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Theorem isgrpd2 12897
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 12897 and ismndd 12838 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 5882 . . . . 5  |-  ( y  =  N  ->  (
y  .+  x )  =  ( N  .+  x ) )
87eqeq1d 2186 . . . 4  |-  ( y  =  N  ->  (
( y  .+  x
)  =  .0.  <->  ( N  .+  x )  =  .0.  ) )
98rspcev 2842 . . 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 12896 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 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   0gc0g 12705   Mndcmnd 12817   Grpcgrp 12877
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 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878  df-grp 12880
This theorem is referenced by: (None)
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