MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expval Unicode version

Theorem expval 11304
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 2388 . . 3  |-  ( ( x  =  A  /\  y  =  N )  ->  ( y  =  0  <-> 
N  =  0 ) )
31breq2d 4158 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  ( 0  <  y  <->  0  <  N ) )
4 simpl 444 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  N )  ->  x  =  A )
54sneqd 3763 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  N )  ->  { x }  =  { A } )
65xpeq2d 4835 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { A } ) )
76seqeq3d 11251 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  seq  1 (  x.  ,  ( NN  X.  { x } ) )  =  seq  1
(  x.  ,  ( NN  X.  { A } ) ) )
87, 1fveq12d 5667 . . . 4  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
91negeqd 9225 . . . . . 6  |-  ( ( x  =  A  /\  y  =  N )  -> 
-u y  =  -u N )
107, 9fveq12d 5667 . . . . 5  |-  ( ( x  =  A  /\  y  =  N )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { x }
) ) `  -u y
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) )
1110oveq2d 6029 . . . 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 3697 . . 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 3695 . 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 11303 . 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 9012 . . 3  |-  1  e.  _V
16 fvex 5675 . . . 4  |-  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N )  e.  _V
17 ovex 6038 . . . 4  |-  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) )  e. 
_V
1816, 17ifex 3733 . . 3  |-  if ( 0  <  N , 
(  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
) ,  ( 1  /  (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) )  e.  _V
1915, 18ifex 3733 . 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 6136 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 3675   {csn 3750   class class class wbr 4146    X. cxp 4809   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    x. cmul 8921    < clt 9046   -ucneg 9217    / cdiv 9602   NNcn 9925   ZZcz 10207    seq cseq 11243   ^cexp 11302
This theorem is referenced by:  expnnval  11305  exp0  11306  expneg  11309
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-1cn 8974
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-recs 6562  df-rdg 6597  df-neg 9219  df-seq 11244  df-exp 11303
  Copyright terms: Public domain W3C validator