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Theorem expival 9422
Description: Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
Assertion
Ref Expression
expival  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
( A ^ N
)  =  if ( N  =  0 ,  1 ,  if ( 0  <  N , 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) ) ) )

Proof of Theorem expival
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iftrue 3364 . . . . 5  |-  ( N  =  0  ->  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) ) )  =  1 )
2 ax-1cn 7035 . . . . 5  |-  1  e.  CC
31, 2syl6eqel 2144 . . . 4  |-  ( N  =  0  ->  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) ) )  e.  CC )
43a1i 9 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
( N  =  0  ->  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) ) )  e.  CC ) )
5 elnnz 8312 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
6 elnnuz 8605 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
75, 6bitr3i 179 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  0  <  N )  <->  N  e.  ( ZZ>= `  1 )
)
87biimpi 117 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  0  <  N )  ->  N  e.  ( ZZ>= ` 
1 ) )
98adantll 453 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  0  <  N
)  ->  N  e.  ( ZZ>= `  1 )
)
10 cnex 7063 . . . . . . . . . . . 12  |-  CC  e.  _V
1110a1i 9 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  0  <  N
)  ->  CC  e.  _V )
12 simpl 106 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  z  e.  ( ZZ>= ` 
1 ) )  ->  A  e.  CC )
13 elnnuz 8605 . . . . . . . . . . . . . . . 16  |-  ( z  e.  NN  <->  z  e.  ( ZZ>= `  1 )
)
14 fvconst2g 5403 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  z  e.  NN )  ->  ( ( NN  X.  { A } ) `  z )  =  A )
1513, 14sylan2br 276 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  z  e.  ( ZZ>= ` 
1 ) )  -> 
( ( NN  X.  { A } ) `  z )  =  A )
1615eleq1d 2122 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  z  e.  ( ZZ>= ` 
1 ) )  -> 
( ( ( NN 
X.  { A }
) `  z )  e.  CC  <->  A  e.  CC ) )
1712, 16mpbird 160 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  z  e.  ( ZZ>= ` 
1 ) )  -> 
( ( NN  X.  { A } ) `  z )  e.  CC )
1817adantlr 454 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  z  e.  (
ZZ>= `  1 ) )  ->  ( ( NN 
X.  { A }
) `  z )  e.  CC )
1918adantlr 454 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  0  <  N )  /\  z  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  z )  e.  CC )
20 mulcl 7066 . . . . . . . . . . . 12  |-  ( ( z  e.  CC  /\  w  e.  CC )  ->  ( z  x.  w
)  e.  CC )
2120adantl 266 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  0  <  N )  /\  (
z  e.  CC  /\  w  e.  CC )
)  ->  ( z  x.  w )  e.  CC )
229, 11, 19, 21iseqcl 9387 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  0  <  N
)  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N )  e.  CC )
23 iftrue 3364 . . . . . . . . . . . 12  |-  ( 0  <  N  ->  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ,  CC ) `
 N ) )
2423eleq1d 2122 . . . . . . . . . . 11  |-  ( 0  <  N  ->  ( if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC  <->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N )  e.  CC ) )
2524adantl 266 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  0  <  N
)  ->  ( if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC  <->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N )  e.  CC ) )
2622, 25mpbird 160 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  0  <  N
)  ->  if (
0  <  N , 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) )  e.  CC )
2726ex 112 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( 0  <  N  ->  if ( 0  < 
N ,  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC ) )
2827adantr 265 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  -.  N  =  0 )  ->  (
0  <  N  ->  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC ) )
29283adantl3 1073 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  (
0  <  N  ->  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC ) )
30 simpll2 955 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  ZZ )
3130znegcld 8421 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  ZZ )
32 simpr 107 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <  N )
3330zred 8419 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  RR )
34 0red 7086 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  e.  RR )
3533, 34lenltd 7193 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <_  0  <->  -.  0  <  N ) )
3632, 35mpbird 160 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  <_  0 )
37 simplr 490 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  N  =  0 )
3837neneqad 2299 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  =/=  0 )
3938necomd 2306 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  =/=  N )
40 0z 8313 . . . . . . . . . . . . . . . . 17  |-  0  e.  ZZ
41 zltlen 8377 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
4240, 41mpan2 409 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ZZ  ->  ( N  <  0  <->  ( N  <_  0  /\  0  =/= 
N ) ) )
43423ad2ant2 937 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
( N  <  0  <->  ( N  <_  0  /\  0  =/=  N ) ) )
4443ad2antrr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <  0  <->  ( N  <_  0  /\  0  =/= 
