Theorem cmncom 14055

Description: A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
ablcom.b	|- B = ( Base ` G )
ablcom.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
cmncom	|- ( ( G e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )

Proof of Theorem cmncom

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablcom.b	. . . . . 6 |- B = ( Base ` G )
2		ablcom.p	. . . . . 6 |- .+ = ( +g ` G )
3	1, 2	iscmn 14046	. . . . 5 |- ( G e. CMnd <-> ( G e. Mnd /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
4	3	simprbi 275	. . . 4 |- ( G e. CMnd -> A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) )
5		rsp2 2594	. . . . 5 |- ( A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) -> ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) ) )
6	5	imp 124	. . . 4 |- ( ( A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) = ( y .+ x ) )
7	4, 6	sylan 283	. . 3 |- ( ( G e. CMnd /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) = ( y .+ x ) )
8	7	caovcomg 6218	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) ) -> ( X .+ Y ) = ( Y .+ X ) )
9	8	3impb 1226	1 |- ( ( G e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   Mndcmnd 13677  CMndccmn 14037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-cmn 14039

This theorem is referenced by:  ablcom  14056  cmn32  14057  cmn4  14058  cmn12  14059  rinvmod  14062  ghmcmn  14080  subcmnd  14086  gsumfzreidx  14090  gsumfzmptfidmadd  14092  gfsumval  14102  srgcom  14226  crngcom  14257