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Theorem aareccl 20112
Description: The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
aareccl  |-  ( ( A  e.  AA  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  AA )

Proof of Theorem aareccl
Dummy variables  f 
g  k  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elaa 20102 . . . 4  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( f `
 A )  =  0 ) )
21simprbi 451 . . 3  |-  ( A  e.  AA  ->  E. f  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( f `
 A )  =  0 )
32adantr 452 . 2  |-  ( ( A  e.  AA  /\  A  =/=  0 )  ->  E. f  e.  (
(Poly `  ZZ )  \  { 0 p }
) ( f `  A )  =  0 )
4 aacn 20103 . . . . 5  |-  ( A  e.  AA  ->  A  e.  CC )
5 reccl 9619 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
64, 5sylan 458 . . . 4  |-  ( ( A  e.  AA  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
76adantr 452 . . 3  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 1  /  A )  e.  CC )
8 zsscn 10224 . . . . . . 7  |-  ZZ  C_  CC
98a1i 11 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ZZ  C_  CC )
10 simprl 733 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  f  e.  ( (Poly `  ZZ )  \  { 0 p }
) )
11 eldifsn 3872 . . . . . . . . 9  |-  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  <->  ( f  e.  (Poly `  ZZ )  /\  f  =/=  0 p ) )
1210, 11sylib 189 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f  e.  (Poly `  ZZ )  /\  f  =/=  0 p ) )
1312simpld 446 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  f  e.  (Poly `  ZZ ) )
14 dgrcl 20021 . . . . . . 7  |-  ( f  e.  (Poly `  ZZ )  ->  (deg `  f
)  e.  NN0 )
1513, 14syl 16 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  NN0 )
1613adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  f  e.  (Poly `  ZZ ) )
17 0z 10227 . . . . . . . 8  |-  0  e.  ZZ
18 eqid 2389 . . . . . . . . 9  |-  (coeff `  f )  =  (coeff `  f )
1918coef2 20019 . . . . . . . 8  |-  ( ( f  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  f ) : NN0 --> ZZ )
2016, 17, 19sylancl 644 . . . . . . 7  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  (coeff `  f
) : NN0 --> ZZ )
21 fznn0sub 11019 . . . . . . . 8  |-  ( k  e.  ( 0 ... (deg `  f )
)  ->  ( (deg `  f )  -  k
)  e.  NN0 )
2221adantl 453 . . . . . . 7  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( (deg `  f )  -  k
)  e.  NN0 )
2320, 22ffvelrnd 5812 . . . . . 6  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  e.  ZZ )
249, 15, 23elplyd 19990 . . . . 5  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  e.  (Poly `  ZZ ) )
25 0cn 9019 . . . . . 6  |-  0  e.  CC
26 eqid 2389 . . . . . . . . . 10  |-  (coeff `  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) )  =  (coeff `  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) )
2726coefv0 20035 . . . . . . . . 9  |-  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) )  e.  (Poly `  ZZ )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) `  0 )  =  ( (coeff `  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) ) `  0
) )
2824, 27syl 16 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) ) `  0
)  =  ( (coeff `  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) ) `  0
) )
2923zcnd 10310 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  e.  CC )
30 eqidd 2390 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) )
3124, 15, 29, 30coeeq2 20030 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (coeff `  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) )  =  ( k  e.  NN0  |->  if ( k  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) ,  0 ) ) )
3231fveq1d 5672 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (coeff `  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) ) `  0
)  =  ( ( k  e.  NN0  |->  if ( k  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) ,  0 ) ) `
 0 ) )
33 0nn0 10170 . . . . . . . . . 10  |-  0  e.  NN0
34 breq1 4158 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
k  <_  (deg `  f
)  <->  0  <_  (deg `  f ) ) )
35 oveq2 6030 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
(deg `  f )  -  k )  =  ( (deg `  f
)  -  0 ) )
3635fveq2d 5674 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  k ) )  =  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) )
37 eqidd 2390 . . . . . . . . . . . 12  |-  ( k  =  0  ->  0  =  0 )
3834, 36, 37ifbieq12d 3706 . . . . . . . . . . 11  |-  ( k  =  0  ->  if ( k  <_  (deg `  f ) ,  ( (coeff `  f ) `  ( (deg `  f
)  -  k ) ) ,  0 )  =  if ( 0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 ) )
39 eqid 2389 . . . . . . . . . . 11  |-  ( k  e.  NN0  |->  if ( k  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) ,  0 ) )  =  ( k  e. 
