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Metamath Proof Explorer

Theorem cmn4

Description: Commutative/associative law for commutative monoids. (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 cmn4 
|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )

Proof

Step Hyp Ref Expression
1 ablcom.b 
 |-  B = ( Base ` G )
2 ablcom.p 
 |-  .+ = ( +g ` G )
3 simp1 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> G e. CMnd )
4 cmnmnd 
 |-  ( G e. CMnd -> G e. Mnd )
5 3 4 syl 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> G e. Mnd )
6 simp2l 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> X e. B )
7 simp2r 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> Y e. B )
8 simp3l 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> Z e. B )
9 simp3r 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> W e. B )
10 1 2 cmncom 
 |-  ( ( G e. CMnd /\ Y e. B /\ Z e. B ) -> ( Y .+ Z ) = ( Z .+ Y ) )
11 3 7 8 10 syl3anc 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( Y .+ Z ) = ( Z .+ Y ) )
12 1 2 5 6 7 8 9 11 mnd4g 
 |-  ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )