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Theorem grpo2grp 21007
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 2362 . 2  |-  ran  .+  =  ( Base `  K
)
3 grp2grp.p . . 3  |-  ( +g  `  K )  =  .+
43eqcomi 2362 . 2  |-  .+  =  ( +g  `  K )
5 grp2grp.g . . 3  |-  .+  e.  GrpOp
6 eqid 2358 . . . 4  |-  ran  .+  =  ran  .+
76grpocl 20973 . . 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 20976 . . 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 2358 . . . 4  |-  (GId `  .+  )  =  (GId `  .+  )
126, 11grpoidcl 20990 . . 3  |-  (  .+  e.  GrpOp  ->  (GId `  .+  )  e.  ran  .+  )
135, 12ax-mp 8 . 2  |-  (GId `  .+  )  e.  ran  .+
146, 11grpolid 20992 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( (GId `  .+  )  .+  a )  =  a )
155, 14mpan 651 . 2  |-  ( a  e.  ran  .+  ->  ( (GId `  .+  )  .+  a )  =  a )
16 eqid 2358 . . . 4  |-  ( inv `  .+  )  =  ( inv `  .+  )
176, 16grpoinvcl 20999 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
185, 17mpan 651 . 2  |-  ( a  e.  ran  .+  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
196, 11, 16grpolinv 21001 . . 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 14601 1  |-  K  e. 
Grp
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1642    e. wcel 1710   ran crn 4769   ` cfv 5334  (class class class)co 5942   Basecbs 13239   +g cplusg 13299   Grpcgrp 14455   GrpOpcgr 20959  GIdcgi 20960   invcgn 20961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-riota 6388  df-0g 13497  df-mnd 14460  df-grp 14582  df-grpo 20964  df-gid 20965  df-ginv 20966
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