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Theorem elaa2lem 38209
Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 38211. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by AV, 1-Oct-2020.)
Hypotheses
Ref Expression
elaa2lem.a  |-  ( ph  ->  A  e.  AA )
elaa2lem.an0  |-  ( ph  ->  A  =/=  0 )
elaa2lem.g  |-  ( ph  ->  G  e.  (Poly `  ZZ ) )
elaa2lem.gn0  |-  ( ph  ->  G  =/=  0p )
elaa2lem.ga  |-  ( ph  ->  ( G `  A
)  =  0 )
elaa2lem.m  |-  M  = inf ( { n  e. 
NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } ,  RR ,  <  )
elaa2lem.i  |-  I  =  ( k  e.  NN0  |->  ( (coeff `  G ) `  ( k  +  M
) ) )
elaa2lem.f  |-  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( (deg `  G )  -  M
) ) ( ( I `  k )  x.  ( z ^
k ) ) )
Assertion
Ref Expression
elaa2lem  |-  ( ph  ->  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) )
Distinct variable groups:    A, f    A, k, z    f, F   
k, G    n, G    z, G    k, I, z   
k, M    n, M    z, M    ph, k, z
Allowed substitution hints:    ph( f, n)    A( n)    F( z, k, n)    G( f)    I( f, n)    M( f)

Proof of Theorem elaa2lem
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 elaa2lem.f . . . 4  |-  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( (deg `  G )  -  M
) ) ( ( I `  k )  x.  ( z ^
k ) ) )
21a1i 11 . . 3  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (
(deg `  G )  -  M ) ) ( ( I `  k
)  x.  ( z ^ k ) ) ) )
3 zsscn 10969 . . . . 5  |-  ZZ  C_  CC
43a1i 11 . . . 4  |-  ( ph  ->  ZZ  C_  CC )
5 elaa2lem.g . . . . . . . . 9  |-  ( ph  ->  G  e.  (Poly `  ZZ ) )
6 dgrcl 23266 . . . . . . . . 9  |-  ( G  e.  (Poly `  ZZ )  ->  (deg `  G
)  e.  NN0 )
75, 6syl 17 . . . . . . . 8  |-  ( ph  ->  (deg `  G )  e.  NN0 )
87nn0zd 11061 . . . . . . 7  |-  ( ph  ->  (deg `  G )  e.  ZZ )
9 elaa2lem.m . . . . . . . . 9  |-  M  = inf ( { n  e. 
NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } ,  RR ,  <  )
10 ssrab2 3500 . . . . . . . . . 10  |-  { n  e.  NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } 
C_  NN0
11 nn0uz 11217 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
1210, 11sseqtri 3450 . . . . . . . . . . . 12  |-  { n  e.  NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } 
C_  ( ZZ>= `  0
)
1312a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 }  C_  ( ZZ>=
`  0 ) )
14 elaa2lem.gn0 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  =/=  0p )
1514neneqd 2648 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  -.  G  =  0p )
16 eqid 2471 . . . . . . . . . . . . . . . . . 18  |-  (deg `  G )  =  (deg
`  G )
17 eqid 2471 . . . . . . . . . . . . . . . . . 18  |-  (coeff `  G )  =  (coeff `  G )
1816, 17dgreq0 23298 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  (Poly `  ZZ )  ->  ( G  =  0p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
195, 18syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G  =  0p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
2015, 19mtbid 307 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  ( (coeff `  G ) `  (deg `  G ) )  =  0 )
2120neqned 2650 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  G
) `  (deg `  G
) )  =/=  0
)
227, 21jca 541 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (deg `  G
)  e.  NN0  /\  ( (coeff `  G ) `  (deg `  G )
)  =/=  0 ) )
23 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( n  =  (deg `  G
)  ->  ( (coeff `  G ) `  n
)  =  ( (coeff `  G ) `  (deg `  G ) ) )
2423neeq1d 2702 . . . . . . . . . . . . . 14  |-  ( n  =  (deg `  G
)  ->  ( (
(coeff `  G ) `  n )  =/=  0  <->  ( (coeff `  G ) `  (deg `  G )
)  =/=  0 ) )
2524elrab 3184 . . . . . . . . . . . . 13  |-  ( (deg
`  G )  e. 
