Theorem iscmn 14046

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 5675	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		iscmn.b	. . . . 5 |- B = ( Base ` G )
3	1, 2	eqtr4di 2285	. . . 4 |- ( g = G -> ( Base ` g ) = B )
4		raleq 2743	. . . . 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 2755	. . . 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 5675	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		iscmn.p	. . . . . . 7 |- .+ = ( +g ` G )
9	7, 8	eqtr4di 2285	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
10	9	oveqd 6075	. . . . 5 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
11	9	oveqd 6075	. . . . 5 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
12	10, 11	eqeq12d 2249	. . . 4 |- ( g = G -> ( ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) <-> ( x .+ y ) = ( y .+ x ) ) )
13	12	2ralbidv 2568	. . 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 14039	. 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 2979	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 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:  isabl2  14047  cmnpropd  14048  iscmnd  14051  cmnmnd  14054  cmncom  14055  ghmcmn  14080  iscrng2  14258