Theorem ismgm 17243

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		fvexd 6203	. . 3 |- ( m = M -> ( Base ` m ) e. _V )
2		fveq2 6191	. . . 4 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
3		ismgm.b	. . . 4 |- B = ( Base ` M )
4	2, 3	syl6eqr 2674	. . 3 |- ( m = M -> ( Base ` m ) = B )
5		fvexd 6203	. . . 4 |- ( ( m = M /\ b = B ) -> ( +g ` m ) e. _V )
6		fveq2 6191	. . . . . 6 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
7	6	adantr 481	. . . . 5 |- ( ( m = M /\ b = B ) -> ( +g ` m ) = ( +g ` M ) )
8		ismgm.o	. . . . 5 |- .o. = ( +g ` M )
9	7, 8	syl6eqr 2674	. . . 4 |- ( ( m = M /\ b = B ) -> ( +g ` m ) = .o. )
10		simplr 792	. . . . 5 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> b = B )
11		oveq 6656	. . . . . . . 8 |- ( o = .o. -> ( x o y ) = ( x .o. y ) )
12	11	adantl 482	. . . . . . 7 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> ( x o y ) = ( x .o. y ) )
13	12, 10	eleq12d 2695	. . . . . 6 |- ( ( ( m = M /\ b = B ) /\ o = .o. ) -> ( ( x o y ) e. b <-> ( x .o. y ) e. B ) )
14	10, 13	raleqbidv 3152	. . . . 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 ) )
15	10, 14	raleqbidv 3152	. . . 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 ) )
16	5, 9, 15	sbcied2 3473	. . 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 ) )
17	1, 4, 16	sbcied2 3473	. 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 ) )
18		df-mgm 17242	. 2 |- Mgm = { m | [. ( Base ` m ) / b ]. [. ( +g ` m ) / o ]. A. x e. b A. y e. b ( x o y ) e. b }
19	17, 18	elab2g 3353	1 |- ( M e. V -> ( M e. Mgm <-> A. x e. B A. y e. B ( x .o. y ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242

This theorem is referenced by:  ismgmn0  17244  mgmcl  17245  issstrmgm  17252  mgm0  17255  issgrpv  17286  0mgm  41774  ismgmd  41776  mgm2mgm  41863  lidlmmgm  41925