Theorem ismgm 13439

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 13140	. . . . 5 |- Base Fn _V
2		vex 2805	. . . . 5 |- m e. _V
3		funfvex 5656	. . . . . 6 |- ( ( Fun Base /\ m e. dom Base ) -> ( Base ` m ) e. _V )
4	3	funfni 5432	. . . . 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 5639	. . . 4 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
8		ismgm.b	. . . 4 |- B = ( Base ` M )
9	7, 8	eqtr4di 2282	. . 3 |- ( m = M -> ( Base ` m ) = B )
10		plusgslid 13194	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
11	10	slotex 13108	. . . . . 6 |- ( m e. _V -> ( +g ` m ) e. _V )
12	11	elv 2806	. . . . 5 |- ( +g ` m ) e. _V
13	12	a1i 9	. . . 4 |- ( ( m = M /\ b = B ) -> ( +g ` m ) e. _V )
14		fveq2 5639	. . . . . 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 2282	. . . 4 |- ( ( m = M /\ b = B ) -> ( +g ` m ) = .o. )
18		simplr 529	. . . . 5 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> b = B )
19		oveq 6023	. . . . . . . 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 2302	. . . . . 6 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> ( ( x o y ) e. b <-> ( x .o. y ) e. B ) )
22	18, 21	raleqbidv 2746	. . . . 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 2746	. . . 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 3069	. . 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 3069	. 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 13438	. 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 2953	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 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   [.wsbc 3031   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   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-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-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-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  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-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  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:  ismgmn0  13440  mgmcl  13441  mgm0  13451  issgrpv  13486  rnglidlmmgm  14509