N ) ) )
4536, 39, 44mpbir2and 862 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  <  0 )
4633lt0neg1d 7581 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <  0  <->  0  <  -u N ) )
4745, 46mpbid 139 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  <  -u N )
48 elnnz 8312 . . . . . . . . . . . 12  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
4931, 47, 48sylanbrc 402 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  NN )
50 elnnuz 8605 . . . . . . . . . . 11  |-  ( -u N  e.  NN  <->  -u N  e.  ( ZZ>= `  1 )
)
5149, 50sylib 131 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  ( ZZ>= `  1 )
)
5210a1i 9 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  CC  e.  _V )
53173ad2antl1 1077 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  z  e.  (
ZZ>= `  1 ) )  ->  ( ( NN 
X.  { A }
) `  z )  e.  CC )
5453adantlr 454 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  z  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  z )  e.  CC )
5554adantlr 454 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  /\  z  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { A }
) `  z )  e.  CC )
5620adantl 266 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  /\  (
z  e.  CC  /\  w  e.  CC )
)  ->  ( z  x.  w )  e.  CC )
5751, 52, 55, 56iseqcl 9387 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N )  e.  CC )
58 simpll1 954 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A  e.  CC )
59 expivallem 9421 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A #  0  /\  -u N  e.  NN )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N ) #  0 )
60593com23 1121 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  -u N  e.  NN  /\  A #  0 )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N ) #  0 )
61603expia 1117 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u N  e.  NN )  ->  ( A #  0  ->  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) #  0 ) )
6258, 49, 61syl2anc 397 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( A #  0  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N ) #  0 ) )
6339neneqd 2241 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  =  N )
64 ioran 679 . . . . . . . . . . . . 13  |-  ( -.  ( 0  <  N  \/  0  =  N
)  <->  ( -.  0  <  N  /\  -.  0  =  N ) )
6532, 63, 64sylanbrc 402 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  ( 0  <  N  \/  0  =  N
) )
66 zleloe 8349 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  ( 0  <  N  \/  0  =  N )
) )
6740, 66mpan 408 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  (
0  <_  N  <->  ( 0  <  N  \/  0  =  N ) ) )
68673ad2ant2 937 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
( 0  <_  N  <->  ( 0  <  N  \/  0  =  N )
) )
6968ad2antrr 465 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (
0  <_  N  <->  ( 0  <  N  \/  0  =  N ) ) )
7065, 69mtbird 608 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <_  N )
7170pm2.21d 559 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (
0  <_  N  ->  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) #  0 ) )
72 simpll3 956 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( A #  0  \/  0  <_  N ) )
7362, 71, 72mpjaod 648 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N ) #  0 )
7457, 73recclapd 7832 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N ) )  e.  CC )
75 iffalse 3367 . . . . . . . . . 10  |-  ( -.  0  <  N  ->  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  =  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N ) ) )
7675eleq1d 2122 . . . . . . . . 9  |-  ( -.  0  <  N  -> 
( if ( 0  <  N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) )  e.  CC  <->  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u N ) )  e.  CC ) )
7776adantl 266 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC  <->  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) )  e.  CC ) )
7874, 77mpbird 160 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC )
7978ex 112 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  ( -.  0  <  N  ->  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC ) )
80 zdclt 8376 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  -> DECID  0  <  N )
8140, 80mpan 408 . . . . . . . . . 10  |-  ( N  e.  ZZ  -> DECID  0  <  N )
82 df-dc 754 . . . . . . . . . 10  |-  (DECID  0  < 
N  <->  ( 0  < 
N  \/  -.  0  <  N ) )
8381, 82sylib 131 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
0  <  N  \/  -.  0  <  N ) )
8483adantl 266 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( 0  <  N  \/  -.  0  <  N
) )
8584adantr 265 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ )  /\  -.  N  =  0 )  ->  (
0  <  N  \/  -.  0  <  N ) )
86853adantl3 1073 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  (
0  <  N  \/  -.  0  <  N ) )
8729, 79, 86mpjaod 648 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC )
88 iffalse 3367 . . . . . . 7  |-  ( -.  N  =  0  ->  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) ) )  =  if ( 0  <  N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) ) )
8988eleq1d 2122 . . . . . 6  |-  ( -.  N  =  0  -> 
( if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) ) )  e.  CC  <->  if ( 0  < 
N ,  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) )  e.  CC ) )
