Theorem iscmn 13423

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 5558	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
2		iscmn.b	. . . . 5 |- B = ( Base ` G )
3	1, 2	eqtr4di 2247	. . . 4 |- ( g = G -> ( Base ` g ) = B )
4		raleq 2693	. . . . 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 2705	. . . 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 5558	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		iscmn.p	. . . . . . 7 |- .+ = ( +g ` G )
9	7, 8	eqtr4di 2247	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
10	9	oveqd 5939	. . . . 5 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
11	9	oveqd 5939	. . . . 5 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
12	10, 11	eqeq12d 2211	. . . 4 |- ( g = G -> ( ( x ( +g ` g ) y ) = ( y ( +g ` g ) x ) <-> ( x .+ y ) = ( y .+ x ) ) )
13	12	2ralbidv 2521	. . 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 13416	. 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 2923	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 1364   e. wcel 2167   A.wral 2475   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   Mndcmnd 13057  CMndccmn 13414

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-cmn 13416

This theorem is referenced by:  isabl2  13424  cmnpropd  13425  iscmnd  13428  cmnmnd  13431  cmncom  13432  ghmcmn  13457  iscrng2  13571