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Theorem exp3val 10250
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 7750 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  N  =  0 )  ->  1  e.  CC )
2 simp1 966 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  A  e.  CC )
3 nnuz 9317 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
4 1zzd 9039 . . . . . . . 8  |-  ( A  e.  CC  ->  1  e.  ZZ )
5 fvconst2g 5602 . . . . . . . . 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 2194 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  e.  CC )
8 mulcl 7715 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
98adantl 275 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
103, 4, 7, 9seqf 10189 . . . . . . 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 479 . . . . 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 967 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  ->  N  e.  ZZ )
1413ad2antrr 479 . . . . . 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 9022 . . . . . 6  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
1714, 15, 16sylanbrc 413 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  0  <  N )  ->  N  e.  NN )
1812, 17ffvelrnd 5524 . . . 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 479 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  seq 1 (  x.  , 
( NN  X.  { A } ) ) : NN --> CC )
2013ad2antrr 479 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  ZZ )
2120znegcld 9133 . . . . . . 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 504 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  N  =  0 )
24 eqcom 2119 . . . . . . . . . . . 12  |-  ( N  =  0  <->  0  =  N )
2523, 24sylnib 650 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  =  N )
26 ioran 726 . . . . . . . . . . 11  |-  ( -.  ( 0  <  N  \/  0  =  N
)  <->  ( -.  0  <  N  /\  -.  0  =  N ) )
2722, 25, 26sylanbrc 413 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  ( 0  <  N  \/  0  =  N
) )
28 0zd 9024 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  0  e.  ZZ )
29 zleloe 9059 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  ( 0  <  N  \/  0  =  N )
) )
3028, 20, 29syl2anc 408 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  (
0  <_  N  <->  ( 0  <  N  \/  0  =  N ) ) )
3127, 30mtbird 647 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -.  0  <_  N )
32 zltnle 9058 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <  0  <->  -.  0  <_  N )
)
3320, 28, 32syl2anc 408 . . . . . . . . 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 9131 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  N  e.  RR )
3635lt0neg1d 8245 . . . . . . . 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 9022 . . . . . . 7  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
3921, 37, 38sylanbrc 413 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  -u N  e.  NN )
4019, 39ffvelrnd 5524 . . . . 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 479 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A  e.  CC )
42 simpll3 1007 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  ( A #  0  \/  0  <_  N ) )
4331, 42ecased 1312 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0
)  /\  -.  0  <  N )  ->  A #  0 )
4441, 43, 39exp3vallem 10249 . . . . 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 8508 . . . 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 9024 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  0  e.  ZZ )
47 simpl2 970 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  ->  N  e.  ZZ )
48 zdclt 9086 . . . . 5  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  -> DECID  0  <  N )
4946, 47, 48syl2anc 408 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  /\  -.  N  =  0 )  -> DECID  0  <  N )
5018, 45, 49ifcldadc 3471 . . 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 9024 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> 
0  e.  ZZ )
52 zdceq 9084 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
5313, 51, 52syl2anc 408 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N ) )  -> DECID  N  =  0 )
541, 50, 53ifcldadc 3471 . 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 3508 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
5655xpeq2d 4533 . . . . . . 7  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
5756seqeq3d 10181 . . . . . 6  |-  ( x  =  A  ->  seq 1 (  x.  , 
( NN  X.  {
x } ) )  =  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) )
5857fveq1d 5391 . . . . 5  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  y
) )
5957fveq1d 5391 . . . . . 6  |-  ( x  =  A  ->  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  -u y
) )
6059oveq2d 5758 . . . . 5  |-  ( x  =  A  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( 1  / 
(  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
6158, 60ifeq12d 3461 . . . 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 3460 . . 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 2124 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
64 breq2 3903 . . . . 5  |-  ( y  =  N  ->  (
0  <  y  <->  0  <  N ) )
65 fveq2 5389 . . . . 5  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  y )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N ) )
66 negeq 7923 . . . . . . 7  |-  ( y  =  N  ->  -u y  =  -u N )
6766fveq2d 5393 . . . . . 6  |-  ( y  =  N  ->  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 -u N ) )
6867oveq2d 5758 . . . . 5  |-  ( y  =  N  ->  (
1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( 1  /  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  -u N ) ) )
6964, 65, 68ifbieq12d 3468 . . . 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 3466 . . 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 10248 . . 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 5873 . 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 1246 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 682  DECID wdc 804    /\ w3a 947    = wceq 1316    e. wcel 1465   ifcif 3444   {csn 3497   class class class wbr 3899    X. cxp 4507   -->wf 5089   ` cfv 5093  (class class class)co 5742   CCcc 7586   0cc0 7588   1c1 7589    x. cmul 7593    < clt 7768    <_ cle 7769   -ucneg 7902   # cap 8310    / cdiv 8399   NNcn 8684   ZZcz 9012    seqcseq 10173   ^cexp 10247
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283  df-seqfrec 10174  df-exp 10248
This theorem is referenced by:  expnnval  10251  exp0  10252  expnegap0  10256
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