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Metamath Proof Explorer

Theorem ablpncan2

Description: Cancellation law for subtraction in an Abelian group. (Contributed by NM, 2-Oct-2014)

Ref Expression
Hypotheses ablsubadd.b 
|- B = ( Base ` G )
ablsubadd.p 
|- .+ = ( +g ` G )
ablsubadd.m 
|- .- = ( -g ` G )
Assertion ablpncan2 
|- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- X ) = Y )

Proof

Step Hyp Ref Expression
1 ablsubadd.b 
 |-  B = ( Base ` G )
2 ablsubadd.p 
 |-  .+ = ( +g ` G )
3 ablsubadd.m 
 |-  .- = ( -g ` G )
4 simp1 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> G e. Abel )
5 simp2 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> X e. B )
6 simp3 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> Y e. B )
7 1 2 3 abladdsub 
 |-  ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ X e. B ) ) -> ( ( X .+ Y ) .- X ) = ( ( X .- X ) .+ Y ) )
8 4 5 6 5 7 syl13anc 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- X ) = ( ( X .- X ) .+ Y ) )
9 ablgrp 
 |-  ( G e. Abel -> G e. Grp )
10 4 9 syl 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> G e. Grp )
11 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
12 1 11 3 grpsubid 
 |-  ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = ( 0g ` G ) )
13 10 5 12 syl2anc 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( X .- X ) = ( 0g ` G ) )
14 13 oveq1d 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( ( X .- X ) .+ Y ) = ( ( 0g ` G ) .+ Y ) )
15 1 2 11 grplid 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
16 10 6 15 syl2anc 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
17 8 14 16 3eqtrd 
 |-  ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- X ) = Y )