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Theorem expval 11888
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 461 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  y  =  N )
21eqeq1d 2451 . . 3  |-  ( ( x  =  A  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
31breq2d 4325 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  ( 0  <  y  <->  0  <  N ) )
4 simpl 457 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  N )  ->  x  =  A )
54sneqd 3910 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  N )  ->  { x }  =  { A } )
65xpeq2d 4885 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { A } ) )
76seqeq3d 11835 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  seq 1 (  x.  ,  ( NN  X.  { x } ) )  =  seq 1
(  x.  ,  ( NN  X.  { A } ) ) )
87, 1fveq12d 5718 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { x }
) ) `  y
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
91negeqd 9625 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  -> 
-u y  =  -u N )
107, 9fveq12d 5718 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { x }
) ) `  -u y
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) )
1110oveq2d 6128 . . . 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 3837 . . 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 3835 . 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 11887 . 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 9402 . . 3  |-  1  e.  _V
16 fvex 5722 . . . 4  |-  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N )  e.  _V
17 ovex 6137 . . . 4  |-  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) )  e. 
_V
1816, 17ifex 3879 . . 3  |-  if ( 0  <  N , 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) )  e.  _V
1915, 18ifex 3879 . 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 6242 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ifcif 3812   {csn 3898   class class class wbr 4313    X. cxp 4859   ` cfv 5439  (class class class)co 6112   CCcc 9301   0cc0 9303   1c1 9304    x. cmul 9308    < clt 9439   -ucneg 9617    / cdiv 10014   NNcn 10343   ZZcz 10667    seqcseq 11827   ^cexp 11886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-1cn 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-recs 6853  df-rdg 6887  df-neg 9619  df-seq 11828  df-exp 11887
This theorem is referenced by:  expnnval  11889  exp0  11890  expneg  11894
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