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Theorem isgrpd2e 14504
Description: Deduce a group from its properties. In this version of isgrpd2 14505, 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 2626 . . 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 5875 . . . . . 6  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  G
) x ) )
7 isgrpd2.z . . . . . 6  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
86, 7eqeq12d 2297 . . . . 5  |-  ( ph  ->  ( ( y  .+  x )  =  .0.  <->  ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )
94, 8rexeqbidv 2749 . . . 4  |-  ( ph  ->  ( E. y  e.  B  ( y  .+  x )  =  .0.  <->  E. y  e.  ( Base `  G ) ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )
104, 9raleqbidv 2748 . . 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 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
13 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2283 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
1512, 13, 14isgrp 14493 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362
This theorem is referenced by:  isgrpd2  14505  isgrpde  14506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-grp 14489
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