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Theorem isgrpd 14868
Description: Deduce a group from its properties. Unlike isgrpd2 14866, 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 6124 . . . . 5  |-  ( y  =  N  ->  (
y  .+  x )  =  ( N  .+  x ) )
109eqeq1d 2451 . . . 4  |-  ( y  =  N  ->  (
( y  .+  x
)  =  .0.  <->  ( N  .+  x )  =  .0.  ) )
1110rspcev 3061 . . 3  |-  ( ( N  e.  B  /\  ( N  .+  x )  =  .0.  )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
127, 8, 11syl2anc 644 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
131, 2, 3, 4, 5, 6, 12isgrpde 14867 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   E.wrex 2713   ` cfv 5489  (class class class)co 6117   Basecbs 13507   +g cplusg 13567   Grpcgrp 14723
This theorem is referenced by:  isgrpi  14869  issubg2  14997  symggrp  15141  isdrngd  15898  psrgrp  16500  dchrabl  21076  mendrng  27589  ldualgrplem  30117  tgrpgrplem  31720  erngdvlem1  31959  erngdvlem1-rN  31967  dvhgrp  32079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5453  df-fun 5491  df-fv 5497  df-ov 6120  df-riota 6585  df-0g 13765  df-mnd 14728  df-grp 14850
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