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Theorem isgrpd2e 14819
 Description: Deduce a group from its properties. In this version of isgrpd2 14820, we don't assume there is an expression for the inverse of . (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
isgrpd2.b
isgrpd2.p
isgrpd2.z
isgrpd2.g
isgrpd2e.n
Assertion
Ref Expression
isgrpd2e
Distinct variable groups:   ,,   ,   ,,   ,,   ,,
Allowed substitution hint:   ()

Proof of Theorem isgrpd2e
StepHypRef Expression
1 isgrpd2.g . 2
2 isgrpd2e.n . . . 4
32ralrimiva 2781 . . 3
4 isgrpd2.b . . . 4
5 isgrpd2.p . . . . . . 7
65oveqd 6090 . . . . . 6
7 isgrpd2.z . . . . . 6
86, 7eqeq12d 2449 . . . . 5
94, 8rexeqbidv 2909 . . . 4
104, 9raleqbidv 2908 . . 3
113, 10mpbid 202 . 2
12 eqid 2435 . . 3
13 eqid 2435 . . 3
14 eqid 2435 . . 3
1512, 13, 14isgrp 14808 . 2
161, 11, 15sylanbrc 646 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  cfv 5446  (class class class)co 6073  cbs 13461   cplusg 13521  c0g 13715  cmnd 14676  cgrp 14677 This theorem is referenced by:  isgrpd2  14820  isgrpde  14821 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  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-iota 5410  df-fv 5454  df-ov 6076  df-grp 14804
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