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Metamath Proof Explorer

Theorem ablsubadd

Description: Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014)

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

Proof

Step Hyp Ref Expression
1 ablsubadd.b 
 |-  B = ( Base ` G )
2 ablsubadd.p 
 |-  .+ = ( +g ` G )
3 ablsubadd.m 
 |-  .- = ( -g ` G )
4 ablgrp 
 |-  ( G e. Abel -> G e. Grp )
5 1 2 3 grpsubadd 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Z .+ Y ) = X ) )
6 4 5 sylan 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Z .+ Y ) = X ) )
7 1 2 ablcom 
 |-  ( ( G e. Abel /\ Y e. B /\ Z e. B ) -> ( Y .+ Z ) = ( Z .+ Y ) )
8 7 3adant3r1 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .+ Z ) = ( Z .+ Y ) )
9 8 eqeq1d 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( Y .+ Z ) = X <-> ( Z .+ Y ) = X ) )
10 6 9 bitr4d 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Y .+ Z ) = X ) )