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Metamath Proof Explorer

Theorem mnd12g

Description: Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses mndcl.b 
|- B = ( Base ` G )
mndcl.p 
|- .+ = ( +g ` G )
mnd4g.1 
|- ( ph -> G e. Mnd )
mnd4g.2 
|- ( ph -> X e. B )
mnd4g.3 
|- ( ph -> Y e. B )
mnd4g.4 
|- ( ph -> Z e. B )
mnd12g.5 
|- ( ph -> ( X .+ Y ) = ( Y .+ X ) )
Assertion mnd12g 
|- ( ph -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )

Proof

Step Hyp Ref Expression
1 mndcl.b 
 |-  B = ( Base ` G )
2 mndcl.p 
 |-  .+ = ( +g ` G )
3 mnd4g.1 
 |-  ( ph -> G e. Mnd )
4 mnd4g.2 
 |-  ( ph -> X e. B )
5 mnd4g.3 
 |-  ( ph -> Y e. B )
6 mnd4g.4 
 |-  ( ph -> Z e. B )
7 mnd12g.5 
 |-  ( ph -> ( X .+ Y ) = ( Y .+ X ) )
8 7 oveq1d 
 |-  ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( Y .+ X ) .+ Z ) )
9 1 2 mndass 
 |-  ( ( G e. Mnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
10 3 4 5 6 9 syl13anc 
 |-  ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
11 1 2 mndass 
 |-  ( ( G e. Mnd /\ ( Y e. B /\ X e. B /\ Z e. B ) ) -> ( ( Y .+ X ) .+ Z ) = ( Y .+ ( X .+ Z ) ) )
12 3 5 4 6 11 syl13anc 
 |-  ( ph -> ( ( Y .+ X ) .+ Z ) = ( Y .+ ( X .+ Z ) ) )
13 8 10 12 3eqtr3d 
 |-  ( ph -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )