Theorem ismgm 13587

Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)

Hypotheses

Ref	Expression
ismgm.b	|- B = ( Base ` M )
ismgm.o	|- .o. = ( +g ` M )

Assertion

Ref	Expression
ismgm	|- ( M e. V -> ( M e. Mgm <-> A. x e. B A. y e. B ( x .o. y ) e. B ) )

Distinct variable groups:   x, B, y   x, M, y   x, .o. , y

Allowed substitution hints:   V( x, y)

Proof of Theorem ismgm

Dummy variables b m o are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 13288	. . . . 5 |- Base Fn _V
2		vex 2818	. . . . 5 |- m e. _V
3		funfvex 5689	. . . . . 6 |- ( ( Fun Base /\ m e. dom Base ) -> ( Base ` m ) e. _V )
4	3	funfni 5460	. . . . 5 |- ( ( Base Fn _V /\ m e. _V ) -> ( Base ` m ) e. _V )
5	1, 2, 4	mp2an 426	. . . 4 |- ( Base ` m ) e. _V
6	5	a1i 9	. . 3 |- ( m = M -> ( Base ` m ) e. _V )
7		fveq2 5672	. . . 4 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
8		ismgm.b	. . . 4 |- B = ( Base ` M )
9	7, 8	eqtr4di 2285	. . 3 |- ( m = M -> ( Base ` m ) = B )
10		plusgslid 13342	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
11	10	slotex 13256	. . . . . 6 |- ( m e. _V -> ( +g ` m ) e. _V )
12	11	elv 2819	. . . . 5 |- ( +g ` m ) e. _V
13	12	a1i 9	. . . 4 |- ( ( m = M /\ b = B ) -> ( +g ` m ) e. _V )
14		fveq2 5672	. . . . . 6 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
15	14	adantr 276	. . . . 5 |- ( ( m = M /\ b = B ) -> ( +g ` m ) = ( +g ` M ) )
16		ismgm.o	. . . . 5 |- .o. = ( +g ` M )
17	15, 16	eqtr4di 2285	. . . 4 |- ( ( m = M /\ b = B ) -> ( +g ` m ) = .o. )
18		simplr 529	. . . . 5 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> b = B )
19		oveq 6058	. . . . . . . 8 |- ( o = .o. -> ( x o y ) = ( x .o. y ) )
20	19	adantl 277	. . . . . . 7 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> ( x o y ) = ( x .o. y ) )
21	20, 18	eleq12d 2305	. . . . . 6 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> ( ( x o y ) e. b <-> ( x .o. y ) e. B ) )
22	18, 21	raleqbidv 2759	. . . . 5 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> ( A. y e. b ( x o y ) e. b <-> A. y e. B ( x .o. y ) e. B ) )
23	18, 22	raleqbidv 2759	. . . 4 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> ( A. x e. b A. y e. b ( x o y ) e. b <-> A. x e. B A. y e. B ( x .o. y ) e. B ) )
24	13, 17, 23	sbcied2 3082	. . 3 |- ( ( m = M /\ b = B ) -> ( [. ( +g ` m ) / o ]. A. x e. b A. y e. b ( x o y ) e. b <-> A. x e. B A. y e. B ( x .o. y ) e. B ) )
25	6, 9, 24	sbcied2 3082	. 2 |- ( m = M -> ( [. ( Base ` m ) / b ]. [. ( +g ` m ) / o ]. A. x e. b A. y e. b ( x o y ) e. b <-> A. x e. B A. y e. B ( x .o. y ) e. B ) )
26		df-mgm 13586	. 2 |- Mgm = { m | [. ( Base ` m ) / b ]. [. ( +g ` m ) / o ]. A. x e. b A. y e. b ( x o y ) e. b }
27	25, 26	elab2g 2966	1 |- ( M e. V -> ( M e. Mgm <-> A. x e. B A. y e. B ( x .o. y ) e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   [.wsbc 3044   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307  Mgmcmgm 13584

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-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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-ov 6055  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-plusg 13320  df-mgm 13586

This theorem is referenced by:  ismgmn0  13588  mgmcl  13589  mgm0  13599  issgrpv  13634  rnglidlmmgm  14661