Theorem opifismgmdc 13691

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 3678	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> if ( ps , C , D ) e. B )
5	4	ralrimivva 2632	. . . . 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 535	. . . 4 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> a e. B )
8		simprr 537	. . . 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 6443	. . . 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 1278	. . 3 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` M ) b ) e. B )
12	11	ralrimivva 2632	. 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 2238	. . . . 5 |- ( +g ` M ) = ( +g ` M )
16	14, 15	ismgmn0 13678	. . . 4 |- ( x e. B -> ( M e. Mgm <-> A. a e. B A. b e. B ( a ( +g ` M ) b ) e. B ) )
17	16	exlimiv 1651	. . 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   ifcif 3638   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   Basecbs 13352   +g cplusg 13431  Mgmcmgm 13674

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-mgm 13676

This theorem is used by: (None)