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Theorem grpo2grp 23853
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 2463 . 2  |-  ran  .+  =  ( Base `  K
)
3 grp2grp.p . . 3  |-  ( +g  `  K )  =  .+
43eqcomi 2463 . 2  |-  .+  =  ( +g  `  K )
5 grp2grp.g . . 3  |-  .+  e.  GrpOp
6 eqid 2451 . . . 4  |-  ran  .+  =  ran  .+
76grpocl 23819 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  /\  b  e.  ran  .+  )  ->  ( a  .+  b )  e.  ran  .+  )
85, 7mp3an1 1302 . 2  |-  ( ( a  e.  ran  .+  /\  b  e.  ran  .+  )  ->  ( a  .+  b )  e.  ran  .+  )
96grpoass 23822 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  (
a  e.  ran  .+  /\  b  e.  ran  .+  /\  c  e.  ran  .+  ) )  ->  (
( a  .+  b
)  .+  c )  =  ( a  .+  ( b  .+  c
) ) )
105, 9mpan 670 . 2  |-  ( ( a  e.  ran  .+  /\  b  e.  ran  .+  /\  c  e.  ran  .+  )  ->  ( ( a 
.+  b )  .+  c )  =  ( a  .+  ( b 
.+  c ) ) )
11 eqid 2451 . . . 4  |-  (GId `  .+  )  =  (GId `  .+  )
126, 11grpoidcl 23836 . . 3  |-  (  .+  e.  GrpOp  ->  (GId `  .+  )  e.  ran  .+  )
135, 12ax-mp 5 . 2  |-  (GId `  .+  )  e.  ran  .+
146, 11grpolid 23838 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( (GId `  .+  )  .+  a )  =  a )
155, 14mpan 670 . 2  |-  ( a  e.  ran  .+  ->  ( (GId `  .+  )  .+  a )  =  a )
16 eqid 2451 . . . 4  |-  ( inv `  .+  )  =  ( inv `  .+  )
176, 16grpoinvcl 23845 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
185, 17mpan 670 . 2  |-  ( a  e.  ran  .+  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
196, 11, 16grpolinv 23847 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( ( inv `  .+  ) `  a )  .+  a
)  =  (GId `  .+  ) )
205, 19mpan 670 . 2  |-  ( a  e.  ran  .+  ->  ( ( ( inv `  .+  ) `  a )  .+  a
)  =  (GId `  .+  ) )
212, 4, 8, 10, 13, 15, 18, 20isgrpi 15663 1  |-  K  e. 
Grp
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 965    = wceq 1370    e. wcel 1758   ran crn 4936   ` cfv 5513  (class class class)co 6187   Basecbs 14273   +g cplusg 14337   Grpcgrp 15509   GrpOpcgr 23805  GIdcgi 23806   invcgn 23807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-0g 14479  df-mnd 15514  df-grp 15644  df-grpo 23810  df-gid 23811  df-ginv 23812
This theorem is referenced by: (None)
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