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Theorem expval 11312
Description: Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expval  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A ^ N
)  =  if ( N  =  0 ,  1 ,  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) ) ) )

Proof of Theorem expval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  y  =  N )
21eqeq1d 2396 . . 3  |-  ( ( x  =  A  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
31breq2d 4166 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  ( 0  <  y  <->  0  <  N ) )
4 simpl 444 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  N )  ->  x  =  A )
54sneqd 3771 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  N )  ->  { x }  =  { A } )
65xpeq2d 4843 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { A } ) )
76seqeq3d 11259 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  seq  1 (  x.  ,  ( NN  X.  { x } ) )  =  seq  1
(  x.  ,  ( NN  X.  { A } ) ) )
87, 1fveq12d 5675 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
91negeqd 9233 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  -> 
-u y  =  -u N )
107, 9fveq12d 5675 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  -u y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) )
1110oveq2d 6037 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  ( 1  /  (  seq  1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( 1  / 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u N
) ) )
123, 8, 11ifbieq12d 3705 . . 3  |-  ( ( x  =  A  /\  y  =  N )  ->  if ( 0  < 
y ,  (  seq  1 (  x.  , 
( NN  X.  {
x } ) ) `
 y ) ,  ( 1  /  (  seq  1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) ) )  =  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) ) )
132, 12ifbieq2d 3703 . 2  |-  ( ( x  =  A  /\  y  =  N )  ->  if ( y  =  0 ,  1 ,  if ( 0  < 
y ,  (  seq  1 (  x.  , 
( NN  X.  {
x } ) ) `
 y ) ,  ( 1  /  (  seq  1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) )  =  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  N ) ,  ( 1  /  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) ) ) )
14 df-exp 11311 . 2  |-  ^  =  ( x  e.  CC ,  y  e.  ZZ  |->  if ( y  =  0 ,  1 ,  if ( 0  <  y ,  (  seq  1
(  x.  ,  ( NN  X.  { x } ) ) `  y ) ,  ( 1  /  (  seq  1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) )
15 1ex 9020 . . 3  |-  1  e.  _V
16 fvex 5683 . . . 4  |-  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N )  e.  _V
17 ovex 6046 . . . 4  |-  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) )  e. 
_V
1816, 17ifex 3741 . . 3  |-  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) )  e.  _V
1915, 18ifex 3741 . 2  |-  if ( N  =  0 ,  1 ,  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) ) )  e.  _V
2013, 14, 19ovmpt2a 6144 1  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A ^ N
)  =  if ( N  =  0 ,  1 ,  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ifcif 3683   {csn 3758   class class class wbr 4154    X. cxp 4817   ` cfv 5395  (class class class)co 6021   CCcc 8922   0cc0 8924   1c1 8925    x. cmul 8929    < clt 9054   -ucneg 9225    / cdiv 9610   NNcn 9933   ZZcz 10215    seq cseq 11251   ^cexp 11310
This theorem is referenced by:  expnnval  11313  exp0  11314  expneg  11317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-1cn 8982
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-recs 6570  df-rdg 6605  df-neg 9227  df-seq 11252  df-exp 11311
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