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Theorem gx0 20944
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 10051 . . 3  |-  0  e.  ZZ
2 gx0.1 . . . 4  |-  X  =  ran  G
3 gx0.2 . . . 4  |-  U  =  (GId `  G )
4 eqid 2296 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
5 gx0.3 . . . 4  |-  P  =  ( ^g `  G
)
62, 3, 4, 5gxval 20941 . . 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 1266 . 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 2296 . . 3  |-  0  =  0
9 iftrue 3584 . . 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 2344 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A P 0 )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ifcif 3578   {csn 3653   class class class wbr 4039    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    < clt 8883   -ucneg 9054   NNcn 9762   ZZcz 10040    seq cseq 11062   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871   ^gcgx 20873
This theorem is referenced by:  gxnn0neg  20946  gxnn0suc  20947  gxcl  20948  gxcom  20952  gxinv  20953  gxid  20956  gxnn0add  20957  gxnn0mul  20960  gxdi  20979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-neg 9056  df-z 10041  df-seq 11063  df-gx 20878
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