NN0  |->  if ( k  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) ,  0 ) )
40 fvex 5684 . . . . . . . . . . . 12  |-  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) )  e.  _V
41 c0ex 9020 . . . . . . . . . . . 12  |-  0  e.  _V
4240, 41ifex 3742 . . . . . . . . . . 11  |-  if ( 0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )  e. 
_V
4338, 39, 42fvmpt 5747 . . . . . . . . . 10  |-  ( 0  e.  NN0  ->  ( ( k  e.  NN0  |->  if ( k  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) ,  0 ) ) `
 0 )  =  if ( 0  <_ 
(deg `  f ) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 ) )
4433, 43ax-mp 8 . . . . . . . . 9  |-  ( ( k  e.  NN0  |->  if ( k  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) ,  0 ) ) `
 0 )  =  if ( 0  <_ 
(deg `  f ) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )
4515nn0ge0d 10211 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  0  <_  (deg `  f ) )
46 iftrue 3690 . . . . . . . . . . 11  |-  ( 0  <_  (deg `  f
)  ->  if (
0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )  =  ( (coeff `  f
) `  ( (deg `  f )  -  0 ) ) )
4745, 46syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  if ( 0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )  =  ( (coeff `  f
) `  ( (deg `  f )  -  0 ) ) )
4815nn0cnd 10210 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  CC )
4948subid1d 9334 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (deg `  f )  -  0 )  =  (deg `  f ) )
5049fveq2d 5674 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) )  =  ( (coeff `  f ) `  (deg `  f ) ) )
5147, 50eqtrd 2421 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  if ( 0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )  =  ( (coeff `  f
) `  (deg `  f
) ) )
5244, 51syl5eq 2433 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( ( k  e.  NN0  |->  if ( k  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) ,  0 ) ) `
 0 )  =  ( (coeff `  f
) `  (deg `  f
) ) )
5328, 32, 523eqtrd 2425 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) ) `  0
)  =  ( (coeff `  f ) `  (deg `  f ) ) )
5412simprd 450 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  f  =/=  0 p )
55 eqid 2389 . . . . . . . . . . 11  |-  (deg `  f )  =  (deg
`  f )
5655, 18dgreq0 20052 . . . . . . . . . 10  |-  ( f  e.  (Poly `  ZZ )  ->  ( f  =  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =  0 ) )
5713, 56syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f  =  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =  0 ) )
5857necon3bid 2587 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f  =/=  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 ) )
5954, 58mpbid 202 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 )
6053, 59eqnetrd 2570 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) ) `  0
)  =/=  0 )
61 ne0p 19995 . . . . . 6  |-  ( ( 0  e.  CC  /\  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) ) `  0
)  =/=  0 )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  =/=  0 p )
6225, 60, 61sylancr 645 . . . . 5  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  =/=  0 p )
63 eldifsn 3872 . . . . 5  |-  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) )  e.  ( (Poly `  ZZ )  \  {
0 p } )  <-> 
( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  e.  (Poly `  ZZ )  /\  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) )  =/=  0 p ) )
6424, 62, 63sylanbrc 646 . . . 4  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  e.  ( (Poly `  ZZ )  \  { 0 p }
) )
65 oveq1 6029 . . . . . . . . 9  |-  ( z  =  ( 1  /  A )  ->  (
z ^ k )  =  ( ( 1  /  A ) ^
k ) )
6665oveq2d 6038 . . . . . . . 8  |-  ( z  =  ( 1  /  A )  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) )  =  ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( 1  /  A ) ^ k ) ) )
6766sumeq2sdv 12427 . . . . . . 7  |-  ( z  =  ( 1  /  A )  ->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) )  =  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) ) )
