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Theorem ismgm 15999
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.
StepHypRef Expression
1 fvex 5882 . . . 4  |-  ( Base `  m )  e.  _V
21a1i 11 . . 3  |-  ( m  =  M  ->  ( Base `  m )  e. 
_V )
3 fveq2 5872 . . . 4  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
4 ismgm.b . . . 4  |-  B  =  ( Base `  M
)
53, 4syl6eqr 2516 . . 3  |-  ( m  =  M  ->  ( Base `  m )  =  B )
6 fvex 5882 . . . . 5  |-  ( +g  `  m )  e.  _V
76a1i 11 . . . 4  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  e.  _V )
8 fveq2 5872 . . . . . 6  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
98adantr 465 . . . . 5  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  =  ( +g  `  M ) )
10 ismgm.o . . . . 5  |-  .o.  =  ( +g  `  M )
119, 10syl6eqr 2516 . . . 4  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  =  .o.  )
12 simplr 755 . . . . 5  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  b  =  B )
13 oveq 6302 . . . . . . . 8  |-  ( o  =  .o.  ->  (
x o y )  =  ( x  .o.  y ) )
1413adantl 466 . . . . . . 7  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  (
x o y )  =  ( x  .o.  y ) )
1514, 12eleq12d 2539 . . . . . 6  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  (
( x o y )  e.  b  <->  ( x  .o.  y )  e.  B
) )
1612, 15raleqbidv 3068 . . . . 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
) )
1712, 16raleqbidv 3068 . . . 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
) )
187, 11, 17sbcied2 3365 . . 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
) )
192, 5, 18sbcied2 3365 . 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
) )
20 df-mgm 15998 . 2  |- Mgm  =  {
m  |  [. ( Base `  m )  / 
b ]. [. ( +g  `  m )  /  o ]. A. x  e.  b 
A. y  e.  b  ( x o y )  e.  b }
2119, 20elab2g 3248 1  |-  ( M  e.  V  ->  ( M  e. Mgm  <->  A. x  e.  B  A. y  e.  B  ( x  .o.  y
)  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   [.wsbc 3327   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711  Mgmcmgm 15996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-mgm 15998
This theorem is referenced by:  ismgmn0  16000  mgmcl  16001  issgrpv  16039  0mgm  32682  ismgmd  32684  mgm2mgm  32771  lidlmmgm  32833
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