Theorem ismgm 13000

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 12736	. . . . 5 |- Base Fn _V
2		vex 2766	. . . . 5 |- m e. _V
3		funfvex 5575	. . . . . 6 |- ( ( Fun Base /\ m e. dom Base ) -> ( Base ` m ) e. _V )
4	3	funfni 5358	. . . . 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 5558	. . . 4 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
8		ismgm.b	. . . 4 |- B = ( Base ` M )
9	7, 8	eqtr4di 2247	. . 3 |- ( m = M -> ( Base ` m ) = B )
10		plusgslid 12790	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
11	10	slotex 12705	. . . . . 6 |- ( m e. _V -> ( +g ` m ) e. _V )
12	11	elv 2767	. . . . 5 |- ( +g ` m ) e. _V
13	12	a1i 9	. . . 4 |- ( ( m = M /\ b = B ) -> ( +g ` m ) e. _V )
14		fveq2 5558	. . . . . 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 2247	. . . 4 |- ( ( m = M /\ b = B ) -> ( +g ` m ) = .o. )
18		simplr 528	. . . . 5 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> b = B )
19		oveq 5928	. . . . . . . 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 2267	. . . . . 6 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> ( ( x o y ) e. b <-> ( x .o. y ) e. B ) )
22	18, 21	raleqbidv 2709	. . . . 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 2709	. . . 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 3027	. . 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 3027	. 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 12999	. 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 2911	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 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   [.wsbc 2989   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755  Mgmcmgm 12997

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mgm 12999

This theorem is referenced by:  ismgmn0  13001  mgmcl  13002  mgm0  13012  issgrpv  13047  rnglidlmmgm  14052