68 eqid 2389 . . . . . . 7  |-  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) )
69 sumex 12410 . . . . . . 7  |-  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) )  e.  _V
7067, 68, 69fvmpt 5747 . . . . . 6  |-  ( ( 1  /  A )  e.  CC  ->  (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) `  ( 1  /  A ) )  =  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) ) )
717, 70syl 16 . . . . 5  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) ) `  (
1  /  A ) )  =  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) ) )
7218coef3 20020 . . . . . . . . . . 11  |-  ( f  e.  (Poly `  ZZ )  ->  (coeff `  f
) : NN0 --> CC )
7313, 72syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (coeff `  f
) : NN0 --> CC )
74 elfznn0 11017 . . . . . . . . . 10  |-  ( n  e.  ( 0 ... (deg `  f )
)  ->  n  e.  NN0 )
75 ffvelrn 5809 . . . . . . . . . 10  |-  ( ( (coeff `  f ) : NN0 --> CC  /\  n  e.  NN0 )  ->  (
(coeff `  f ) `  n )  e.  CC )
7673, 74, 75syl2an 464 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( (coeff `  f ) `  n
)  e.  CC )
774ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  A  e.  CC )
78 expcl 11328 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  n  e.  NN0 )  -> 
( A ^ n
)  e.  CC )
7977, 74, 78syl2an 464 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
n )  e.  CC )
8076, 79mulcld 9043 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( (coeff `  f ) `  n
)  x.  ( A ^ n ) )  e.  CC )
8177, 15expcld 11452 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( A ^
(deg `  f )
)  e.  CC )
8281adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
(deg `  f )
)  e.  CC )
83 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  A  =/=  0
)
8415nn0zd 10307 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  ZZ )
8577, 83, 84expne0d 11458 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( A ^
(deg `  f )
)  =/=  0 )
8685adantr 452 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
(deg `  f )
)  =/=  0 )
8780, 82, 86divcld 9724 . . . . . . 7  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( ( (coeff `  f ) `  n )  x.  ( A ^ n ) )  /  ( A ^
(deg `  f )
) )  e.  CC )
88 fveq2 5670 . . . . . . . . 9  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( (coeff `  f ) `  n
)  =  ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) ) )
89 oveq2 6030 . . . . . . . . 9  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( A ^ n )  =  ( A ^ (
( 0  +  (deg
`  f ) )  -  k ) ) )
9088, 89oveq12d 6040 . . . . . . . 8  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( (
(coeff `  f ) `  n )  x.  ( A ^ n ) )  =  ( ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) )  x.  ( A ^
( ( 0  +  (deg `  f )
)  -  k ) ) ) )
9190oveq1d 6037 . . . . . . 7  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( (
( (coeff `  f
) `  n )  x.  ( A ^ n
) )  /  ( A ^ (deg `  f
) ) )  =  ( ( ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) )  x.  ( A ^
( ( 0  +  (deg `  f )
)  -  k ) ) )  /  ( A ^ (deg `  f
) ) ) )
9287, 91fsumrev2 12494 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  sum_ n  e.  ( 0 ... (deg `  f ) ) ( ( ( (coeff `  f ) `  n
)  x.  ( A ^ n ) )  /  ( A ^
(deg `  f )
) )  =  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( ( (coeff `  f
) `  ( (
0  +  (deg `  f ) )  -  k ) )  x.  ( A ^ (
( 0  +  (deg
`  f ) )  -  k ) ) )  /  ( A ^ (deg `  f
) ) ) )
9348adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  (deg `  f
)  e.  CC )
9493addid2d 9201 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( 0  +  (deg `  f )
)  =  (deg `  f ) )
9594oveq1d 6037 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( 0  +  (deg `  f
) )  -  k
)  =  ( (deg
`  f )  -  k ) )
9695fveq2d 5674 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) )  =  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) )