9089adantl 266 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  ( if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) ) )  e.  CC  <->  if (
0  <  N , 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) )  e.  CC ) )
9187, 90mpbird 160 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) ) )  e.  CC )
9291ex 112 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
( -.  N  =  0  ->  if ( N  =  0 , 
1 ,  if ( 0  <  N , 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) ) )  e.  CC ) )
93 zdceq 8374 . . . . . 6  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
9440, 93mpan2 409 . . . . 5  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
95 df-dc 754 . . . . 5  |-  (DECID  N  =  0  <->  ( N  =  0  \/  -.  N  =  0 ) )
9694, 95sylib 131 . . . 4  |-  ( N  e.  ZZ  ->  ( N  =  0  \/  -.  N  =  0
) )
97963ad2ant2 937 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
( N  =  0  \/  -.  N  =  0 ) )
984, 92, 97mpjaod 648 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) ) )  e.  CC )
99 sneq 3414 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
10099xpeq2d 4397 . . . . . . 7  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
101 iseqeq3 9380 . . . . . . 7  |-  ( ( NN  X.  { x } )  =  ( NN  X.  { A } )  ->  seq 1 (  x.  , 
( NN  X.  {
x } ) ,  CC )  =  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) )
102100, 101syl 14 . . . . . 6  |-  ( x  =  A  ->  seq 1 (  x.  , 
( NN  X.  {
x } ) ,  CC )  =  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) )
103102fveq1d 5208 . . . . 5  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ,  CC ) `  y
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  y )
)
104102fveq1d 5208 . . . . . 6  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ,  CC ) `  -u y
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u y ) )
105104oveq2d 5556 . . . . 5  |-  ( x  =  A  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ,  CC ) `  -u y
) )  =  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u y ) ) )
106103, 105ifeq12d 3375 . . . 4  |-  ( x  =  A  ->  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { x } ) ,  CC ) `  y ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { x } ) ,  CC ) `  -u y ) ) )  =  if ( 0  <  y ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  y
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u y ) ) ) )
107106ifeq2d 3374 . . 3  |-  ( x  =  A  ->  if ( y  =  0 ,  1 ,  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { x } ) ,  CC ) `  y ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { x } ) ,  CC ) `  -u y ) ) ) )  =  if ( y  =  0 ,  1 ,  if ( 0  <  y ,  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ,  CC ) `
 y ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u y ) ) ) ) )
108 eqeq1 2062 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
109 breq2 3796 . . . . 5  |-  ( y  =  N  ->  (
0  <  y  <->  0  <  N ) )
110 fveq2 5206 . . . . 5  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  y )  =  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N )
)
111 negeq 7267 . . . . . . 7  |-  ( y  =  N  ->  -u y  =  -u N )
112111fveq2d 5210 . . . . . 6  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u y )  =  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) )
113112oveq2d 5556 . . . . 5  |-  ( y  =  N  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  -u y ) )  =  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) )
114109, 110, 113ifbieq12d 3382 . . . 4  |-  ( y  =  N  ->  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  y ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u y
) ) )  =  if ( 0  < 
N ,  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) ) )
115108, 114ifbieq2d 3380 . . 3  |-  ( y  =  N  ->  if ( y  =  0 ,  1 ,  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  y ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u y
) ) ) )  =  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) ) ) )
116 df-iexp 9420 . . 3  |-  ^  =  ( x  e.  CC ,  y  e.  ZZ  |->  if ( y  =  0 ,  1 ,  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { x } ) ,  CC ) `  y ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { x } ) ,  CC ) `  -u y ) ) ) ) )
117107, 115, 116ovmpt2g 5663 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N ) ,  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N
) ) ) )  e.  CC )  -> 
( A ^ N
)  =  if ( N  =  0 ,  1 ,  if ( 0  <  N , 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) ) ) )
11898, 117syld3an3 1191 1  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
( A ^ N
)  =  if ( N  =  0 ,  1 ,  if ( 0  <  N , 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  -u N ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639  DECID wdc 753    /\ w3a 896    = wceq 1259    e. wcel 1409    =/= wne 2220   _Vcvv 2574   ifcif 3359   {csn 3403   class class class wbr 3792    X. cxp 4371   ` cfv 4930  (class class class)co 5540   CCcc 6945   0cc0 6947   1c1 6948    x. cmul 6952    < clt 7119    <_ cle 7120   -ucneg 7246   # cap 7646    / cdiv 7725   NNcn 7990   ZZcz 8302   ZZ>=cuz 8569    seqcseq 9375   ^cexp 9419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-iseq 9376  df-iexp 9420
This theorem is referenced by:  expinnval  9423  exp0  9424  expnegap0  9428
  Copyright terms: Public domain W3C validator