Theorem iscmn 13830

Description: The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
iscmn	|- ( G e. CMnd <-> ( G e. Mnd /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )

Distinct variable groups:   x, y, B   x, G, y

Allowed substitution hints:   .+ ( x, y)

Proof of Theorem iscmn

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5627	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		iscmn.b	. . . . 5 |- B = ( Base ` G )
3	1, 2	eqtr4di 2280	. . . 4 |- ( g = G -> ( Base ` g ) = B )
4		raleq 2728	. . . . 5 |- ( ( Base ` g ) = B -> ( A. y e. ( Base ` g ) ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) <-> A. y e. B ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) ) )
5	4	raleqbi1dv 2740	. . . 4 |- ( ( Base ` g ) = B -> ( A. x e. ( Base ` g ) A. y e. ( Base ` g ) ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) <-> A. x e. B A. y e. B ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) ) )
6	3, 5	syl 14	. . 3 |- ( g = G -> ( A. x e. ( Base ` g ) A. y e. ( Base ` g ) ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) <-> A. x e. B A. y e. B ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) ) )
7		fveq2 5627	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		iscmn.p	. . . . . . 7 |- .+ = ( +g ` G )
9	7, 8	eqtr4di 2280	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
10	9	oveqd 6018	. . . . 5 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
11	9	oveqd 6018	. . . . 5 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
12	10, 11	eqeq12d 2244	. . . 4 |- ( g = G -> ( ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) <-> ( x .+ y ) = ( y .+ x ) ) )
13	12	2ralbidv 2554	. . 3 |- ( g = G -> ( A. x e. B A. y e. B ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) <-> A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
14	6, 13	bitrd 188	. 2 |- ( g = G -> ( A. x e. ( Base ` g ) A. y e. ( Base ` g ) ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) <-> A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
15		df-cmn 13823	. 2 |- CMnd = { g e. Mnd | A. x e. ( Base ` g ) A. y e. ( Base ` g ) ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) }
16	14, 15	elrab2 2962	1 |- ( G e. CMnd <-> ( G e. Mnd /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = 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:  isabl2  13831  cmnpropd  13832  iscmnd  13835  cmnmnd  13838  cmncom  13839  ghmcmn  13864  iscrng2  13978