Theorem cmncom 12901

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 12892	. . . . 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 2525	. . . . 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 6020	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) ) -> ( X .+ Y ) = ( Y .+ X ) )
9	8	3impb 1199	1 |- ( ( G e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2146   A.wral 2453   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   Mndcmnd 12682  CMndccmn 12884

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868  df-cmn 12886

This theorem is referenced by:  ablcom  12902  cmn32  12903  cmn4  12904  cmn12  12905  rinvmod  12908  srgcom  12959