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Theorem expval 11106
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 2291 . . 3  |-  ( ( x  =  A  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
31breq2d 4035 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  ( 0  <  y  <->  0  <  N ) )
4 simpl 443 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  N )  ->  x  =  A )
54sneqd 3653 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  N )  ->  { x }  =  { A } )
65xpeq2d 4713 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { A } ) )
76seqeq3d 11054 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  seq  1 (  x.  ,  ( NN  X.  { x } ) )  =  seq  1
(  x.  ,  ( NN  X.  { A } ) ) )
87, 1fveq12d 5531 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
91negeqd 9046 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  -> 
-u y  =  -u N )
107, 9fveq12d 5531 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  -u y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) )
1110oveq2d 5874 . . . 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 3587 . . 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 3585 . 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 11105 . 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 8833 . . 3  |-  1  e.  _V
16 fvex 5539 . . . 4  |-  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N )  e.  _V
17 ovex 5883 . . . 4  |-  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) )  e. 
_V
1816, 17ifex 3623 . . 3  |-  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) )  e.  _V
1915, 18ifex 3623 . 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 5978 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 1623    e. wcel 1684   ifcif 3565   {csn 3640   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867   -ucneg 9038    / cdiv 9423   NNcn 9746   ZZcz 10024    seq cseq 11046   ^cexp 11104
This theorem is referenced by:  expnnval  11107  exp0  11108  expneg  11111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-1cn 8795
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-neg 9040  df-seq 11047  df-exp 11105
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