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Theorem expval 11122
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 2304 . . 3  |-  ( ( x  =  A  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
31breq2d 4051 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  ( 0  <  y  <->  0  <  N ) )
4 simpl 443 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  N )  ->  x  =  A )
54sneqd 3666 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  N )  ->  { x }  =  { A } )
65xpeq2d 4729 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { A } ) )
76seqeq3d 11070 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  seq  1 (  x.  ,  ( NN  X.  { x } ) )  =  seq  1
(  x.  ,  ( NN  X.  { A } ) ) )
87, 1fveq12d 5547 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
91negeqd 9062 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  -> 
-u y  =  -u N )
107, 9fveq12d 5547 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  -u y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) )
1110oveq2d 5890 . . . 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 3600 . . 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 3598 . 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 11121 . 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 8849 . . 3  |-  1  e.  _V
16 fvex 5555 . . . 4  |-  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N )  e.  _V
17 ovex 5899 . . . 4  |-  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) )  e. 
_V
1816, 17ifex 3636 . . 3  |-  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) )  e.  _V
1915, 18ifex 3636 . 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 5994 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 1632    e. wcel 1696   ifcif 3578   {csn 3653   class class class wbr 4039    X. cxp 4703   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883   -ucneg 9054    / cdiv 9439   NNcn 9762   ZZcz 10040    seq cseq 11062   ^cexp 11120
This theorem is referenced by:  expnnval  11123  exp0  11124  expneg  11127
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-1cn 8811
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-neg 9056  df-seq 11063  df-exp 11121
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