Theorem iscmnd 13965

Description: Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
iscmnd.b	|- ( ph -> B = ( Base ` G ) )
iscmnd.p	|- ( ph -> .+ = ( +g ` G ) )
iscmnd.g	|- ( ph -> G e. Mnd )
iscmnd.c	|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )

Assertion

Ref	Expression
iscmnd	|- ( ph -> G e. CMnd )

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

Allowed substitution hints:   .+ ( x, y)

Proof of Theorem iscmnd

Step	Hyp	Ref	Expression
1		iscmnd.g	. . 3 |- ( ph -> G e. Mnd )
2		iscmnd.c	. . . . 5 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )
3	2	3expib 1233	. . . 4 |- ( ph -> ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) ) )
4	3	ralrimivv 2614	. . 3 |- ( ph -> A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) )
5		iscmnd.b	. . . . 5 |- ( ph -> B = ( Base ` G ) )
6		iscmnd.p	. . . . . . . 8 |- ( ph -> .+ = ( +g ` G ) )
7	6	oveqd 6045	. . . . . . 7 |- ( ph -> ( x .+ y ) = ( x ( +g ` G ) y ) )
8	6	oveqd 6045	. . . . . . 7 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
9	7, 8	eqeq12d 2246	. . . . . 6 |- ( ph -> ( ( x .+ y ) = ( y .+ x ) <-> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
10	5, 9	raleqbidv 2747	. . . . 5 |- ( ph -> ( A. y e. B ( x .+ y ) = ( y .+ x ) <-> A. y e. ( Base ` G ) ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
11	5, 10	raleqbidv 2747	. . . 4 |- ( ph -> ( A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) <-> A. x e. ( Base ` G ) A. y e. ( Base ` G ) ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
12	11	anbi2d 464	. . 3 |- ( ph -> ( ( G e. Mnd /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) <-> ( G e. Mnd /\ A. x e. ( Base ` G ) A. y e. ( Base ` G ) ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) ) )
13	1, 4, 12	mpbi2and 952	. 2 |- ( ph -> ( G e. Mnd /\ A. x e. ( Base ` G ) A. y e. ( Base ` G ) ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
14		eqid 2231	. . 3 |- ( Base ` G ) = ( Base ` G )
15		eqid 2231	. . 3 |- ( +g ` G ) = ( +g ` G )
16	14, 15	iscmn 13960	. 2 |- ( G e. CMnd <-> ( G e. Mnd /\ A. x e. ( Base ` G ) A. y e. ( Base ` G ) ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
17	13, 16	sylibr 134	1 |- ( ph -> G e. CMnd )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   Mndcmnd 13579  CMndccmn 13951

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 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031  df-cmn 13953

This theorem is referenced by:  isabld  13966  subcmnd  14000  iscrngd  14136