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Theorem aareccl 19706
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 19696 . . . 4  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( f `
 A )  =  0 ) )
21simprbi 450 . . 3  |-  ( A  e.  AA  ->  E. f  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( f `
 A )  =  0 )
32adantr 451 . 2  |-  ( ( A  e.  AA  /\  A  =/=  0 )  ->  E. f  e.  (
(Poly `  ZZ )  \  { 0 p }
) ( f `  A )  =  0 )
4 aacn 19697 . . . . . . 7  |-  ( A  e.  AA  ->  A  e.  CC )
5 reccl 9431 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
64, 5sylan 457 . . . . . 6  |-  ( ( A  e.  AA  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
76adantr 451 . . . . 5  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 1  /  A )  e.  CC )
8 zsscn 10032 . . . . . . . . 9  |-  ZZ  C_  CC
98a1i 10 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ZZ  C_  CC )
10 simprl 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  f  e.  ( (Poly `  ZZ )  \  { 0 p }
) )
11 eldifsn 3749 . . . . . . . . . . 11  |-  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  <->  ( f  e.  (Poly `  ZZ )  /\  f  =/=  0 p ) )
1210, 11sylib 188 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f  e.  (Poly `  ZZ )  /\  f  =/=  0 p ) )
1312simpld 445 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  f  e.  (Poly `  ZZ ) )
14 dgrcl 19615 . . . . . . . . 9  |-  ( f  e.  (Poly `  ZZ )  ->  (deg `  f
)  e.  NN0 )
1513, 14syl 15 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  NN0 )
1613adantr 451 . . . . . . . . . 10  |-  ( ( ( ( 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 10035 . . . . . . . . . 10  |-  0  e.  ZZ
18 eqid 2283 . . . . . . . . . . 11  |-  (coeff `  f )  =  (coeff `  f )
1918coef2 19613 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  f ) : NN0 --> ZZ )
2016, 17, 19sylancl 643 . . . . . . . . 9  |-  ( ( ( ( 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 10824 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... (deg `  f )
)  ->  ( (deg `  f )  -  k
)  e.  NN0 )
2221adantl 452 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( (deg `  f )  -  k
)  e.  NN0 )
23 ffvelrn 5663 . . . . . . . . 9  |-  ( ( (coeff `  f ) : NN0 --> ZZ  /\  (
(deg `  f )  -  k )  e. 
NN0 )  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  k ) )  e.  ZZ )
2420, 22, 23syl2anc 642 . . . . . . . 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 ) )  e.  ZZ )
259, 15, 24elplyd 19584 . . . . . . 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 ) ) )  e.  (Poly `  ZZ ) )
26 0cn 8831 . . . . . . . 8  |-  0  e.  CC
27 eqid 2283 . . . . . . . . . . . 12  |-  (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 ) ) ) )
2827coefv0 19629 . . . . . . . . . . 11  |-  ( ( 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
) )
2925, 28syl 15 . . . . . . . . . 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 ) ) ) `  0
)  =  ( (coeff `  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( z ^ k ) ) ) ) `  0
) )
3024zcnd 10118 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )
31 eqidd 2284 . . . . . . . . . . . 12  |-  ( ( ( 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 ) ) ) )
3225, 15, 30, 31coeeq2 19624 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) )
3332fveq1d 5527 . . . . . . . . . 10  |-  ( ( ( 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 ) )
34 0nn0 9980 . . . . . . . . . . . 12  |-  0  e.  NN0
35 breq1 4026 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
k  <_  (deg `  f
)  <->  0  <_  (deg `  f ) ) )
36 oveq2 5866 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
(deg `  f )  -  k )  =  ( (deg `  f
)  -  0 ) )
3736fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  k ) )  =  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) )
38 eqidd 2284 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  0  =  0 )
3935, 37, 38ifbieq12d 3587 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  if ( k  <_  (deg `  f ) ,  ( (coeff `  f ) `  ( (deg `  f
)  -  k ) ) ,  0 )  =  if ( 0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 ) )
40 eqid 2283 . . . . . . . . . . . . 13  |-  ( 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 ) )
41 fvex 5539 . . . . . . . . . . . . . 14  |-  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) )  e.  _V
42 c0ex 8832 . . . . . . . . . . . . . 14  |-  0  e.  _V
4341, 42ifex 3623 . . . . . . . . . . . . 13  |-  if ( 0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )  e. 
