Theorem cmncom 13036

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 13027	. . . . 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 2527	. . . . 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 6027	. 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 2148   A.wral 2455   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   Mndcmnd 12749  CMndccmn 13019

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-ov 5875  df-cmn 13021

This theorem is referenced by:  ablcom  13037  cmn32  13038  cmn4  13039  cmn12  13040  rinvmod  13043  subcmnd  13060  srgcom  13097  crngcom  13128