Theorem opifismgmdc 12602

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 3555	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> if ( ps , C , D ) e. B )
5	4	ralrimivva 2548	. . . . 5 |- ( ph -> A. x e. B A. y e. B if ( ps , C , D ) e. B )
6	5	adantr 274	. . . 4 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> A. x e. B A. y e. B if ( ps , C , D ) e. B )
7		simprl 521	. . . 4 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> a e. B )
8		simprr 522	. . . 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 6175	. . . 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 1228	. . 3 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` M ) b ) e. B )
12	11	ralrimivva 2548	. 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 2165	. . . . 5 |- ( +g ` M ) = ( +g ` M )
16	14, 15	ismgmn0 12589	. . . 4 |- ( x e. B -> ( M e. Mgm <-> A. a e. B A. b e. B ( a ( +g ` M ) b ) e. B ) )
17	16	exlimiv 1586	. . 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 166	1 |- ( ph -> M e. Mgm )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 824   = wceq 1343   E.wex 1480   e. wcel 2136   A.wral 2444   ifcif 3520   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   Basecbs 12394   +g cplusg 12457  Mgmcmgm 12585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-inn 8858  df-2 8916  df-ndx 12397  df-slot 12398  df-base 12400  df-plusg 12470  df-mgm 12587

This theorem is referenced by: (None)