_V
4439, 40, 43fvmpt 5602 . . . . . . . . . . . 12  |-  ( 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 ) )
4534, 44ax-mp 8 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  |->  if ( k  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  k ) ) ,  0 ) ) `
 0 )  =  if ( 0  <_ 
(deg `  f ) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )
4615nn0ge0d 10021 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  0  <_  (deg `  f ) )
47 iftrue 3571 . . . . . . . . . . . . 13  |-  ( 0  <_  (deg `  f
)  ->  if (
0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )  =  ( (coeff `  f
) `  ( (deg `  f )  -  0 ) ) )
4846, 47syl 15 . . . . . . . . . . . 12  |-  ( ( ( 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 ) ) )
4915nn0cnd 10020 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  CC )
5049subid1d 9146 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (deg `  f )  -  0 )  =  (deg `  f ) )
5150fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) )  =  ( (coeff `  f ) `  (deg `  f ) ) )
5248, 51eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( 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
) ) )
5345, 52syl5eq 2327 . . . . . . . . . 10  |-  ( ( ( 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
) ) )
5429, 33, 533eqtrd 2319 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
5512simprd 449 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  f  =/=  0 p )
56 eqid 2283 . . . . . . . . . . . . 13  |-  (deg `  f )  =  (deg
`  f )
5756, 18dgreq0 19646 . . . . . . . . . . . 12  |-  ( f  e.  (Poly `  ZZ )  ->  ( f  =  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =  0 ) )
5813, 57syl 15 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f  =  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =  0 ) )
5958necon3bid 2481 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f  =/=  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 ) )
6055, 59mpbid 201 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 )
6154, 60eqnetrd 2464 . . . . . . . 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
)  =/=  0 )
62 ne0p 19589 . . . . . . . 8  |-  ( ( 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 )
6326, 61, 62sylancr 644 . . . . . . 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 p )
64 eldifsn 3749 . . . . . . 7  |-  ( ( 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 ) )
6525, 63, 64sylanbrc 645 . . . . . 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 ) ) )  e.  ( (Poly `  ZZ )  \  { 0 p }
) )
66 oveq1 5865 . . . . . . . . . . 11  |-  ( z  =  ( 1  /  A )  ->  (
z ^ k )  =  ( ( 1  /  A ) ^
k ) )
6766oveq2d 5874 . . . . . . . . . 10  |-  ( z  =  ( 1  /  A )  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) )  =  ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( 1  /  A ) ^ k ) ) )
6867sumeq2sdv 12177 . . . . . . . . 9  |-  ( 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 ) ) )
69 eqid 2283 . . . . . . . . 9  |-  ( 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 ) ) )
70 sumex 12160 . . . . . . . . 9  |-  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) )  e.  _V
7168, 69, 70fvmpt 5602 . . . . . . . 8  |-  ( ( 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 ) ) )
727, 71syl 15 . . . . . . 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 ) ) ) `  (
1  /  A ) )  =  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) ) )
7318coef3 19614 . . . . . . . . . . . . 13  |-  ( f  e.  (Poly `  ZZ )  ->  (coeff `  f
) : NN0 --> CC )
7413, 73syl 15 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (coeff `  f
) : NN0 --> CC )
75 elfznn0 10822 . . . . . . . . . . . 12  |-  ( n  e.  ( 0 ... (deg `  f )
)  ->  n  e.  NN0 )
76 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) : NN0 --> CC  /\  n  e.  NN0 )  ->  (
(coeff `  f ) `  n )  e.  CC )
7774, 75, 76syl2an 463 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( (coeff `  f ) `  n
)  e.  CC )
784ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  A  e.  CC )
79 expcl 11121 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  n  e.  NN0 )  -> 
( A ^ n
)  e.  CC )
