Theorem opifismgmdc 13576

Description: A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)

Hypotheses

Ref	Expression
opifismgm.b	|- B = ( Base ` M )
opifismgm.p	|- ( +g ` M ) = ( x e. B , y e. B |-> if ( ps , C , D ) )
opifismgmdc.dc	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> DECID ps )
opifismgm.m	|- ( ph -> E. x x e. B )
opifismgm.c	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> C e. B )
opifismgm.d	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> D e. B )

Assertion

Ref	Expression
opifismgmdc	|- ( ph -> M e. Mgm )

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

Allowed substitution hints:   ps( x, y)   C( x, y)   D( x, y)   M( y)

Proof of Theorem opifismgmdc

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opifismgm.c	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> C e. B )
2		opifismgm.d	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> D e. B )
3		opifismgmdc.dc	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> DECID ps )
4	1, 2, 3	ifcldcd 3659	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> if ( ps , C , D ) e. B )
5	4	ralrimivva 2624	. . . . 5 |- ( ph -> A. x e. B A. y e. B if ( ps , C , D ) e. B )
6	5	adantr 276	. . . 4 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> A. x e. B A. y e. B if ( ps , C , D ) e. B )
7		simprl 531	. . . 4 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> a e. B )
8		simprr 533	. . . 4 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> b e. B )
9		opifismgm.p	. . . . 5 |- ( +g ` M ) = ( x e. B , y e. B |-> if ( ps , C , D ) )
10	9	ovmpoelrn 6402	. . . 4 |- ( ( A. x e. B A. y e. B if ( ps , C , D ) e. B /\ a e. B /\ b e. B ) -> ( a ( +g ` M ) b ) e. B )
11	6, 7, 8, 10	syl3anc 1274	. . 3 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` M ) b ) e. B )
12	11	ralrimivva 2624	. 2 |- ( ph -> A. a e. B A. b e. B ( a ( +g ` M ) b ) e. B )
13		opifismgm.m	. . 3 |- ( ph -> E. x x e. B )
14		opifismgm.b	. . . . 5 |- B = ( Base ` M )
15		eqid 2232	. . . . 5 |- ( +g ` M ) = ( +g ` M )
16	14, 15	ismgmn0 13563	. . . 4 |- ( x e. B -> ( M e. Mgm <-> A. a e. B A. b e. B ( a ( +g ` M ) b ) e. B ) )
17	16	exlimiv 1647	. . 3 |- ( E. x x e. B -> ( M e. Mgm <-> A. a e. B A. b e. B ( a ( +g ` M ) b ) e. B ) )
18	13, 17	syl 14	. 2 |- ( ph -> ( M e. Mgm <-> A. a e. B A. b e. B ( a ( +g ` M ) b ) e. B ) )
19	12, 18	mpbird 167	1 |- ( ph -> M e. Mgm )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   ifcif 3619   ` cfv 5351  (class class class)co 6049   e. cmpo 6051   Basecbs 13204   +g cplusg 13282  Mgmcmgm 13559

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-in2 620  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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-inn 9237  df-2 9295  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-mgm 13561

This theorem is referenced by: (None)