Theorem iscmnd 13749

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 2589	. . 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 5984	. . . . . . 7 |- ( ph -> ( x .+ y ) = ( x ( +g ` G ) y ) )
8	6	oveqd 5984	. . . . . . 7 |- ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
9	7, 8	eqeq12d 2222	. . . . . 6 |- ( ph -> ( ( x .+ y ) = ( y .+ x ) <-> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
10	5, 9	raleqbidv 2721	. . . . 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 2721	. . . 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 2207	. . 3 |- ( Base ` G ) = ( Base ` G )
15		eqid 2207	. . 3 |- ( +g ` G ) = ( +g ` G )
16	14, 15	iscmn 13744	. 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 2178   A.wral 2486   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   Mndcmnd 13363  CMndccmn 13735

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970  df-cmn 13737

This theorem is referenced by:  isabld  13750  subcmnd  13784  iscrngd  13919