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Theorem isgrpd2 16200
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 2457, but we make an exception for theorems such as isgrpd2 16200, ismndd 16070, and islmodd 17645 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 6303 . . . . 5  |-  ( y  =  N  ->  (
y  .+  x )  =  ( N  .+  x ) )
87eqeq1d 2459 . . . 4  |-  ( y  =  N  ->  (
( y  .+  x
)  =  .0.  <->  ( N  .+  x )  =  .0.  ) )
98rspcev 3210 . . 3  |-  ( ( N  e.  B  /\  ( N  .+  x )  =  .0.  )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
105, 6, 9syl2anc 661 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
111, 2, 3, 4, 10isgrpd2e 16199 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   0gc0g 14857   Mndcmnd 16046   Grpcgrp 16180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-grp 16184
This theorem is referenced by:  prdsgrpd  16306  oppggrp  16519
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