Theorem iscmnd 13428

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 1208	. . . 4 |- ( ph -> ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) ) )
4	3	ralrimivv 2578	. . 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 5939	. . . . . . 7 |- ( ph -> ( x .+ y ) = ( x ( +g ` G ) y ) )
8	6	oveqd 5939	. . . . . . 7 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
9	7, 8	eqeq12d 2211	. . . . . 6 |- ( ph -> ( ( x .+ y ) = ( y .+ x ) <-> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
10	5, 9	raleqbidv 2709	. . . . 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 2709	. . . 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 945	. 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 2196	. . 3 |- ( Base ` G ) = ( Base ` G )
15		eqid 2196	. . 3 |- ( +g ` G ) = ( +g ` G )
16	14, 15	iscmn 13423	. 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 980   = 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:  isabld  13429  subcmnd  13463  iscrngd  13598