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Theorem aareccl 19722
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 19712 . . . 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 19713 . . . . . . 7  |-  ( A  e.  AA  ->  A  e.  CC )
5 reccl 9447 . . . . . . 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 10048 . . . . . . . . 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 3762 . . . . . . . . . . 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 19631 . . . . . . . . 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 10051 . . . . . . . . . 10  |-  0  e.  ZZ
18 eqid 2296 . . . . . . . . . . 11  |-  (coeff `  f )  =  (coeff `  f )
1918coef2 19629 . . . . . . . . . 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 10840 . . . . . . . . . 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 5679 . . . . . . . . 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 19600 . . . . . . 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 8847 . . . . . . . 8  |-  0  e.  CC
27 eqid 2296 . . . . . . . . . . . 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 19645 . . . . . . . . . . 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 10134 . . . . . . . . . . . 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 2297 . . . . . . . . . . . 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 19640 . . . . . . . . . . 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 5543 . . . . . . . . . 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 9996 . . . . . . . . . . . 12  |-  0  e.  NN0
35 breq1 4042 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
k  <_  (deg `  f
)  <->  0  <_  (deg `  f ) ) )
36 oveq2 5882 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
(deg `  f )  -  k )  =  ( (deg `  f
)  -  0 ) )
3736fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  k ) )  =  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) )
38 eqidd 2297 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  0  =  0 )
3935, 37, 38ifbieq12d 3600 . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . 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 5555 . . . . . . . . . . . . . 14  |-  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) )  e.  _V
42 c0ex 8848 . . . . . . . . . . . . . 14  |-  0  e.  _V
4341, 42ifex 3636 . . . . . . . . . . . . 13  |-  if ( 0  <_  (deg `  f
) ,  ( (coeff `  f ) `  (
(deg `  f )  -  0 ) ) ,  0 )  e. 
_V
4439, 40, 43fvmpt 5618 . . . . . . . . . . . 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 10037 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  0  <_  (deg `  f ) )
47 iftrue 3584 . . . . . . . . . . . . 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 10036 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  CC )
5049subid1d 9162 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( (deg `  f )  -  0 )  =  (deg `  f ) )
5150fveq2d 5545 . . . . . . . . . . . 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 2328 . . . . . . . . . . 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 2340 . . . . . . . . . 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 2332 . . . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  (deg `  f )  =  (deg
`  f )
5756, 18dgreq0 19662 . . . . . . . . . . . 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 2494 . . . . . . . . . 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 2477 . . . . . . . 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 19605 . . . . . . . 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 3762 . . . . . . 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 5881 . . . . . . . . . . 11  |-  ( z  =  ( 1  /  A )  ->  (
z ^ k )  =  ( ( 1  /  A ) ^
k ) )
6766oveq2d 5890 . . . . . . . . . 10  |-  ( z  =  ( 1  /  A )  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
z ^ k ) )  =  ( ( (coeff `  f ) `  ( (deg `  f
)  -  k ) )  x.  ( ( 1  /  A ) ^ k ) ) )
6867sumeq2sdv 12193 . . . . . . . . 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 2296 . . . . . . . . 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 12176 . . . . . . . . 9  |-  sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  ( (deg `  f )  -  k
) )  x.  (
( 1  /  A
) ^ k ) )  e.  _V
7168, 69, 70fvmpt 5618 . . . . . . . 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 19630 . . . . . . . . . . . . 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 10838 . . . . . . . . . . . 12  |-  ( n  e.  ( 0 ... (deg `  f )
)  ->  n  e.  NN0 )
76 ffvelrn 5679 . . . . . . . . . . . 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 11137 . . . . . . . . . . . 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 8871 . . . . . . . . . 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 11261 . . . . . . . . . . 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 10131 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  (deg `  f
)  e.  ZZ )
8678, 84, 85expne0d 11267 . . . . . . . . . . 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 9552 . . . . . . . . 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 5541 . . . . . . . . . . 11  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( (coeff `  f ) `  n
)  =  ( (coeff `  f ) `  (
( 0  +  (deg
`  f ) )  -  k ) ) )
90 oveq2 5882 . . . . . . . . . . 11  |-  ( n  =  ( ( 0  +  (deg `  f
) )  -  k
)  ->  ( A ^ n )  =  ( A ^ (
( 0  +  (deg
`  f ) )  -  k ) ) )
9189, 90oveq12d 5892 . . . . . . . . . 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 5889 . . . . . . . . 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 12260 . . . . . . . 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 9029 . . . . . . . . . . . . . 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 5889 . . . . . . . . . . . . 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 5545 . . . . . . . . . . . 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 5890 . . . . . . . . . . . . 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 10838 . . . . . . . . . . . . . . . 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 10131 . . . . . . . . . . . . . 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 11272 . . . . . . . . . . . . 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 2328 . . . . . . . . . . . 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 5892 . . . . . . . . . . 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 5889 . . . . . . . . . 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 11137 . . . . . . . . . . . . 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 11267 . . . . . . . . . . . 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 9552 . . . . . . . . . . 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 9587 . . . . . . . . . 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 9550 . . . . . . . . . . . . 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 5889 . . . . . . . . . . . 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 9577 . . . . . . . . . . . 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 11269 . . . . . . . . . . . 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 2338 . . . . . . . . . . 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 5890 . . . . . . . . . 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 2332 . . . . . . . . 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 12192 . . . . . . . 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 2328 . . . . . . 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 19637 . . . . . . . . . . 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 2330 . . . . . . . . 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 5889 . . . . . . . 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 11051 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 0 ... (deg `  f )
)  e.  Fin )
131130, 82, 81, 86fsumdivc 12264 . . . . . . . 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 9551 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  A  =/=  0 )  /\  ( f  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  (
f `  A )  =  0 ) )  ->  ( 0  / 
( A ^ (deg `  f ) ) )  =  0 )
133129, 131, 1323eqtr3d 2336 . . . . . . 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 2334 . . . . . 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 5540 . . . . . . . 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 2304 . . . . . . 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 2897 . . . . . 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 19712 . . . . 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 2680 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162    C_ wss 3165   ifcif 3578   {csn 3653   class class class wbr 4039    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053    / cdiv 9439   NN0cn0 9981   ZZcz 10040   ...cfz 10798   ^cexp 11120   sum_csu 12174   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585   AAcaa 19710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589  df-aa 19711
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