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Theorem aareccl 20235
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 20225 . . . 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 20226 . . . . 5  |-  ( A  e.  AA  ->  A  e.  CC )
5 reccl 9677 . . . . 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 10282 . . . . . . 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 3919 . . . . . . . . 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 20144 . . . . . . 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 10285 . . . . . . . 8  |-  0  e.  ZZ
18 eqid 2435 . . . . . . . . 9  |-  (coeff `  f )  =  (coeff `  f )
1918coef2 20142 . . . . . . . 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 11077 . . . . . . . 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 5863 . . . . . 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 20113 . . . . 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 9076 . . . . . 6  |-  0  e.  CC
26 eqid 2435 . . . . . . . . . 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 20158 . . . . . . . . 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 10368 . . . . . . . . . 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 2436 . . . . . . . . . 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 20153 . . . . . . . . 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 5722 . . . . . . . 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 10228 . . . . . . . . . 10  |-  0  e.  NN0
34 breq1 4207 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
k  <_  (deg `  f
)  <->  0  <_  (deg `  f ) ) )
35 oveq2 6081 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
(deg `  f )  -  k )  =  ( (deg `  f
)  -  0 ) )
3635fveq2d 5724 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  k ) )  =  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) )
37 eqidd 2436 . . . . . . . . . . . 12  |-  ( k  =  0  ->  0  =  0 )
3834, 36, 37ifbieq12d 3753 . . . . . . . . . . 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 2435 . . . . . . . . . . 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 5734 . . . . . . . . . . . 12  |-  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) )  e.  _V
41 c0ex 9077 . . . . . . . . . . . 12  |-  0  e.  _V
4240, 41ifex 3789 . . . . . . . . . . 11  |-  if ( 0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )  e. 
_V
4338, 39, 42fvmpt 5798 . . . . . . . . . 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 10269 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  0  <_  (deg `  f ) )
46 iftrue 3737 . . . . . . . . . . 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 10268 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  CC )
4948subid1d 9392 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (deg `  f )  -  0 )  =  (deg `  f ) )
5049fveq2d 5724 . . . . . . . . . 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 2467 . . . . . . . . 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 2479 . . . . . . . 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 2471 . . . . . . 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 2435 . . . . . . . . . . 11  |-  (deg `  f )  =  (deg
`  f )
5655, 18dgreq0 20175 . . . . . . . . . 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 2633 . . . . . . . 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 2616 . . . . . 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 20118 . . . . . 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 3919 . . . . 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 6080 . . . . . . . . 9  |-  ( z  =  ( 1  /  A )  ->  (
z ^ k )  =  ( ( 1  /  A ) ^
k ) )
6665oveq2d 6089 . . . . . . . 8  |-  ( z  =  ( 1  /  A )  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) )  =  ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( 1  /  A ) ^ k ) ) )
6766sumeq2sdv 12490 . . . . . . 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 2435 . . . . . . 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 12473 . . . . . . 7  |-  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) )  e.  _V
7067, 68, 69fvmpt 5798 . . . . . 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 20143 . . . . . . . . . . 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 11075 . . . . . . . . . 10  |-  ( n  e.  ( 0 ... (deg `  f )
)  ->  n  e.  NN0 )
75 ffvelrn 5860 . . . . . . . . . 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 11391 . . . . . . . . . 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 9100 . . . . . . . 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 11515 . . . . . . . . 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 10365 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  ZZ )
8577, 83, 84expne0d 11521 . . . . . . . . 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 9782 . . . . . . 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 5720 . . . . . . . . 9  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( (coeff `  f ) `  n
)  =  ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) ) )
89 oveq2 6081 . . . . . . . . 9  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( A ^ n )  =  ( A ^ (
( 0  +  (deg
`  f ) )  -  k ) ) )
9088, 89oveq12d 6091 . . . . . . . 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 6088 . . . . . . 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 12557 . . . . . 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 9259 . . . . . . . . . . . 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 6088 . . . . . . . . . . 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 5724 . . . . . . . . . 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 6089 . . . . . . . . . . 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 11075 . . . . . . . . . . . . . 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 10365 . . . . . . . . . . . 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 11526 . . . . . . . . . . 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 2467 . . . . . . . . . 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 6091 . . . . . . . . 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 6088 . . . . . . . 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 11391 . . . . . . . . . . 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 11521 . . . . . . . . . 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 9782 . . . . . . . . 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 9817 . . . . . . . 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 9780 . . . . . . . . . . 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 6088 . . . . . . . . . 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 9807 . . . . . . . . . 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 11523 . . . . . . . . . 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 2477 . . . . . . . . 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 6089 . . . . . . . 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 2471 . . . . . . 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 12489 . . . . . 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 2467 . . . . 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 20150 . . . . . . . . 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 2469 . . . . . . 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 6088 . . . . . 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 11304 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 0 ... (deg `  f )
)  e.  Fin )
130129, 81, 80, 85fsumdivc 12561 . . . . . 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 9781 . . . . . 6  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 0  / 
( A ^ (deg `  f ) ) )  =  0 )
132128, 130, 1313eqtr3d 2475 . . . . 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 2473 . . . 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 5719 . . . . . 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 2443 . . . . 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 3044 . . . 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 20225 . . 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 2826 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309    C_ wss 3312   ifcif 3731   {csn 3806   class class class wbr 4204    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    <_ cle 9113    - cmin 9283    / cdiv 9669   NN0cn0 10213   ZZcz 10274   ...cfz 11035   ^cexp 11374   sum_csu 12471   0 pc0p 19553  Polycply 20095  coeffccoe 20097  degcdgr 20098   AAcaa 20223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102  df-aa 20224
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