{ n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 }  <->  ( (deg `  G )  e.  NN0  /\  ( (coeff `  G
) `  (deg `  G
) )  =/=  0
) )
2622, 25sylibr 217 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  e.  { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 } )
27 ne0i 3728 . . . . . . . . . . . 12  |-  ( (deg
`  G )  e. 
{ n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 }  ->  { n  e.  NN0  |  ( (coeff `  G ) `  n
)  =/=  0 }  =/=  (/) )
2826, 27syl 17 . . . . . . . . . . 11  |-  ( ph  ->  { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 }  =/=  (/) )
29 infssuzcl 11268 . . . . . . . . . . 11  |-  ( ( { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 }  C_  ( ZZ>=
`  0 )  /\  { n  e.  NN0  | 
( (coeff `  G
) `  n )  =/=  0 }  =/=  (/) )  -> inf ( { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 } ,  RR ,  <  )  e.  {
n  e.  NN0  | 
( (coeff `  G
) `  n )  =/=  0 } )
3013, 28, 29syl2anc 673 . . . . . . . . . 10  |-  ( ph  -> inf ( { n  e. 
NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } ,  RR ,  <  )  e.  { n  e. 
NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } )
3110, 30sseldi 3416 . . . . . . . . 9  |-  ( ph  -> inf ( { n  e. 
NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } ,  RR ,  <  )  e.  NN0 )
329, 31syl5eqel 2553 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
3332nn0zd 11061 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
348, 33zsubcld 11068 . . . . . 6  |-  ( ph  ->  ( (deg `  G
)  -  M )  e.  ZZ )
359a1i 11 . . . . . . . 8  |-  ( ph  ->  M  = inf ( { n  e.  NN0  | 
( (coeff `  G
) `  n )  =/=  0 } ,  RR ,  <  ) )
36 infssuzle 11267 . . . . . . . . 9  |-  ( ( { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 }  C_  ( ZZ>=
`  0 )  /\  (deg `  G )  e. 
{ n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 } )  -> inf ( { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 } ,  RR ,  <  )  <_  (deg `  G ) )
3713, 26, 36syl2anc 673 . . . . . . . 8  |-  ( ph  -> inf ( { n  e. 
NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } ,  RR ,  <  )  <_  (deg `  G
) )
3835, 37eqbrtrd 4416 . . . . . . 7  |-  ( ph  ->  M  <_  (deg `  G
) )
397nn0red 10950 . . . . . . . 8  |-  ( ph  ->  (deg `  G )  e.  RR )
4032nn0red 10950 . . . . . . . 8  |-  ( ph  ->  M  e.  RR )
4139, 40subge0d 10224 . . . . . . 7  |-  ( ph  ->  ( 0  <_  (
(deg `  G )  -  M )  <->  M  <_  (deg
`  G ) ) )
4238, 41mpbird 240 . . . . . 6  |-  ( ph  ->  0  <_  ( (deg `  G )  -  M
) )
4334, 42jca 541 . . . . 5  |-  ( ph  ->  ( ( (deg `  G )  -  M
)  e.  ZZ  /\  0  <_  ( (deg `  G )  -  M
) ) )
44 elnn0z 10974 . . . . 5  |-  ( ( (deg `  G )  -  M )  e.  NN0  <->  (
( (deg `  G
)  -  M )  e.  ZZ  /\  0  <_  ( (deg `  G
)  -  M ) ) )
4543, 44sylibr 217 . . . 4  |-  ( ph  ->  ( (deg `  G
)  -  M )  e.  NN0 )
46 id 22 . . . . . . . . 9  |-  ( G  e.  (Poly `  ZZ )  ->  G  e.  (Poly `  ZZ ) )
47 0zd 10973 . . . . . . . . 9  |-  ( G  e.  (Poly `  ZZ )  ->  0  e.  ZZ )
4817coef2 23264 . . . . . . . . 9  |-  ( ( G  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  G ) : NN0 --> ZZ )
4946, 47, 48syl2anc 673 . . . . . . . 8  |-  ( G  e.  (Poly `  ZZ )  ->  (coeff `  G
) : NN0 --> ZZ )
505, 49syl 17 . . . . . . 7  |-  ( ph  ->  (coeff `  G ) : NN0 --> ZZ )
5150adantr 472 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  (coeff `  G
) : NN0 --> ZZ )
