Theorem iscmnd 13634

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 1209	. . . 4 |- ( ph -> ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) ) )
4	3	ralrimivv 2587	. . 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 5961	. . . . . . 7 |- ( ph -> ( x .+ y ) = ( x ( +g ` G ) y ) )
8	6	oveqd 5961	. . . . . . 7 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
9	7, 8	eqeq12d 2220	. . . . . 6 |- ( ph -> ( ( x .+ y ) = ( y .+ x ) <-> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
10	5, 9	raleqbidv 2718	. . . . 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 2718	. . . 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 946	. 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 2205	. . 3 |- ( Base ` G ) = ( Base ` G )
15		eqid 2205	. . 3 |- ( +g ` G ) = ( +g ` G )
16	14, 15	iscmn 13629	. 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 981   = wceq 1373   e. wcel 2176   A.wral 2484   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   Mndcmnd 13248  CMndccmn 13620

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947  df-cmn 13622

This theorem is referenced by:  isabld  13635  subcmnd  13669  iscrngd  13804