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Theorem isgrpd2e 12937
Description: Deduce a group from its properties. In this version of isgrpd2 12938, 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 2563 . . 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 5908 . . . . . 6  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  G
) x ) )
7 isgrpd2.z . . . . . 6  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
86, 7eqeq12d 2204 . . . . 5  |-  ( ph  ->  ( ( y  .+  x )  =  .0.  <->  ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )
94, 8rexeqbidv 2699 . . . 4  |-  ( ph  ->  ( E. y  e.  B  ( y  .+  x )  =  .0.  <->  E. y  e.  ( Base `  G ) ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )
104, 9raleqbidv 2698 . . 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 147 . 2  |-  ( ph  ->  A. x  e.  (
Base `  G ) E. y  e.  ( Base `  G ) ( y ( +g  `  G
) x )  =  ( 0g `  G
) )
12 eqid 2189 . . 3  |-  ( Base `  G )  =  (
Base `  G )
13 eqid 2189 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2189 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
1512, 13, 14isgrp 12923 . 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 417 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   ` cfv 5231  (class class class)co 5891   Basecbs 12486   +g cplusg 12561   0gc0g 12733   Mndcmnd 12849   Grpcgrp 12917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5239  df-ov 5894  df-grp 12920
This theorem is referenced by:  isgrpd2  12938  isgrpde  12939
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