Theorem mgm2mgm 41863

Description: Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)

Assertion

Ref	Expression
mgm2mgm	|- ( M e. MgmALT <-> M e. Mgm )

Proof of Theorem mgm2mgm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` M ) = ( Base ` M )
2		eqid 2622	. . . . 5 |- ( +g ` M ) = ( +g ` M )
3	1, 2	ismgmALT 41859	. . . 4 |- ( M e. MgmALT -> ( M e. MgmALT <-> ( +g ` M ) clLaw ( Base ` M ) ) )
4		fvex 6201	. . . . . 6 |- ( +g ` M ) e. _V
5		fvex 6201	. . . . . 6 |- ( Base ` M ) e. _V
6		iscllaw 41825	. . . . . 6 |- ( ( ( +g ` M ) e. _V /\ ( Base ` M ) e. _V ) -> ( ( +g ` M ) clLaw ( Base ` M ) <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) ) )
7	4, 5, 6	mp2an 708	. . . . 5 |- ( ( +g ` M ) clLaw ( Base ` M ) <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) )
8	1, 2	ismgm 17243	. . . . . 6 |- ( M e. MgmALT -> ( M e. Mgm <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) ) )
9	8	biimprd 238	. . . . 5 |- ( M e. MgmALT -> ( A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) -> M e. Mgm ) )
10	7, 9	syl5bi 232	. . . 4 |- ( M e. MgmALT -> ( ( +g ` M ) clLaw ( Base ` M ) -> M e. Mgm ) )
11	3, 10	sylbid 230	. . 3 |- ( M e. MgmALT -> ( M e. MgmALT -> M e. Mgm ) )
12	11	pm2.43i 52	. 2 |- ( M e. MgmALT -> M e. Mgm )
13		mgmplusgiopALT 41830	. . 3 |- ( M e. Mgm -> ( +g ` M ) clLaw ( Base ` M ) )
14	1, 2	ismgmALT 41859	. . 3 |- ( M e. Mgm -> ( M e. MgmALT <-> ( +g ` M ) clLaw ( Base ` M ) ) )
15	13, 14	mpbird 247	. 2 |- ( M e. Mgm -> M e. MgmALT )
16	12, 15	impbii 199	1 |- ( M e. MgmALT <-> M e. Mgm )

Syntax hints:   <-> wb 196   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240   clLaw ccllaw 41819  MgmALTcmgm2 41851

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-sep 4781  ax-nul 4789  ax-pr 4906

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-mo 2475  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-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242  df-cllaw 41822  df-mgm2 41855

This theorem is referenced by:  sgrp2sgrp  41864