Metamath Proof Explorer

Theorem cmn32

Description: Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014) (Revised by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses ablcom.b 
|- B = ( Base ` G )
ablcom.p 
|- .+ = ( +g ` G )
Assertion cmn32 
|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )

Proof

Step Hyp Ref Expression
1 ablcom.b 
 |-  B = ( Base ` G )
2 ablcom.p 
 |-  .+ = ( +g ` G )
3 cmnmnd 
 |-  ( G e. CMnd -> G e. Mnd )
4 3 adantr 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> G e. Mnd )
5 simpr1 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
6 simpr2 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
7 simpr3 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
8 1 2 cmncom 
 |-  ( ( G e. CMnd /\ Y e. B /\ Z e. B ) -> ( Y .+ Z ) = ( Z .+ Y ) )
9 8 3adant3r1 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .+ Z ) = ( Z .+ Y ) )
10 1 2 4 5 6 7 9 mnd32g 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )