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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  |-  ( 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 2442 . 2  |-  ran  .+  =  ( Base `  K
)
3 grp2grp.p . . 3  |-  ( +g  `  K )  =  .+
43eqcomi 2442 . 2  |-  .+  =  ( +g  `  K )
5 grp2grp.g . . 3  |-  .+  e.  GrpOp
6 eqid 2438 . . . 4  |-  ran  .+  =  ran  .+
76grpocl 21790 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  /\  b  e.  ran  .+  )  ->  ( a  .+  b )  e.  ran  .+  )
85, 7mp3an1 1267 . 2  |-  ( ( a  e.  ran  .+  /\  b  e.  ran  .+  )  ->  ( a  .+  b )  e.  ran  .+  )
96grpoass 21793 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  (
a  e.  ran  .+  /\  b  e.  ran  .+  /\  c  e.  ran  .+  ) )  ->  (
( a  .+  b
)  .+  c )  =  ( a  .+  ( b  .+  c
) ) )
105, 9mpan 653 . 2  |-  ( ( a  e.  ran  .+  /\  b  e.  ran  .+  /\  c  e.  ran  .+  )  ->  ( ( a 
.+  b )  .+  c )  =  ( a  .+  ( b 
.+  c ) ) )
11 eqid 2438 . . . 4  |-  (GId `  .+  )  =  (GId `  .+  )
126, 11grpoidcl 21807 . . 3  |-  (  .+  e.  GrpOp  ->  (GId `  .+  )  e.  ran  .+  )
135, 12ax-mp 8 . 2  |-  (GId `  .+  )  e.  ran  .+
146, 11grpolid 21809 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( (GId `  .+  )  .+  a )  =  a )
155, 14mpan 653 . 2  |-  ( a  e.  ran  .+  ->  ( (GId `  .+  )  .+  a )  =  a )
16 eqid 2438 . . . 4  |-  ( inv `  .+  )  =  ( inv `  .+  )
176, 16grpoinvcl 21816 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
185, 17mpan 653 . 2  |-  ( a  e.  ran  .+  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
196, 11, 16grpolinv 21818 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( ( inv `  .+  ) `  a )  .+  a
)  =  (GId `  .+  ) )
205, 19mpan 653 . 2  |-  ( a  e.  ran  .+  ->  ( ( ( inv `  .+  ) `  a )  .+  a
)  =  (GId `  .+  ) )
212, 4, 8, 10, 13, 15, 18, 20isgrpi 14833 1  |-  K  e. 
Grp
Colors of variables: wff set class
Syntax hints:    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4881   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Grpcgrp 14687   GrpOpcgr 21776  GIdcgi 21777   invcgn 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
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