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Theorem isgrpd2 14521
Description: Deduce a group from its properties.  N (negative) is normally dependent on  x i.e. read it as  N ( x ). Note: normally we don't use a  ph antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2296, but we make an exception for theorems such as isgrpd2 14521, ismndd 14412, and islmodd 15649 since theorems using them often rewrite the structure components. (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 )
isgrpd2.n  |-  ( (
ph  /\  x  e.  B )  ->  N  e.  B )
isgrpd2.j  |-  ( (
ph  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )
Assertion
Ref Expression
isgrpd2  |-  ( ph  ->  G  e.  Grp )
Distinct variable groups:    x,  .+    x, B    x, G    ph, x
Allowed substitution hints:    N( x)    .0. ( x)

Proof of Theorem isgrpd2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 isgrpd2.b . 2  |-  ( ph  ->  B  =  ( Base `  G ) )
2 isgrpd2.p . 2  |-  ( ph  ->  .+  =  ( +g  `  G ) )
3 isgrpd2.z . 2  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
4 isgrpd2.g . 2  |-  ( ph  ->  G  e.  Mnd )
5 isgrpd2.n . . 3  |-  ( (
ph  /\  x  e.  B )  ->  N  e.  B )
6 isgrpd2.j . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )
7 oveq1 5881 . . . . 5  |-  ( y  =  N  ->  (
y  .+  x )  =  ( N  .+  x ) )
87eqeq1d 2304 . . . 4  |-  ( y  =  N  ->  (
( y  .+  x
)  =  .0.  <->  ( N  .+  x )  =  .0.  ) )
98rspcev 2897 . . 3  |-  ( ( N  e.  B  /\  ( N  .+  x )  =  .0.  )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
105, 6, 9syl2anc 642 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
111, 2, 3, 4, 10isgrpd2e 14520 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   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:  prdsgrpd  14620  oppggrp  14846
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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