52 simpr 468 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
5332adantr 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  M  e.  NN0 )
5452, 53nn0addcld 10953 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( k  +  M )  e.  NN0 )
5551, 54ffvelrnd 6038 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (coeff `  G ) `  (
k  +  M ) )  e.  ZZ )
56 elaa2lem.i . . . . 5  |-  I  =  ( k  e.  NN0  |->  ( (coeff `  G ) `  ( k  +  M
) ) )
5755, 56fmptd 6061 . . . 4  |-  ( ph  ->  I : NN0 --> ZZ )
58 elplyr 23234 . . . 4  |-  ( ( ZZ  C_  CC  /\  (
(deg `  G )  -  M )  e.  NN0  /\  I : NN0 --> ZZ )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( (deg
`  G )  -  M ) ) ( ( I `  k
)  x.  ( z ^ k ) ) )  e.  (Poly `  ZZ ) )
594, 45, 57, 58syl3anc 1292 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( (deg `  G )  -  M
) ) ( ( I `  k )  x.  ( z ^
k ) ) )  e.  (Poly `  ZZ ) )
602, 59eqeltrd 2549 . 2  |-  ( ph  ->  F  e.  (Poly `  ZZ ) )
61 simpr 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  k  <_  ( (deg `  G
)  -  M ) )  ->  k  <_  ( (deg `  G )  -  M ) )
6261iftrued 3880 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  k  <_  ( (deg `  G
)  -  M ) )  ->  if (
k  <_  ( (deg `  G )  -  M
) ,  ( (coeff `  G ) `  (
k  +  M ) ) ,  0 )  =  ( (coeff `  G ) `  (
k  +  M ) ) )
63 iffalse 3881 . . . . . . . . . . 11  |-  ( -.  k  <_  ( (deg `  G )  -  M
)  ->  if (
k  <_  ( (deg `  G )  -  M
) ,  ( (coeff `  G ) `  (
k  +  M ) ) ,  0 )  =  0 )
6463adantl 473 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  if ( k  <_  (
(deg `  G )  -  M ) ,  ( (coeff `  G ) `  ( k  +  M
) ) ,  0 )  =  0 )
65 simpr 468 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  -.  k  <_  ( (deg `  G )  -  M
) )
6639ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (deg `  G )  e.  RR )
6740ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  M  e.  RR )
6866, 67resubcld 10068 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (
(deg `  G )  -  M )  e.  RR )
69 nn0re 10902 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN0  ->  k  e.  RR )
7069ad2antlr 741 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  k  e.  RR )
7168, 70ltnled 9799 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (
( (deg `  G
)  -  M )  <  k  <->  -.  k  <_  ( (deg `  G
)  -  M ) ) )
7265, 71mpbird 240 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (
(deg `  G )  -  M )  <  k
)
7366, 67, 70ltsubaddd 10230 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (
( (deg `  G
)  -  M )  <  k  <->  (deg `  G
)  <  ( k  +  M ) ) )
7472, 73mpbid 215 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (deg `  G )  <  (
k  +  M ) )
75 olc 391 . . . . . . . . . . . . 13  |-  ( (deg
`  G )  < 
( k  +  M
)  ->  ( G  =  0p  \/  (deg `  G )  <  ( k  +  M
) ) )
7674, 75syl 17 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  ( G  =  0p  \/  (deg `  G )  <  ( k  +  M
) ) )
775ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  G  e.  (Poly `  ZZ )
)
7854adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (
k  +  M )  e.  NN0 )
7916, 17dgrlt 23299 . . . . . . . . . . . . 13  |-  ( ( G  e.  (Poly `  ZZ )  /\  (
k  +  M )  e.  NN0 )  -> 
( ( G  =  0p  \/  (deg `  G )  <  (
k  +  M ) )  <->  ( (deg `  G )  <_  (
k  +  M )  /\  ( (coeff `  G ) `  (
k  +  M ) )  =  0 ) ) )
8077, 78, 79syl2anc 673 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (
