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Theorem isgrpd2e 12915
Description: Deduce a group from its properties. In this version of isgrpd2 12916, 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 2560 . . 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 5905 . . . . . 6  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  G
) x ) )
7 isgrpd2.z . . . . . 6  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
86, 7eqeq12d 2202 . . . . 5  |-  ( ph  ->  ( ( y  .+  x )  =  .0.  <->  ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )
94, 8rexeqbidv 2696 . . . 4  |-  ( ph  ->  ( E. y  e.  B  ( y  .+  x )  =  .0.  <->  E. y  e.  ( Base `  G ) ( y ( +g  `  G
) x )  =  ( 0g `  G
) ) )
104, 9raleqbidv 2695 . . 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 2187 . . 3  |-  ( Base `  G )  =  (
Base `  G )
13 eqid 2187 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2187 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
1512, 13, 14isgrp 12902 . 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 1363    e. wcel 2158   A.wral 2465   E.wrex 2466   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   0gc0g 12722   Mndcmnd 12836   Grpcgrp 12896
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891  df-grp 12899
This theorem is referenced by:  isgrpd2  12916  isgrpde  12917
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