Theorem cmncom 13839

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 13830	. . . . 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 2580	. . . . 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 6161	. 2 |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) ) -> ( X .+ Y ) = ( Y .+ X ) )
9	8	3impb 1223	1 |- ( ( G e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   Mndcmnd 13449  CMndccmn 13821

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004  df-cmn 13823

This theorem is referenced by:  ablcom  13840  cmn32  13841  cmn4  13842  cmn12  13843  rinvmod  13846  ghmcmn  13864  subcmnd  13870  gsumfzreidx  13874  gsumfzmptfidmadd  13876  srgcom  13946  crngcom  13977