( G  =  0p  \/  (deg `  G )  <  (
k  +  M ) )  <->  ( (deg `  G )  <_  (
k  +  M )  /\  ( (coeff `  G ) `  (
k  +  M ) )  =  0 ) ) )
8176, 80mpbid 215 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (
(deg `  G )  <_  ( k  +  M
)  /\  ( (coeff `  G ) `  (
k  +  M ) )  =  0 ) )
8281simprd 470 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  (
(coeff `  G ) `  ( k  +  M
) )  =  0 )
8364, 82eqtr4d 2508 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  -.  k  <_  ( (deg `  G )  -  M
) )  ->  if ( k  <_  (
(deg `  G )  -  M ) ,  ( (coeff `  G ) `  ( k  +  M
) ) ,  0 )  =  ( (coeff `  G ) `  (
k  +  M ) ) )
8462, 83pm2.61dan 808 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  if (
k  <_  ( (deg `  G )  -  M
) ,  ( (coeff `  G ) `  (
k  +  M ) ) ,  0 )  =  ( (coeff `  G ) `  (
k  +  M ) ) )
8584mpteq2dva 4482 . . . . . . 7  |-  ( ph  ->  ( k  e.  NN0  |->  if ( k  <_  (
(deg `  G )  -  M ) ,  ( (coeff `  G ) `  ( k  +  M
) ) ,  0 ) )  =  ( k  e.  NN0  |->  ( (coeff `  G ) `  (
k  +  M ) ) ) )
8650, 4fssd 5750 . . . . . . . . . 10  |-  ( ph  ->  (coeff `  G ) : NN0 --> CC )
8786adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  (coeff `  G
) : NN0 --> CC )
88 elfznn0 11913 . . . . . . . . . . 11  |-  ( k  e.  ( 0 ... ( (deg `  G
)  -  M ) )  ->  k  e.  NN0 )
8988adantl 473 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  k  e.  NN0 )
9032adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  M  e.  NN0 )
9189, 90nn0addcld 10953 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( k  +  M )  e.  NN0 )
9287, 91ffvelrnd 6038 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( (coeff `  G ) `  (
k  +  M ) )  e.  CC )
93 eqidd 2472 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... ( (deg `  G )  -  M
) )  =  ( 0 ... ( (deg
`  G )  -  M ) ) )
94 simpl 464 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ph )
9556a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  I  =  ( k  e.  NN0  |->  ( (coeff `  G ) `  (
k  +  M ) ) ) )
9695, 55fvmpt2d 5974 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( I `  k )  =  ( (coeff `  G ) `  ( k  +  M
) ) )
9794, 89, 96syl2anc 673 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( I `  k )  =  ( (coeff `  G ) `  ( k  +  M
) ) )
9897adantlr 729 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( I `  k )  =  ( (coeff `  G ) `  ( k  +  M
) ) )
9998oveq1d 6323 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( ( I `
 k )  x.  ( z ^ k
) )  =  ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( z ^ k
) ) )
10093, 99sumeq12rdv 13850 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... (
(deg `  G )  -  M ) ) ( ( I `  k
)  x.  ( z ^ k ) )  =  sum_ k  e.  ( 0 ... ( (deg
`  G )  -  M ) ) ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( z ^ k
) ) )
101100mpteq2dva 4482 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( (deg `  G )  -  M
) ) ( ( I `  k )  x.  ( z ^
k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( (deg
`  G )  -  M ) ) ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( z ^ k
) ) ) )
1022, 101eqtrd 2505 . . . . . . . 8  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (
(deg `  G )  -  M ) ) ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( z ^ k
) ) ) )
10360, 45, 92, 102coeeq2 23275 . . . . . . 7  |-  ( ph  ->  (coeff `  F )  =  ( k  e. 