8078, 75, 79syl2an 463 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
n )  e.  CC )
8177, 80mulcld 8855 . . . . . . . . . 10  |-  ( ( ( ( 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 )
8278, 15expcld 11245 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( A ^
(deg `  f )
)  e.  CC )
8382adantr 451 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
(deg `  f )
)  e.  CC )
84 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  A  =/=  0
)
8515nn0zd 10115 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  ZZ )
8678, 84, 85expne0d 11251 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( A ^
(deg `  f )
)  =/=  0 )
8786adantr 451 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  n  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
(deg `  f )
)  =/=  0 )
8881, 83, 87divcld 9536 . . . . . . . . 9  |-  ( ( ( ( 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 )
89 fveq2 5525 . . . . . . . . . . 11  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( (coeff `  f ) `  n
)  =  ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) ) )
90 oveq2 5866 . . . . . . . . . . 11  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( A ^ n )  =  ( A ^ (
( 0  +  (deg
`  f ) )  -  k ) ) )
9189, 90oveq12d 5876 . . . . . . . . . 10  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( (
(coeff `  f ) `  n )  x.  ( A ^ n ) )  =  ( ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) )  x.  ( A ^
( ( 0  +  (deg `  f )
)  -  k ) ) ) )
9291oveq1d 5873 . . . . . . . . 9  |-  ( 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
) ) ) )
9388, 92fsumrev2 12244 . . . . . . . 8  |-  ( ( ( 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
) ) ) )
9449adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  (deg `  f
)  e.  CC )
9594addid2d 9013 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( 0  +  (deg `  f )
)  =  (deg `  f ) )
9695oveq1d 5873 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) )
9796fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) )
9896oveq2d 5874 . . . . . . . . . . . . 13  |-  ( ( ( ( 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
) ) )
9978adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  A  e.  CC )
10084adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  A  =/=  0
)
101 elfznn0 10822 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( 0 ... (deg `  f )
)  ->  k  e.  NN0 )
102101adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  k  e.  NN0 )
103102nn0zd 10115 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  k  e.  ZZ )
10485adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  (deg `  f
)  e.  ZZ )
10599, 100, 103, 104expsubd 11256 . . . . . . . . . . . . 13  |-  ( ( ( ( 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
) ) )
10698, 105eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( ( ( 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
) ) )
10797, 106oveq12d 5876 . . . . . . . . . . 11  |-  ( ( ( ( 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
) ) ) )
108107oveq1d 5873 . . . . . . . . . 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 ) )  x.  ( A ^ ( ( 0  +  (deg `  f
) )  -  k
) ) )  / 
( A ^ (deg `  f ) ) )  =  ( ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( A ^ (deg `  f ) )  / 
( A ^ k
) ) )  / 
( A ^ (deg `  f ) ) ) )
10982adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
(deg `  f )
)  e.  CC )
110 expcl 11121 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
11178, 101, 110syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
k )  e.  CC )
11299, 100, 103expne0d 11251 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
k )  =/=  0
)
113109, 111, 112divcld 9536 . . . . . . . . . . 11  |-  ( ( ( ( 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 )
11486adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  (
f  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( f `  A )  =  0 ) )  /\  k  e.  ( 0 ... (deg `  f ) ) )  ->  ( A ^
(deg `  f )
)  =/=  0 )
11530, 113, 109, 114divassd 9571 . . . . . . . . . 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 ) )  x.  ( ( A ^ (deg `  f ) )  / 
( A ^ k
) ) )  / 
