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Metamath Proof Explorer

Theorem grppnpcan2

Description: Cancellation law for mixed addition and subtraction. ( pnpcan2 analog.) (Contributed by NM, 15-Feb-2008) (Revised by Mario Carneiro, 2-Dec-2014)

Ref Expression
Hypotheses grpsubadd.b 
|- B = ( Base ` G )
grpsubadd.p 
|- .+ = ( +g ` G )
grpsubadd.m 
|- .- = ( -g ` G )
Assertion grppnpcan2 
|- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- ( Y .+ Z ) ) = ( X .- Y ) )

Proof

Step Hyp Ref Expression
1 grpsubadd.b 
 |-  B = ( Base ` G )
2 grpsubadd.p 
 |-  .+ = ( +g ` G )
3 grpsubadd.m 
 |-  .- = ( -g ` G )
4 simpl 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> G e. Grp )
5 1 2 grpcl 
 |-  ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( X .+ Z ) e. B )
6 5 3adant3r2 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ Z ) e. B )
7 simpr3 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
8 simpr2 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
9 1 2 3 grpsubsub4 
 |-  ( ( G e. Grp /\ ( ( X .+ Z ) e. B /\ Z e. B /\ Y e. B ) ) -> ( ( ( X .+ Z ) .- Z ) .- Y ) = ( ( X .+ Z ) .- ( Y .+ Z ) ) )
10 4 6 7 8 9 syl13anc 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( X .+ Z ) .- Z ) .- Y ) = ( ( X .+ Z ) .- ( Y .+ Z ) ) )
11 1 2 3 grppncan 
 |-  ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( X .+ Z ) .- Z ) = X )
12 11 3adant3r2 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- Z ) = X )
13 12 oveq1d 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( X .+ Z ) .- Z ) .- Y ) = ( X .- Y ) )
14 10 13 eqtr3d 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- ( Y .+ Z ) ) = ( X .- Y ) )