Y ) = ( X .+ ( Y .- Y ) ) )
9 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
10 1 9 3 grpsubid 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( Y .- Y ) = ( 0g ` G ) )
11 10 oveq2d 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( X .+ ( Y .- Y ) ) = ( X .+ ( 0g ` G ) ) )
12 11 3adant2 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( Y .- Y ) ) = ( X .+ ( 0g ` G ) ) )
13 1 2 9 grprid 
 |-  ( ( G e. Grp /\ X e. B ) -> ( X .+ ( 0g ` G ) ) = X )
14 13 3adant3 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( 0g ` G ) ) = X )
15 8 12 14 3eqtrd 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- Y ) = X )