Metamath Proof Explorer

Theorem cmn12

Description: Commutative/associative law for Abelian monoids. (Contributed by Stefan O'Rear, 5-Sep-2015) (Revised by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses ablcom.b 
|- B = ( Base ` G )
ablcom.p 
|- .+ = ( +g ` G )
Assertion cmn12 
|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )

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 /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
9 8 3adant3r3 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ Y ) = ( Y .+ X ) )
10 1 2 4 5 6 7 9 mnd12g 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )