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Theorem isgrpd 14822
Description: Deduce a group from its properties. Unlike isgrpd2 14820, this one goes straight from the base properties rather than going through  Mnd.  N (negative) is normally dependent on  x i.e. read it as  N ( x ). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrpd.b  |-  ( ph  ->  B  =  ( Base `  G ) )
isgrpd.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
isgrpd.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
isgrpd.a  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
isgrpd.z  |-  ( ph  ->  .0.  e.  B )
isgrpd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
isgrpd.n  |-  ( (
ph  /\  x  e.  B )  ->  N  e.  B )
isgrpd.j  |-  ( (
ph  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )
Assertion
Ref Expression
isgrpd  |-  ( ph  ->  G  e.  Grp )
Distinct variable groups:    x, y,
z,  .+    x,  .0. , y,
z    x, B, y, z   
y, N    ph, x, y, z    x, G, y, z
Allowed substitution hints:    N( x, z)

Proof of Theorem isgrpd
StepHypRef Expression
1 isgrpd.b . 2  |-  ( ph  ->  B  =  ( Base `  G ) )
2 isgrpd.p . 2  |-  ( ph  ->  .+  =  ( +g  `  G ) )
3 isgrpd.c . 2  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
4 isgrpd.a . 2  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
5 isgrpd.z . 2  |-  ( ph  ->  .0.  e.  B )
6 isgrpd.i . 2  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
7 isgrpd.n . . 3  |-  ( (
ph  /\  x  e.  B )  ->  N  e.  B )
8 isgrpd.j . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )
9 oveq1 6080 . . . . 5  |-  ( y  =  N  ->  (
y  .+  x )  =  ( N  .+  x ) )
109eqeq1d 2443 . . . 4  |-  ( y  =  N  ->  (
( y  .+  x
)  =  .0.  <->  ( N  .+  x )  =  .0.  ) )
1110rspcev 3044 . . 3  |-  ( ( N  e.  B  /\  ( N  .+  x )  =  .0.  )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
127, 8, 11syl2anc 643 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
131, 2, 3, 4, 5, 6, 12isgrpde 14821 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677
This theorem is referenced by:  isgrpi  14823  issubg2  14951  symggrp  15095  isdrngd  15852  psrgrp  16454  dchrabl  21030  mendrng  27468  ldualgrplem  29880  tgrpgrplem  31483  erngdvlem1  31722  erngdvlem1-rN  31730  dvhgrp  31842
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719  df-mnd 14682  df-grp 14804
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