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Metamath Proof Explorer

Theorem grpsubsub

Description: Double group subtraction. (Contributed by NM, 24-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 grpsubsub 
|- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .- ( Y .- Z ) ) = ( X .+ ( Z .- Y ) ) )

Proof

Step Hyp Ref Expression
1 grpsubadd.b 
 |-  B = ( Base ` G )
2 grpsubadd.p 
 |-  .+ = ( +g ` G )
3 grpsubadd.m 
 |-  .- = ( -g ` G )
4 simpr1 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
5 1 3 grpsubcl 
 |-  ( ( G e. Grp /\ Y e. B /\ Z e. B ) -> ( Y .- Z ) e. B )
6 5 3adant3r1 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .- Z ) e. B )
7 eqid 
 |-  ( invg ` G ) = ( invg ` G )
8 1 2 7 3 grpsubval 
 |-  ( ( X e. B /\ ( Y .- Z ) e. B ) -> ( X .- ( Y .- Z ) ) = ( X .+ ( ( invg ` G ) ` ( Y .- Z ) ) ) )
9 4 6 8 syl2anc 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .- ( Y .- Z ) ) = ( X .+ ( ( invg ` G ) ` ( Y .- Z ) ) ) )
10 1 3 7 grpinvsub 
 |-  ( ( G e. Grp /\ Y e. B /\ Z e. B ) -> ( ( invg ` G ) ` ( Y .- Z ) ) = ( Z .- Y ) )
11 10 3adant3r1 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( invg ` G ) ` ( Y .- Z ) ) = ( Z .- Y ) )
12 11 oveq2d 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ ( ( invg ` G ) ` ( Y .- Z ) ) ) = ( X .+ ( Z .- Y ) ) )
13 9 12 eqtrd 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .- ( Y .- Z ) ) = ( X .+ ( Z .- Y ) ) )