Theorem opifismgmdc 13453

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 3643	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> if ( ps , C , D ) e. B )
5	4	ralrimivva 2614	. . . . 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 6371	. . . 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 1273	. . 3 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` M ) b ) e. B )
12	11	ralrimivva 2614	. 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 2231	. . . . 5 |- ( +g ` M ) = ( +g ` M )
16	14, 15	ismgmn0 13440	. . . 4 |- ( x e. B -> ( M e. Mgm <-> A. a e. B A. b e. B ( a ( +g ` M ) b ) e. B ) )
17	16	exlimiv 1646	. . 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 841   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   ifcif 3605   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   Basecbs 13081   +g cplusg 13159  Mgmcmgm 13436

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-mgm 13438

This theorem is referenced by: (None)