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Theorem exp3val 10235
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 7746 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  N  =  0 )  ->  1  e.  CC )
2 simp1 964 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  A  e.  CC )
3 nnuz 9310 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
4 1zzd 9032 . . . . . . . 8  |-  ( A  e.  CC  ->  1  e.  ZZ )
5 fvconst2g 5600 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
6 simpl 108 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  A  e.  CC )
75, 6eqeltrd 2192 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  e.  CC )
8 mulcl 7711 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98adantl 273 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
103, 4, 7, 9seqf 10174 . . . . . . 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 477 . . . . 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 965 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  N  e.  ZZ )
1413ad2antrr 477 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  N  e.  ZZ )
15 simpr 109 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  0  <  N )
16 elnnz 9015 . . . . . 6  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
1714, 15, 16sylanbrc 411 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  N  e.  NN )
1812, 17ffvelrnd 5522 . . . 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 477 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  seq 1 (  x.  , 
( NN  X.  { A } ) ) : NN --> CC )
2013ad2antrr 477 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  ZZ )
2120znegcld 9126 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  ZZ )
22 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <  N )
23 simplr 502 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  N  =  0 )
24 eqcom 2117 . . . . . . . . . . . 12  |-  ( N  =  0  <->  0  =  N )
2523, 24sylnib 648 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  =  N )
26 ioran 724 . . . . . . . . . . 11  |-  ( -.  ( 0  <  N  \/  0  =  N
)  <->  ( -.  0  <  N  /\  -.  0  =  N ) )
2722, 25, 26sylanbrc 411 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  ( 0  <  N  \/  0  =  N
) )
28 0zd 9017 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  e.  ZZ )
29 zleloe 9052 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  ( 0  <  N  \/  0  =  N )
) )
3028, 20, 29syl2anc 406 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (
0  <_  N  <->  ( 0  <  N  \/  0  =  N ) ) )
3127, 30mtbird 645 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <_  N )
32 zltnle 9051 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <  0  <->  -.  0  <_  N )
)
3320, 28, 32syl2anc 406 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <  0  <->  -.  0  <_  N ) )
3431, 33mpbird 166 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  <  0 )
3520zred 9124 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  RR )
3635lt0neg1d 8241 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( N  <  0  <->  0  <  -u N ) )
3734, 36mpbid 146 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  <  -u N )
38 elnnz 9015 . . . . . . 7  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
3921, 37, 38sylanbrc 411 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  NN )
4019, 39ffvelrnd 5522 . . . . 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 477 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A  e.  CC )
42 simpll3 1005 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( A #  0  \/  0  <_  N ) )
4331, 42ecased 1310 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A #  0 )
4441, 43, 39exp3vallem 10234 . . . . 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 8501 . . . 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 9017 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  0  e.  ZZ )
47 simpl2 968 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  N  e.  ZZ )
48 zdclt 9079 . . . . 5  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  -> DECID  0  <  N )
4946, 47, 48syl2anc 406 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  -> DECID  0  <  N )
5018, 45, 49ifcldadc 3469 . . 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 9017 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
0  e.  ZZ )
52 zdceq 9077 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
5313, 51, 52syl2anc 406 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> DECID  N  =  0 )
541, 50, 53ifcldadc 3469 . 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 3506 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
5655xpeq2d 4531 . . . . . . 7  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
5756seqeq3d 10166 . . . . . 6  |-  ( x  =  A  ->  seq 1 (  x.  , 
( NN  X.  {
x } ) )  =  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) )
5857fveq1d 5389 . . . . 5  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  y
) )
5957fveq1d 5389 . . . . . 6  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u y
) )
6059oveq2d 5756 . . . . 5  |-  ( x  =  A  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
6158, 60ifeq12d 3459 . . . 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 3458 . . 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 2122 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
64 breq2 3901 . . . . 5  |-  ( y  =  N  ->  (
0  <  y  <->  0  <  N ) )
65 fveq2 5387 . . . . 5  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  y )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) )
66 negeq 7919 . . . . . . 7  |-  ( y  =  N  ->  -u y  =  -u N )
6766fveq2d 5391 . . . . . 6  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 -u N ) )
6867oveq2d 5756 . . . . 5  |-  ( y  =  N  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) )
6964, 65, 68ifbieq12d 3466 . . . 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 3464 . . 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 10233 . . 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 5871 . 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 1244 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 103    <-> wb 104    \/ wo 680  DECID wdc 802    /\ w3a 945    = wceq 1314    e. wcel 1463   ifcif 3442   {csn 3495   class class class wbr 3897    X. cxp 4505   -->wf 5087   ` cfv 5091  (class class class)co 5740   CCcc 7582   0cc0 7584   1c1 7585    x. cmul 7589    < clt 7764    <_ cle 7765   -ucneg 7898   # cap 8306    / cdiv 8392   NNcn 8677   ZZcz 9005    seqcseq 10158   ^cexp 10232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-n0 8929  df-z 9006  df-uz 9276  df-seqfrec 10159  df-exp 10233
This theorem is referenced by:  expnnval  10236  exp0  10237  expnegap0  10241
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