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Theorem isgrpd2 14828
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 2436, but we make an exception for theorems such as isgrpd2 14828, ismndd 14719, and islmodd 15956 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 6088 . . . . 5  |-  ( y  =  N  ->  (
y  .+  x )  =  ( N  .+  x ) )
87eqeq1d 2444 . . . 4  |-  ( y  =  N  ->  (
( y  .+  x
)  =  .0.  <->  ( N  .+  x )  =  .0.  ) )
98rspcev 3052 . . 3  |-  ( ( N  e.  B  /\  ( N  .+  x )  =  .0.  )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
105, 6, 9syl2anc 643 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
111, 2, 3, 4, 10isgrpd2e 14827 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Mndcmnd 14684   Grpcgrp 14685
This theorem is referenced by:  prdsgrpd  14927  oppggrp  15153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-grp 14812
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