Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgm2mgm Structured version   Unicode version

Theorem mgm2mgm 32923
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 2454 . . . . 5  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2454 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
31, 2ismgmALT 32919 . . . 4  |-  ( M  e. MgmALT  ->  ( M  e. MgmALT  <->  ( +g  `  M ) clLaw 
( Base `  M )
) )
4 fvex 5858 . . . . . 6  |-  ( +g  `  M )  e.  _V
5 fvex 5858 . . . . . 6  |-  ( Base `  M )  e.  _V
6 iscllaw 32885 . . . . . 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 670 . . . . 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 16072 . . . . . 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 32890 . . 3  |-  ( M  e. Mgm  ->  ( +g  `  M
) clLaw  ( Base `  M
) )
141, 2ismgmALT 32919 . . 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 1823   A.wral 2804   _Vcvv 3106   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784  Mgmcmgm 16069   clLaw ccllaw 32879  MgmALTcmgm2 32911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-iota 5534  df-fv 5578  df-ov 6273  df-mgm 16071  df-cllaw 32882  df-mgm2 32915
This theorem is referenced by:  sgrp2sgrp  32924
  Copyright terms: Public domain W3C validator