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Theorem gx0 21851
Description: The result of the group power operator when the exponent is zero. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
gx0.1  |-  X  =  ran  G
gx0.2  |-  U  =  (GId `  G )
gx0.3  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gx0  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  U )

Proof of Theorem gx0
StepHypRef Expression
1 0z 10295 . . 3  |-  0  e.  ZZ
2 gx0.1 . . . 4  |-  X  =  ran  G
3 gx0.2 . . . 4  |-  U  =  (GId `  G )
4 eqid 2438 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
5 gx0.3 . . . 4  |-  P  =  ( ^g `  G
)
62, 3, 4, 5gxval 21848 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  0  e.  ZZ )  ->  ( A P 0 )  =  if ( 0  =  0 ,  U ,  if ( 0  <  0 ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) ` 
0 ) ,  ( ( inv `  G
) `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u 0 ) ) ) ) )
71, 6mp3an3 1269 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  if ( 0  =  0 ,  U ,  if ( 0  <  0 ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) ` 
0 ) ,  ( ( inv `  G
) `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u 0 ) ) ) ) )
8 eqid 2438 . . 3  |-  0  =  0
9 iftrue 3747 . . 3  |-  ( 0  =  0  ->  if ( 0  =  0 ,  U ,  if ( 0  <  0 ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) ` 
0 ) ,  ( ( inv `  G
) `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u 0 ) ) ) )  =  U )
108, 9ax-mp 8 . 2  |-  if ( 0  =  0 ,  U ,  if ( 0  <  0 ,  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  0
) ,  ( ( inv `  G ) `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u 0 ) ) ) )  =  U
117, 10syl6eq 2486 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   ifcif 3741   {csn 3816   class class class wbr 4214    X. cxp 4878   ran crn 4881   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993    < clt 9122   -ucneg 9294   NNcn 10002   ZZcz 10284    seq cseq 11325   GrpOpcgr 21776  GIdcgi 21777   invcgn 21778   ^gcgx 21780
This theorem is referenced by:  gxnn0neg  21853  gxnn0suc  21854  gxcl  21855  gxcom  21859  gxinv  21860  gxid  21863  gxnn0add  21864  gxnn0mul  21867  gxdi  21886
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  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-i2m1 9060  ax-1ne0 9061  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-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-pw 3803  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-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-neg 9296  df-z 10285  df-seq 11326  df-gx 21785
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