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Theorem grpo2grp 20901
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  |-  ( Base `  K )  =  ran  .+
grp2grp.p  |-  ( +g  `  K )  =  .+
grp2grp.g  |-  .+  e.  GrpOp
Assertion
Ref Expression
grpo2grp  |-  K  e. 
Grp

Proof of Theorem grpo2grp
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grp2grp.a . . 3  |-  ( Base `  K )  =  ran  .+
21eqcomi 2287 . 2  |-  ran  .+  =  ( Base `  K
)
3 grp2grp.p . . 3  |-  ( +g  `  K )  =  .+
43eqcomi 2287 . 2  |-  .+  =  ( +g  `  K )
5 grp2grp.g . . 3  |-  .+  e.  GrpOp
6 eqid 2283 . . . 4  |-  ran  .+  =  ran  .+
76grpocl 20867 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  /\  b  e.  ran  .+  )  ->  ( a  .+  b )  e.  ran  .+  )
85, 7mp3an1 1264 . 2  |-  ( ( a  e.  ran  .+  /\  b  e.  ran  .+  )  ->  ( a  .+  b )  e.  ran  .+  )
96grpoass 20870 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  (
a  e.  ran  .+  /\  b  e.  ran  .+  /\  c  e.  ran  .+  ) )  ->  (
( a  .+  b
)  .+  c )  =  ( a  .+  ( b  .+  c
) ) )
105, 9mpan 651 . 2  |-  ( ( a  e.  ran  .+  /\  b  e.  ran  .+  /\  c  e.  ran  .+  )  ->  ( ( a 
.+  b )  .+  c )  =  ( a  .+  ( b 
.+  c ) ) )
11 eqid 2283 . . . 4  |-  (GId `  .+  )  =  (GId `  .+  )
126, 11grpoidcl 20884 . . 3  |-  (  .+  e.  GrpOp  ->  (GId `  .+  )  e.  ran  .+  )
135, 12ax-mp 8 . 2  |-  (GId `  .+  )  e.  ran  .+
146, 11grpolid 20886 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( (GId `  .+  )  .+  a )  =  a )
155, 14mpan 651 . 2  |-  ( a  e.  ran  .+  ->  ( (GId `  .+  )  .+  a )  =  a )
16 eqid 2283 . . . 4  |-  ( inv `  .+  )  =  ( inv `  .+  )
176, 16grpoinvcl 20893 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
185, 17mpan 651 . 2  |-  ( a  e.  ran  .+  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
196, 11, 16grpolinv 20895 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( ( inv `  .+  ) `  a )  .+  a
)  =  (GId `  .+  ) )
205, 19mpan 651 . 2  |-  ( a  e.  ran  .+  ->  ( ( ( inv `  .+  ) `  a )  .+  a
)  =  (GId `  .+  ) )
212, 4, 8, 10, 13, 15, 18, 20isgrpi 14508 1  |-  K  e. 
Grp
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-grpo 20858  df-gid 20859  df-ginv 20860
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