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Theorem isgrpd2e 14520
Description: Deduce a group from its properties. In this version of isgrpd2 14521, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
isgrpd2.b  |-  ( ph  ->  B  =  ( Base `  G ) )
isgrpd2.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
isgrpd2.z  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
isgrpd2.g  |-  ( ph  ->  G  e.  Mnd )
isgrpd2e.n  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
Assertion
Ref Expression
isgrpd2e  |-  ( ph  ->  G  e.  Grp )
Distinct variable groups:    x, y,  .+    y,  .0.    x, B, y   
x, G, y    ph, x, y
Allowed substitution hint:    .0. ( x)

Proof of Theorem isgrpd2e
StepHypRef Expression
1 isgrpd2.g . 2  |-  ( ph  ->  G  e.  Mnd )
2 isgrpd2e.n . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
32ralrimiva 2639 . . 3  |-  ( ph  ->  A. x  e.  B  E. y  e.  B  ( y  .+  x
)  =  .0.  )
4 isgrpd2.b . . . 4  |-  ( ph  ->  B  =  ( Base `  G ) )
5 isgrpd2.p . . . . . . 7  |-  ( ph  ->  .+  =  ( +g  `  G ) )
65oveqd 5891 . . . . . 6  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  G
) x ) )
7 isgrpd2.z . . . . . 6  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
86, 7eqeq12d 2310 . . . . 5  |-  ( ph  ->  ( ( y  .+  x )  =  .0.  <->  ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )
94, 8rexeqbidv 2762 . . . 4  |-  ( ph  ->  ( E. y  e.  B  ( y  .+  x )  =  .0.  <->  E. y  e.  ( Base `  G ) ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )
104, 9raleqbidv 2761 . . 3  |-  ( ph  ->  ( A. x  e.  B  E. y  e.  B  ( y  .+  x )  =  .0.  <->  A. x  e.  ( Base `  G ) E. y  e.  ( Base `  G
) ( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
113, 10mpbid 201 . 2  |-  ( ph  ->  A. x  e.  (
Base `  G ) E. y  e.  ( Base `  G ) ( y ( +g  `  G
) x )  =  ( 0g `  G
) )
12 eqid 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
13 eqid 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2296 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
1512, 13, 14isgrp 14509 . 2  |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. x  e.  ( Base `  G
) E. y  e.  ( Base `  G
) ( y ( +g  `  G ) x )  =  ( 0g `  G ) ) )
161, 11, 15sylanbrc 645 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   Grpcgrp 14378
This theorem is referenced by:  isgrpd2  14521  isgrpde  14522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-grp 14505
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