Theorem iscmnd 18205

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 1268	. . . 4 |- ( ph -> ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) ) )
4	3	ralrimivv 2970	. . 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 6667	. . . . . . 7 |- ( ph -> ( x .+ y ) = ( x ( +g ` G ) y ) )
8	6	oveqd 6667	. . . . . . 7 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
9	7, 8	eqeq12d 2637	. . . . . 6 |- ( ph -> ( ( x .+ y ) = ( y .+ x ) <-> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
10	5, 9	raleqbidv 3152	. . . . 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 3152	. . . 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 740	. . 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 956	. 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 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
15		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
16	14, 15	iscmn 18200	. 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 224	1 |- ( ph -> G e. CMnd )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cmn 18195

This theorem is referenced by:  isabld  18206  subcmn  18242  prdscmnd  18264  iscrngd  18586  psrcrng  19413  xrsmcmn  19769  2zrngacmnd  41942