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Theorem expval 11108
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 447 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  y  =  N )
21eqeq1d 2293 . . 3  |-  ( ( x  =  A  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
31breq2d 4037 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  ( 0  <  y  <->  0  <  N ) )
4 simpl 443 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  N )  ->  x  =  A )
54sneqd 3655 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  N )  ->  { x }  =  { A } )
65xpeq2d 4715 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { A } ) )
76seqeq3d 11056 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  seq  1 (  x.  ,  ( NN  X.  { x } ) )  =  seq  1
(  x.  ,  ( NN  X.  { A } ) ) )
87, 1fveq12d 5533 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
91negeqd 9048 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  -> 
-u y  =  -u N )
107, 9fveq12d 5533 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  -u y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) )
1110oveq2d 5876 . . . 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 3589 . . 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 3587 . 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 11107 . 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 8835 . . 3  |-  1  e.  _V
16 fvex 5541 . . . 4  |-  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N )  e.  _V
17 ovex 5885 . . . 4  |-  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) )  e. 
_V
1816, 17ifex 3625 . . 3  |-  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) )  e.  _V
1915, 18ifex 3625 . 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 5980 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 358    = wceq 1625    e. wcel 1686   ifcif 3567   {csn 3642   class class class wbr 4025    X. cxp 4689   ` cfv 5257  (class class class)co 5860   CCcc 8737   0cc0 8739   1c1 8740    x. cmul 8744    < clt 8869   -ucneg 9040    / cdiv 9425   NNcn 9748   ZZcz 10026    seq cseq 11048   ^cexp 11106
This theorem is referenced by:  expnnval  11109  exp0  11110  expneg  11113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-1cn 8797
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-recs 6390  df-rdg 6425  df-neg 9042  df-seq 11049  df-exp 11107
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