Theorem opifismgmdc 12625

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 3561	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> if ( ps , C , D ) e. B )
5	4	ralrimivva 2552	. . . . 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 526	. . . 4 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> a e. B )
8		simprr 527	. . . 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 6186	. . . 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 1233	. . 3 |- ( ( ph /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` M ) b ) e. B )
12	11	ralrimivva 2552	. 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 2170	. . . . 5 |- ( +g ` M ) = ( +g ` M )
16	14, 15	ismgmn0 12612	. . . 4 |- ( x e. B -> ( M e. Mgm <-> A. a e. B A. b e. B ( a ( +g ` M ) b ) e. B ) )
17	16	exlimiv 1591	. . 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 829   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   ifcif 3526   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   Basecbs 12416   +g cplusg 12480  Mgmcmgm 12608

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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-inn 8879  df-2 8937  df-ndx 12419  df-slot 12420  df-base 12422  df-plusg 12493  df-mgm 12610

This theorem is referenced by: (None)