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Theorem exp3val 10522
Description: Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
Assertion
Ref Expression
exp3val  |-  ( ( 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 } ) ) `
 N ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) ) ) )

Proof of Theorem exp3val
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1cnd 7973 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  N  =  0 )  ->  1  e.  CC )
2 simp1 997 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  A  e.  CC )
3 nnuz 9563 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
4 1zzd 9280 . . . . . . . 8  |-  ( A  e.  CC  ->  1  e.  ZZ )
5 fvconst2g 5731 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
6 simpl 109 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  A  e.  CC )
75, 6eqeltrd 2254 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  e.  CC )
8 mulcl 7938 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98adantl 277 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
103, 4, 7, 9seqf 10461 . . . . . . 7  |-  ( A  e.  CC  ->  seq 1 (  x.  , 
( NN  X.  { A } ) ) : NN --> CC )
112, 10syl 14 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  seq 1 (  x.  , 
( NN  X.  { A } ) ) : NN --> CC )
1211ad2antrr 488 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  seq 1
(  x.  ,  ( NN  X.  { A } ) ) : NN --> CC )
13 simp2 998 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  N  e.  ZZ )
1413ad2antrr 488 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  N  e.  ZZ )
15 simpr 110 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  0  <  N )
16 elnnz 9263 . . . . . 6  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
1714, 15, 16sylanbrc 417 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  N  e.  NN )
1812, 17ffvelcdmd 5653 . . . 4  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N )  e.  CC )
1911ad2antrr 488 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  seq 1 (  x.  , 
( NN  X.  { A } ) ) : NN --> CC )
2013ad2antrr 488 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  ZZ )
2120znegcld 9377 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  ZZ )
22 simpr 110 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <  N )
23 simplr 528 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  N  =  0 )
24 eqcom 2179 . . . . . . . . . . . 12  |-  ( N  =  0  <->  0  =  N )
2523, 24sylnib 676 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  =  N )
26 ioran 752 . . . . . . . . . . 11  |-  ( -.  ( 0  <  N  \/  0  =  N
)  <->  ( -.  0  <  N  /\  -.  0  =  N ) )
2722, 25, 26sylanbrc 417 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  ( 0  <  N  \/  0  =  N
) )
28 0zd 9265 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  e.  ZZ )
29 zleloe 9300 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  ( 0  <  N  \/  0  =  N )
) )
3028, 20, 29syl2anc 411 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (
0  <_  N  <->  ( 0  <  N  \/  0  =  N ) ) )
3127, 30mtbird 673 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <_  N )
32 zltnle 9299 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <  0  <->  -.  0  <_  N )
)
3320, 28, 32syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <  0  <->  -.  0  <_  N ) )
3431, 33mpbird 167 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  <  0 )
3520zred 9375 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  RR )
3635lt0neg1d 8472 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <  0  <->  0  <  -u N ) )
3734, 36mpbid 147 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  <  -u N )
38 elnnz 9263 . . . . . . 7  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
3921, 37, 38sylanbrc 417 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  NN )
4019, 39ffvelcdmd 5653 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N )  e.  CC )
412ad2antrr 488 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A  e.  CC )
42 simpll3 1038 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( A #  0  \/  0  <_  N ) )
4331, 42ecased 1349 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A #  0 )
4441, 43, 39exp3vallem 10521 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) #  0 )
4540, 44recclapd 8738 . . . 4  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) )  e.  CC )
46 0zd 9265 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  0  e.  ZZ )
47 simpl2 1001 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  N  e.  ZZ )
48 zdclt 9330 . . . . 5  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  -> DECID  0  <  N )
4946, 47, 48syl2anc 411 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  -> DECID  0  <  N )
5018, 45, 49ifcldadc 3564 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  N ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) )  e.  CC )
51 0zd 9265 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
0  e.  ZZ )
52 zdceq 9328 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
5313, 51, 52syl2anc 411 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> DECID  N  =  0 )
541, 50, 53ifcldadc 3564 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  N ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) ) )  e.  CC )
55 sneq 3604 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
5655xpeq2d 4651 . . . . . . 7  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
5756seqeq3d 10453 . . . . . 6  |-  ( x  =  A  ->  seq 1 (  x.  , 
( NN  X.  {
x } ) )  =  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) )
5857fveq1d 5518 . . . . 5  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  y
) )
5957fveq1d 5518 . . . . . 6  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u y
) )
6059oveq2d 5891 . . . . 5  |-  ( x  =  A  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
6158, 60ifeq12d 3554 . . . 4  |-  ( x  =  A  ->  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { x } ) ) `  y ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) ) )  =  if ( 0  <  y ,  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  y
) ,  ( 1  /  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  -u y ) ) ) )
6261ifeq2d 3553 . . 3  |-  ( x  =  A  ->  if ( y  =  0 ,  1 ,  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { x } ) ) `  y ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) )  =  if ( y  =  0 ,  1 ,  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  y ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y ) ) ) ) )
63 eqeq1 2184 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
64 breq2 4008 . . . . 5  |-  ( y  =  N  ->  (
0  <  y  <->  0  <  N ) )
65 fveq2 5516 . . . . 5  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  y )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) )
66 negeq 8150 . . . . . . 7  |-  ( y  =  N  ->  -u y  =  -u N )
6766fveq2d 5520 . . . . . 6  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 -u N ) )
6867oveq2d 5891 . . . . 5  |-  ( y  =  N  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) )
6964, 65, 68ifbieq12d 3561 . . . 4  |-  ( y  =  N  ->  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  y ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y ) ) )  =  if ( 0  <  N ,  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) ) )
7063, 69ifbieq2d 3559 . . 3  |-  ( y  =  N  ->  if ( y  =  0 ,  1 ,  if ( 0  <  y ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  y ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -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 ) ) ) ) )
71 df-exp 10520 . . 3  |-  ^  =  ( 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 ) ) ) ) )
7262, 70, 71ovmpog 6009 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq 1
(  x.  ,  ( NN  X.  { A } ) ) `  N ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) ) )  e.  CC )  ->  ( 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 ) ) ) ) )
7354, 72syld3an3 1283 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 } ) ) `
 N ) ,  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148   ifcif 3535   {csn 3593   class class class wbr 4004    X. cxp 4625   -->wf 5213   ` cfv 5217  (class class class)co 5875   CCcc 7809   0cc0 7811   1c1 7812    x. cmul 7816    < clt 7992    <_ cle 7993   -ucneg 8129   # cap 8538    / cdiv 8629   NNcn 8919   ZZcz 9253    seqcseq 10445   ^cexp 10519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-seqfrec 10446  df-exp 10520
This theorem is referenced by:  expnnval  10523  exp0  10524  expnegap0  10528
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