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Theorem isgrpd 16192
Description: Deduce a group from its properties. Unlike isgrpd2 16190, 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 6203 . . . . 5  |-  ( y  =  N  ->  (
y  .+  x )  =  ( N  .+  x ) )
109eqeq1d 2384 . . . 4  |-  ( y  =  N  ->  (
( y  .+  x
)  =  .0.  <->  ( N  .+  x )  =  .0.  ) )
1110rspcev 3135 . . 3  |-  ( ( N  e.  B  /\  ( N  .+  x )  =  .0.  )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
127, 8, 11syl2anc 659 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
131, 2, 3, 4, 5, 6, 12isgrpde 16191 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   E.wrex 2733   ` cfv 5496  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   Grpcgrp 16170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-riota 6158  df-ov 6199  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174
This theorem is referenced by:  isgrpi  16193  issubg2  16333  symggrp  16542  isdrngd  17534  psrgrp  18164  dchrabl  23646  motgrp  24050  mendring  31309  ldualgrplem  35283  tgrpgrplem  36888  erngdvlem1  37127  erngdvlem1-rN  37135  dvhgrp  37247
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