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Theorem ismgm 16482
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 5873 . . . 4  |-  ( Base `  m )  e.  _V
21a1i 11 . . 3  |-  ( m  =  M  ->  ( Base `  m )  e. 
_V )
3 fveq2 5863 . . . 4  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
4 ismgm.b . . . 4  |-  B  =  ( Base `  M
)
53, 4syl6eqr 2502 . . 3  |-  ( m  =  M  ->  ( Base `  m )  =  B )
6 fvex 5873 . . . . 5  |-  ( +g  `  m )  e.  _V
76a1i 11 . . . 4  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  e.  _V )
8 fveq2 5863 . . . . . 6  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
98adantr 467 . . . . 5  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  =  ( +g  `  M ) )
10 ismgm.o . . . . 5  |-  .o.  =  ( +g  `  M )
119, 10syl6eqr 2502 . . . 4  |-  ( ( m  =  M  /\  b  =  B )  ->  ( +g  `  m
)  =  .o.  )
12 simplr 761 . . . . 5  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  b  =  B )
13 oveq 6294 . . . . . . . 8  |-  ( o  =  .o.  ->  (
x o y )  =  ( x  .o.  y ) )
1413adantl 468 . . . . . . 7  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  (
x o y )  =  ( x  .o.  y ) )
1514, 12eleq12d 2522 . . . . . 6  |-  ( ( ( m  =  M  /\  b  =  B )  /\  o  =  .o.  )  ->  (
( x o y )  e.  b  <->  ( x  .o.  y )  e.  B
) )
1612, 15raleqbidv 3000 . . . . 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 3000 . . . 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 3304 . . 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 3304 . 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 16481 . 2  |- Mgm  =  {
m  |  [. ( Base `  m )  / 
b ]. [. ( +g  `  m )  /  o ]. A. x  e.  b 
A. y  e.  b  ( x o y )  e.  b }
2119, 20elab2g 3186 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 188    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736   _Vcvv 3044   [.wsbc 3266   ` cfv 5581  (class class class)co 6288   Basecbs 15114   +g cplusg 15183  Mgmcmgm 16479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-nul 4533
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-iota 5545  df-fv 5589  df-ov 6291  df-mgm 16481
This theorem is referenced by:  ismgmn0  16483  mgmcl  16484  issgrpv  16522  0mgm  39761  ismgmd  39763  mgm2mgm  39850  lidlmmgm  39912
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