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Theorem isgrpd2 12896
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 2177, but we make an exception for theorems such as isgrpd2 12896 and ismndd 12837 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 2186 . . . 4  |-  ( y  =  N  ->  (
( y  .+  x
)  =  .0.  <->  ( N  .+  x )  =  .0.  ) )
98rspcev 2841 . . 3  |-  ( ( N  e.  B  /\  ( N  .+  x )  =  .0.  )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
105, 6, 9syl2anc 411 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
111, 2, 3, 4, 10isgrpd2e 12895 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   E.wrex 2456   ` cfv 5216  (class class class)co 5874   Basecbs 12461   +g cplusg 12535   0gc0g 12704   Mndcmnd 12816   Grpcgrp 12876
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877  df-grp 12879
This theorem is referenced by: (None)
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