9795oveq2d 6038 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
( ( 0  +  (deg `  f )
)  -  k ) )  =  ( A ^ ( (deg `  f )  -  k
) ) )
9877adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  A  e.  CC )
9983adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  A  =/=  0
)
100 elfznn0 11017 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 0 ... (deg `  f )
)  ->  k  e.  NN0 )
101100adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  k  e.  NN0 )
102101nn0zd 10307 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  k  e.  ZZ )
10384adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  (deg `  f
)  e.  ZZ )
10498, 99, 102, 103expsubd 11463 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
( (deg `  f
)  -  k ) )  =  ( ( A ^ (deg `  f ) )  / 
( A ^ k
) ) )
10597, 104eqtrd 2421 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
( ( 0  +  (deg `  f )
)  -  k ) )  =  ( ( A ^ (deg `  f ) )  / 
( A ^ k
) ) )
10696, 105oveq12d 6040 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) )  x.  ( A ^
( ( 0  +  (deg `  f )
)  -  k ) ) )  =  ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( A ^ (deg `  f ) )  / 
( A ^ k
) ) ) )
107106oveq1d 6037 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( ( (coeff `  f ) `  ( ( 0  +  (deg `  f )
)  -  k ) )  x.  ( A ^ ( ( 0  +  (deg `  f
) )  -  k
) ) )  / 
( A ^ (deg `  f ) ) )  =  ( ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( A ^ (deg `  f ) )  / 
( A ^ k
) ) )  / 
( A ^ (deg `  f ) ) ) )
10881adantr 452 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
(deg `  f )
)  e.  CC )
109 expcl 11328 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
11077, 100, 109syl2an 464 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
k )  e.  CC )
11198, 99, 102expne0d 11458 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
k )  =/=  0
)
112108, 110, 111divcld 9724 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( A ^ (deg `  f
) )  /  ( A ^ k ) )  e.  CC )
11385adantr 452 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
(deg `  f )
)  =/=  0 )
11429, 112, 108, 113divassd 9759 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( A ^ (deg `  f ) )  / 
( A ^ k
) ) )  / 
( A ^ (deg `  f ) ) )  =  ( ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  x.  ( ( ( A ^ (deg `  f ) )  / 
( A ^ k
) )  /  ( A ^ (deg `  f
) ) ) ) )
115108, 113dividd 9722 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( A ^ (deg `  f
) )  /  ( A ^ (deg `  f
) ) )  =  1 )
116115oveq1d 6037 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( ( A ^ (deg `  f ) )  / 
( A ^ (deg `  f ) ) )  /  ( A ^
k ) )  =  ( 1  /  ( A ^ k ) ) )
117108, 110, 108, 111, 113divdiv32d 9749 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( ( A ^ (deg `  f ) )  / 
( A ^ k
) )  /  ( A ^ (deg `  f
) ) )  =  ( ( ( A ^ (deg `  f
) )  /  ( A ^ (deg `  f
) ) )  / 
( A ^ k
) ) )
11898, 99, 102exprecd 11460 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( 1  /  A ) ^
k )  =  ( 1  /  ( A ^ k ) ) )
119116, 117, 1183eqtr4d 2431 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( ( A ^ (deg `  f ) )  / 
( A ^ k
) )  /  ( A ^ (deg `  f
) ) )  =  ( ( 1  /  A ) ^ k
) )
120119oveq2d 6038 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  x.  ( ( ( A ^ (deg `  f ) )  / 
( A ^ k
) )  /  ( A ^ (deg `  f
) ) ) )  =  ( ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  x.  ( ( 1  /  A ) ^
k ) ) )
121107, 114, 1203eqtrd 2425 . . . . . . 7  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( ( ( (coeff `  f ) `  ( ( 0  +  (deg `  f )
)  -  k ) )  x.  ( A ^ ( ( 0  +  (deg `  f
) )  -  k
) ) )  / 
( A ^ (deg `  f ) ) )  =  ( ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  x.  ( ( 1  /  A ) ^
k ) ) )
122121sumeq2dv 12426 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) )  x.  ( A ^
( ( 0  +  (deg `  f )
)  -  k ) ) )  /  ( A ^ (deg `  f
) ) )  = 
sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) ) )
