Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpo2grp Structured version   Unicode version

Theorem grpo2grp 21824
 Description: Convert a group operation to a group structure. (Contributed by NM, 25-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
grp2grp.a
grp2grp.p
grp2grp.g
Assertion
Ref Expression
grpo2grp

Proof of Theorem grpo2grp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grp2grp.a . . 3
21eqcomi 2442 . 2
3 grp2grp.p . . 3
43eqcomi 2442 . 2
5 grp2grp.g . . 3
6 eqid 2438 . . . 4
76grpocl 21790 . . 3
85, 7mp3an1 1267 . 2
96grpoass 21793 . . 3
105, 9mpan 653 . 2
11 eqid 2438 . . . 4 GId GId
126, 11grpoidcl 21807 . . 3 GId
135, 12ax-mp 8 . 2 GId
146, 11grpolid 21809 . . 3 GId
155, 14mpan 653 . 2 GId
16 eqid 2438 . . . 4
176, 16grpoinvcl 21816 . . 3
185, 17mpan 653 . 2
196, 11, 16grpolinv 21818 . . 3 GId
205, 19mpan 653 . 2 GId
212, 4, 8, 10, 13, 15, 18, 20isgrpi 14833 1
 Colors of variables: wff set class Syntax hints:   w3a 937   wceq 1653   wcel 1726   crn 4881  cfv 5456  (class class class)co 6083  cbs 13471   cplusg 13531  cgrp 14687  cgr 21776  GIdcgi 21777  cgn 21778 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-grpo 21781  df-gid 21782  df-ginv 21783
 Copyright terms: Public domain W3C validator