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Theorem ismgm 15747
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 df-mgm 15746 . . 3  |- Mgm  =  {
m  |  [. ( Base `  m )  / 
b ]. [. ( +g  `  m )  /  o ]. A. x  e.  b 
A. y  e.  b  ( x o y )  e.  b }
21eleq2i 2545 . 2  |-  ( M  e. Mgm 
<->  M  e.  { m  |  [. ( Base `  m
)  /  b ]. [. ( +g  `  m
)  /  o ]. A. x  e.  b  A. y  e.  b 
( x o y )  e.  b } )
3 fvex 5882 . . . . 5  |-  ( Base `  m )  e.  _V
43a1i 11 . . . 4  |-  ( m  =  M  ->  ( Base `  m )  e. 
_V )
5 fveq2 5872 . . . . 5  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
6 ismgm.b . . . . 5  |-  B  =  ( Base `  M
)
75, 6syl6eqr 2526 . . . 4  |-  ( m  =  M  ->  ( Base `  m )  =  B )
8 fvex 5882 . . . . . 6  |-  ( +g  `  m )  e.  _V
98a1i 11 . . . . 5  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  e.  _V )
10 fveq2 5872 . . . . . . 7  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
1110adantr 465 . . . . . 6  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  =  ( +g  `  M ) )
12 ismgm.o . . . . . 6  |-  .o.  =  ( +g  `  M )
1311, 12syl6eqr 2526 . . . . 5  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  =  .o.  )
14 simplr 754 . . . . . 6  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  b  =  B )
15 oveq 6301 . . . . . . . . 9  |-  ( o  =  .o.  ->  (
x o y )  =  ( x  .o.  y ) )
1615adantl 466 . . . . . . . 8  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  (
x o y )  =  ( x  .o.  y ) )
1716, 14eleq12d 2549 . . . . . . 7  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  (
( x o y )  e.  b  <->  ( x  .o.  y )  e.  B
) )
1814, 17raleqbidv 3077 . . . . . 6  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  ( A. y  e.  b 
( x o y )  e.  b  <->  A. y  e.  B  ( x  .o.  y )  e.  B
) )
1914, 18raleqbidv 3077 . . . . 5  |-  ( ( ( 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
) )
209, 13, 19sbcied2 3374 . . . 4  |-  ( ( 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
) )
214, 7, 20sbcied2 3374 . . 3  |-  ( 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
) )
2221elabg 3256 . 2  |-  ( M  e.  V  ->  ( M  e.  { 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 ) )
232, 22syl5bb 257 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 1379    e. wcel 1767   {cab 2452   A.wral 2817   _Vcvv 3118   [.wsbc 3336   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   Mgm cmgm 15744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-mgm 15746
This theorem is referenced by:  ismgmn0  15748  mgmcl  15749  issgrpv  15787  mgm2mgm  32310  lidlmmgm  32325
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