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Theorem grpo2grp 25437
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 2467 . 2  |-  ran  .+  =  ( Base `  K
)
3 grp2grp.p . . 3  |-  ( +g  `  K )  =  .+
43eqcomi 2467 . 2  |-  .+  =  ( +g  `  K )
5 grp2grp.g . . 3  |-  .+  e.  GrpOp
6 eqid 2454 . . . 4  |-  ran  .+  =  ran  .+
76grpocl 25403 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  /\  b  e.  ran  .+  )  ->  ( a  .+  b )  e.  ran  .+  )
85, 7mp3an1 1309 . 2  |-  ( ( a  e.  ran  .+  /\  b  e.  ran  .+  )  ->  ( a  .+  b )  e.  ran  .+  )
96grpoass 25406 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  (
a  e.  ran  .+  /\  b  e.  ran  .+  /\  c  e.  ran  .+  ) )  ->  (
( a  .+  b
)  .+  c )  =  ( a  .+  ( b  .+  c
) ) )
105, 9mpan 668 . 2  |-  ( ( a  e.  ran  .+  /\  b  e.  ran  .+  /\  c  e.  ran  .+  )  ->  ( ( a 
.+  b )  .+  c )  =  ( a  .+  ( b 
.+  c ) ) )
11 eqid 2454 . . . 4  |-  (GId `  .+  )  =  (GId `  .+  )
126, 11grpoidcl 25420 . . 3  |-  (  .+  e.  GrpOp  ->  (GId `  .+  )  e.  ran  .+  )
135, 12ax-mp 5 . 2  |-  (GId `  .+  )  e.  ran  .+
146, 11grpolid 25422 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( (GId `  .+  )  .+  a )  =  a )
155, 14mpan 668 . 2  |-  ( a  e.  ran  .+  ->  ( (GId `  .+  )  .+  a )  =  a )
16 eqid 2454 . . . 4  |-  ( inv `  .+  )  =  ( inv `  .+  )
176, 16grpoinvcl 25429 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
185, 17mpan 668 . 2  |-  ( a  e.  ran  .+  ->  ( ( inv `  .+  ) `  a )  e.  ran  .+  )
196, 11, 16grpolinv 25431 . . 3  |-  ( ( 
.+  e.  GrpOp  /\  a  e.  ran  .+  )  ->  ( ( ( inv `  .+  ) `  a )  .+  a
)  =  (GId `  .+  ) )
205, 19mpan 668 . 2  |-  ( a  e.  ran  .+  ->  ( ( ( inv `  .+  ) `  a )  .+  a
)  =  (GId `  .+  ) )
212, 4, 8, 10, 13, 15, 18, 20isgrpi 16278 1  |-  K  e. 
Grp
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 971    = wceq 1398    e. wcel 1823   ran crn 4989   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   Grpcgrp 16255   GrpOpcgr 25389  GIdcgi 25390   invcgn 25391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-grpo 25394  df-gid 25395  df-ginv 25396
This theorem is referenced by: (None)
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