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Theorem exp3val 9957
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 7504 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  N  =  0 )  ->  1  e.  CC )
2 simp1 943 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  A  e.  CC )
3 nnuz 9054 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
4 1zzd 8777 . . . . . . . 8  |-  ( A  e.  CC  ->  1  e.  ZZ )
5 fvconst2g 5511 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
6 simpl 107 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  A  e.  CC )
75, 6eqeltrd 2164 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  e.  CC )
8 mulcl 7469 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98adantl 271 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
103, 4, 7, 9seqf 9880 . . . . . . 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 472 . . . . 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 944 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  N  e.  ZZ )
1413ad2antrr 472 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  N  e.  ZZ )
15 simpr 108 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  0  <  N )
16 elnnz 8760 . . . . . 6  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
1714, 15, 16sylanbrc 408 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  N  e.  NN )
1812, 17ffvelrnd 5435 . . . 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 472 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  seq 1 (  x.  , 
( NN  X.  { A } ) ) : NN --> CC )
2013ad2antrr 472 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  ZZ )
2120znegcld 8870 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  ZZ )
22 simpr 108 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <  N )
23 simplr 497 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  N  =  0 )
24 eqcom 2090 . . . . . . . . . . . 12  |-  ( N  =  0  <->  0  =  N )
2523, 24sylnib 636 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  =  N )
26 ioran 704 . . . . . . . . . . 11  |-  ( -.  ( 0  <  N  \/  0  =  N
)  <->  ( -.  0  <  N  /\  -.  0  =  N ) )
2722, 25, 26sylanbrc 408 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  ( 0  <  N  \/  0  =  N
) )
28 0zd 8762 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  e.  ZZ )
29 zleloe 8797 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  ( 0  <  N  \/  0  =  N )
) )
3028, 20, 29syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (
0  <_  N  <->  ( 0  <  N  \/  0  =  N ) ) )
3127, 30mtbird 633 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <_  N )
32 zltnle 8796 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <  0  <->  -.  0  <_  N )
)
3320, 28, 32syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <  0  <->  -.  0  <_  N ) )
3431, 33mpbird 165 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  <  0 )
3520zred 8868 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  RR )
3635lt0neg1d 7993 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <  0  <->  0  <  -u N ) )
3734, 36mpbid 145 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  <  -u N )
38 elnnz 8760 . . . . . . 7  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
3921, 37, 38sylanbrc 408 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  NN )
4019, 39ffvelrnd 5435 . . . . 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 472 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A  e.  CC )
42 simpll3 984 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( A #  0  \/  0  <_  N ) )
4331, 42ecased 1285 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A #  0 )
4441, 43, 39exp3vallem 9956 . . . . 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 8248 . . . 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 8762 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  0  e.  ZZ )
47 simpl2 947 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  N  e.  ZZ )
48 zdclt 8824 . . . . 5  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  -> DECID  0  <  N )
4946, 47, 48syl2anc 403 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  -> DECID  0  <  N )
5018, 45, 49ifcldadc 3420 . . 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 8762 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
0  e.  ZZ )
52 zdceq 8822 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
5313, 51, 52syl2anc 403 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> DECID  N  =  0 )
541, 50, 53ifcldadc 3420 . 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 3457 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
5655xpeq2d 4462 . . . . . . 7  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
5756seqeq3d 9866 . . . . . 6  |-  ( x  =  A  ->  seq 1 (  x.  , 
( NN  X.  {
x } ) )  =  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) )
5857fveq1d 5307 . . . . 5  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  y
) )
5957fveq1d 5307 . . . . . 6  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u y
) )
6059oveq2d 5668 . . . . 5  |-  ( x  =  A  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
6158, 60ifeq12d 3410 . . . 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 3409 . . 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 2094 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
64 breq2 3849 . . . . 5  |-  ( y  =  N  ->  (
0  <  y  <->  0  <  N ) )
65 fveq2 5305 . . . . 5  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  y )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) )
66 negeq 7675 . . . . . . 7  |-  ( y  =  N  ->  -u y  =  -u N )
6766fveq2d 5309 . . . . . 6  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 -u N ) )
6867oveq2d 5668 . . . . 5  |-  ( y  =  N  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) )
6964, 65, 68ifbieq12d 3417 . . . 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 3415 . . 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 9955 . . 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, 71ovmpt2g 5779 . 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 1219 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 102    <-> wb 103    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289    e. wcel 1438   ifcif 3393   {csn 3446   class class class wbr 3845    X. cxp 4436   -->wf 5011   ` cfv 5015  (class class class)co 5652   CCcc 7348   0cc0 7350   1c1 7351    x. cmul 7355    < clt 7522    <_ cle 7523   -ucneg 7654   # cap 8058    / cdiv 8139   NNcn 8422   ZZcz 8750    seqcseq 9852   ^cexp 9954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-iseq 9853  df-seq3 9854  df-exp 9955
This theorem is referenced by:  expnnval  9958  exp0  9959  expnegap0  9963
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