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Theorem isgrpd 15924
 Description: Deduce a group from its properties. Unlike isgrpd2 15922, this one goes straight from the base properties rather than going through . (negative) is normally dependent on i.e. read it as . (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrpd.b
isgrpd.p
isgrpd.c
isgrpd.a
isgrpd.z
isgrpd.i
isgrpd.n
isgrpd.j
Assertion
Ref Expression
isgrpd
Distinct variable groups:   ,,,   , ,,   ,,,   ,   ,,,   ,,,
Allowed substitution hints:   (,)

Proof of Theorem isgrpd
StepHypRef Expression
1 isgrpd.b . 2
2 isgrpd.p . 2
3 isgrpd.c . 2
4 isgrpd.a . 2
5 isgrpd.z . 2
6 isgrpd.i . 2
7 isgrpd.n . . 3
8 isgrpd.j . . 3
9 oveq1 6301 . . . . 5
109eqeq1d 2469 . . . 4
1110rspcev 3219 . . 3
127, 8, 11syl2anc 661 . 2
131, 2, 3, 4, 5, 6, 12isgrpde 15923 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379   wcel 1767  wrex 2818  cfv 5593  (class class class)co 6294  cbs 14502   cplusg 14567  cgrp 15902 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-riota 6255  df-ov 6297  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906 This theorem is referenced by:  isgrpi  15925  issubg2  16065  symggrp  16274  isdrngd  17269  psrgrp  17898  dchrabl  23372  motgrp  23773  mendring  31038  ldualgrplem  34235  tgrpgrplem  35838  erngdvlem1  36077  erngdvlem1-rN  36085  dvhgrp  36197
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