( A ^ (deg `  f ) ) )  =  ( ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  x.  ( ( ( A ^ (deg `  f ) )  / 
( A ^ k
) )  /  ( A ^ (deg `  f
) ) ) ) )
116109, 114dividd 9534 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )
117116oveq1d 5873 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) )
118109, 111, 109, 112, 114divdiv32d 9561 . . . . . . . . . . . 12  |-  ( ( ( ( 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
) ) )
11999, 100, 103exprecd 11253 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) )
120117, 118, 1193eqtr4d 2325 . . . . . . . . . . 11  |-  ( ( ( ( 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
) )
121120oveq2d 5874 . . . . . . . . . 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 ) )  x.  ( ( ( A ^ (deg `  f ) )  / 
( A ^ k
) )  /  ( A ^ (deg `  f
) ) ) )  =  ( ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  x.  ( ( 1  /  A ) ^
k ) ) )
122108, 115, 1213eqtrd 2319 . . . . . . . . 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
) ) )  / 
( A ^ (deg `  f ) ) )  =  ( ( (coeff `  f ) `  (
(deg `  f )  -  k ) )  x.  ( ( 1  /  A ) ^
k ) ) )
123122sumeq2dv 12176 . . . . . . . 8  |-  ( ( ( 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 ) ) )
12493, 123eqtrd 2315 . . . . . . 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 ) )  /  ( A ^
(deg `  f )
) )  =  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( 1  /  A ) ^ k ) ) )
12518, 56coeid2 19621 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  ZZ )  /\  A  e.  CC )  ->  (
f `  A )  =  sum_ n  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  n )  x.  ( A ^ n
) ) )
12613, 78, 125syl2anc 642 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
127 simprr 733 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( f `  A )  =  0 )
128126, 127eqtr3d 2317 . . . . . . . . 9  |-  ( ( ( 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 )
129128oveq1d 5873 . . . . . . . 8  |-  ( ( ( 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
) ) ) )
130 fzfid 11035 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 0 ... (deg `  f )
)  e.  Fin )
131130, 82, 81, 86fsumdivc 12248 . . . . . . . 8  |-  ( ( ( 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
) ) ) )
13282, 86div0d 9535 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 0  / 
( A ^ (deg `  f ) ) )  =  0 )
133129, 131, 1323eqtr3d 2323 . . . . . . 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 ) )  /  ( A ^
(deg `  f )
) )  =  0 )
13472, 124, 1333eqtr2d 2321 . . . . . 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 ) ) ) `  (
1  /  A ) )  =  0 )
135 fveq1 5524 . . . . . . . 8  |-  ( 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 ) ) )
136135eqeq1d 2291 . . . . . . 7  |-  ( 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 ) )
137136rspcev 2884 . . . . . 6  |-  ( ( ( 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 )
13865, 134, 137syl2anc 642 . . . . 5  |-  ( ( ( 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 )
139 elaa 19696 . . . . 5  |-  ( ( 1  /  A )  e.  AA  <->  ( (
1  /  A )  e.  CC  /\  E. g  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( g `  (
1  /  A ) )  =  0 ) )
1407, 138, 139sylanbrc 645 . . . 4  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 1  /  A )  e.  AA )
141140expr 598 . . 3  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  f  e.  ( (Poly `  ZZ )  \  { 0 p }
) )  ->  (
( f `  A
)  =  0  -> 
( 1  /  A
)  e.  AA ) )
142141rexlimdva 2667 . 2  |-  ( ( A  e.  AA  /\  A  =/=  0 )  -> 
( E. f  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( f `
 A )  =  0  ->  ( 1  /  A )  e.  AA ) )
1433, 142mpd 14 1  |-  ( ( A  e.  AA  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  AA )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149    C_ wss 3152   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037    / cdiv 9423   NN0cn0 9965   ZZcz 10024   ...cfz 10782   ^cexp 11104   sum_csu 12158   0 pc0p 19024  Polycply 19566  coeffccoe 19568  degcdgr 19569   AAcaa 19694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573  df-aa 19695
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