12392, 122eqtrd 2421 . . . . 5  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  sum_ n  e.  ( 0 ... (deg `  f ) ) ( ( ( (coeff `  f ) `  n
)  x.  ( A ^ n ) )  /  ( A ^
(deg `  f )
) )  =  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( 1  /  A ) ^ k ) ) )
12418, 55coeid2 20027 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  ZZ )  /\  A  e.  CC )  ->  (
f `  A )  =  sum_ n  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  n )  x.  ( A ^ n
) ) )
12513, 77, 124syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f `  A )  =  sum_ n  e.  ( 0 ... (deg `  f )
) ( ( (coeff `  f ) `  n
)  x.  ( A ^ n ) ) )
126 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f `  A )  =  0 )
127125, 126eqtr3d 2423 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  sum_ n  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  n )  x.  ( A ^ n
) )  =  0 )
128127oveq1d 6037 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( sum_ n  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  n )  x.  ( A ^ n
) )  /  ( A ^ (deg `  f
) ) )  =  ( 0  /  ( A ^ (deg `  f
) ) ) )
129 fzfid 11241 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 0 ... (deg `  f )
)  e.  Fin )
130129, 81, 80, 85fsumdivc 12498 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( sum_ n  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  n )  x.  ( A ^ n
) )  /  ( A ^ (deg `  f
) ) )  = 
sum_ n  e.  (
0 ... (deg `  f
) ) ( ( ( (coeff `  f
) `  n )  x.  ( A ^ n
) )  /  ( A ^ (deg `  f
) ) ) )
13181, 85div0d 9723 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 0  / 
( A ^ (deg `  f ) ) )  =  0 )
132128, 130, 1313eqtr3d 2429 . . . . 5  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  sum_ n  e.  ( 0 ... (deg `  f ) ) ( ( ( (coeff `  f ) `  n
)  x.  ( A ^ n ) )  /  ( A ^
(deg `  f )
) )  =  0 )
13371, 123, 1323eqtr2d 2427 . . . 4  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) ) `  (
1  /  A ) )  =  0 )
134 fveq1 5669 . . . . . 6  |-  ( g  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  ->  (
g `  ( 1  /  A ) )  =  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) ) `  (
1  /  A ) ) )
135134eqeq1d 2397 . . . . 5  |-  ( g  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) )  ->  (
( g `  (
1  /  A ) )  =  0  <->  (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) `  ( 1  /  A ) )  =  0 ) )
136135rspcev 2997 . . . 4  |-  ( ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) )  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) ) ) `  (
1  /  A ) )  =  0 )  ->  E. g  e.  ( (Poly `  ZZ )  \  { 0 p }
) ( g `  ( 1  /  A
) )  =  0 )
13764, 133, 136syl2anc 643 . . 3  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  E. g  e.  ( (Poly `  ZZ )  \  { 0 p }
) ( g `  ( 1  /  A
) )  =  0 )
138 elaa 20102 . . 3  |-  ( ( 1  /  A )  e.  AA  <->  ( (
1  /  A )  e.  CC  /\  E. g  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( g `  (
1  /  A ) )  =  0 ) )
1397, 137, 138sylanbrc 646 . 2  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 1  /  A )  e.  AA )
1403, 139rexlimddv 2779 1  |-  ( ( A  e.  AA  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  AA )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652    \ cdif 3262    C_ wss 3265   ifcif 3684   {csn 3759   class class class wbr 4155    e. cmpt 4209   -->wf 5392   ` cfv 5396  (class class class)co 6022   CCcc 8923   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    <_ cle 9056    - cmin 9225    / cdiv 9611   NN0cn0 10155   ZZcz 10216   ...cfz 10977   ^cexp 11311   sum_csu 12408   0 pc0p 19430  Polycply 19972  coeffccoe 19974  degcdgr 19975   AAcaa 20100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-fz 10978  df-fzo 11068  df-fl 11131  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-rlim 12212  df-sum 12409  df-0p 19431  df-ply 19976  df-coe 19978  df-dgr 19979  df-aa 20101
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