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Theorem mgm2mgm 32388
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.
StepHypRef Expression
1 eqid 2443 . . . . 5  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2443 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
31, 2ismgmALT 32384 . . . 4  |-  ( M  e. MgmALT  ->  ( M  e. MgmALT  <->  ( +g  `  M ) clLaw 
( Base `  M )
) )
4 fvex 5866 . . . . . 6  |-  ( +g  `  M )  e.  _V
5 fvex 5866 . . . . . 6  |-  ( Base `  M )  e.  _V
6 iscllaw 32350 . . . . . 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 )
) )
74, 5, 6mp2an 672 . . . . 5  |-  ( ( +g  `  M ) clLaw 
( Base `  M )  <->  A. x  e.  ( Base `  M ) A. y  e.  ( Base `  M
) ( x ( +g  `  M ) y )  e.  (
Base `  M )
)
81, 2ismgm 15851 . . . . . 6  |-  ( M  e. MgmALT  ->  ( M  e. Mgm  <->  A. x  e.  ( Base `  M ) A. y  e.  ( Base `  M
) ( x ( +g  `  M ) y )  e.  (
Base `  M )
) )
98biimprd 223 . . . . 5  |-  ( M  e. MgmALT  ->  ( A. x  e.  ( Base `  M
) A. y  e.  ( Base `  M
) ( x ( +g  `  M ) y )  e.  (
Base `  M )  ->  M  e. Mgm ) )
107, 9syl5bi 217 . . . 4  |-  ( M  e. MgmALT  ->  ( ( +g  `  M ) clLaw  ( Base `  M )  ->  M  e. Mgm ) )
113, 10sylbid 215 . . 3  |-  ( M  e. MgmALT  ->  ( M  e. MgmALT  ->  M  e. Mgm ) )
1211pm2.43i 47 . 2  |-  ( M  e. MgmALT  ->  M  e. Mgm )
13 mgmplusgiopALT 32355 . . 3  |-  ( M  e. Mgm  ->  ( +g  `  M
) clLaw  ( Base `  M
) )
141, 2ismgmALT 32384 . . 3  |-  ( M  e. Mgm  ->  ( M  e. MgmALT  <->  ( +g  `  M ) clLaw 
( Base `  M )
) )
1513, 14mpbird 232 . 2  |-  ( M  e. Mgm  ->  M  e. MgmALT )
1612, 15impbii 188 1  |-  ( M  e. MgmALT 
<->  M  e. Mgm )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1804   A.wral 2793   _Vcvv 3095   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14613   +g cplusg 14678  Mgmcmgm 15848   clLaw ccllaw 32344  MgmALTcmgm2 32376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-iota 5541  df-fv 5586  df-ov 6284  df-mgm 15850  df-cllaw 32347  df-mgm2 32380
This theorem is referenced by:  sgrp2sgrp  32389
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