NN0  |->  if ( k  <_  ( (deg `  G )  -  M
) ,  ( (coeff `  G ) `  (
k  +  M ) ) ,  0 ) ) )
10485, 103, 953eqtr4d 2515 . . . . . 6  |-  ( ph  ->  (coeff `  F )  =  I )
105104fveq1d 5881 . . . . 5  |-  ( ph  ->  ( (coeff `  F
) `  0 )  =  ( I ` 
0 ) )
106 oveq1 6315 . . . . . . . . 9  |-  ( k  =  0  ->  (
k  +  M )  =  ( 0  +  M ) )
107106adantl 473 . . . . . . . 8  |-  ( (
ph  /\  k  = 
0 )  ->  (
k  +  M )  =  ( 0  +  M ) )
1083, 33sseldi 3416 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
109108addid2d 9852 . . . . . . . . 9  |-  ( ph  ->  ( 0  +  M
)  =  M )
110109adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  = 
0 )  ->  (
0  +  M )  =  M )
111107, 110eqtrd 2505 . . . . . . 7  |-  ( (
ph  /\  k  = 
0 )  ->  (
k  +  M )  =  M )
112111fveq2d 5883 . . . . . 6  |-  ( (
ph  /\  k  = 
0 )  ->  (
(coeff `  G ) `  ( k  +  M
) )  =  ( (coeff `  G ) `  M ) )
113 0nn0 10908 . . . . . . 7  |-  0  e.  NN0
114113a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  NN0 )
11550, 32ffvelrnd 6038 . . . . . 6  |-  ( ph  ->  ( (coeff `  G
) `  M )  e.  ZZ )
11695, 112, 114, 115fvmptd 5969 . . . . 5  |-  ( ph  ->  ( I `  0
)  =  ( (coeff `  G ) `  M
) )
117 eqidd 2472 . . . . 5  |-  ( ph  ->  ( (coeff `  G
) `  M )  =  ( (coeff `  G ) `  M
) )
118105, 116, 1173eqtrd 2509 . . . 4  |-  ( ph  ->  ( (coeff `  F
) `  0 )  =  ( (coeff `  G ) `  M
) )
11935, 30eqeltrd 2549 . . . . . 6  |-  ( ph  ->  M  e.  { n  e.  NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } )
120 fveq2 5879 . . . . . . . 8  |-  ( n  =  M  ->  (
(coeff `  G ) `  n )  =  ( (coeff `  G ) `  M ) )
121120neeq1d 2702 . . . . . . 7  |-  ( n  =  M  ->  (
( (coeff `  G
) `  n )  =/=  0  <->  ( (coeff `  G ) `  M
)  =/=  0 ) )
122121elrab 3184 . . . . . 6  |-  ( M  e.  { n  e. 
NN0  |  ( (coeff `  G ) `  n
)  =/=  0 }  <-> 
( M  e.  NN0  /\  ( (coeff `  G
) `  M )  =/=  0 ) )
123119, 122sylib 201 . . . . 5  |-  ( ph  ->  ( M  e.  NN0  /\  ( (coeff `  G
) `  M )  =/=  0 ) )
124123simprd 470 . . . 4  |-  ( ph  ->  ( (coeff `  G
) `  M )  =/=  0 )
125118, 124eqnetrd 2710 . . 3  |-  ( ph  ->  ( (coeff `  F
) `  0 )  =/=  0 )
1265, 47syl 17 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
127 aasscn 23350 . . . . . . . . . . 11  |-  AA  C_  CC
128 elaa2lem.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  AA )
129127, 128sseldi 3416 . . . . . . . . . 10  |-  ( ph  ->  A  e.  CC )
13094, 129syl 17 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  A  e.  CC )
131130, 89expcld 12454 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( A ^
k )  e.  CC )
13292, 131mulcld 9681 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( ( (coeff `  G ) `  (
k  +  M ) )  x.  ( A ^ k ) )  e.  CC )
133 oveq1 6315 . . . . . . . . 9  |-  ( k  =  ( j  -  M )  ->  (
k  +  M )  =  ( ( j  -  M )  +  M ) )
134133fveq2d 5883 . . . . . . . 8  |-  ( k  =  ( j  -  M )  ->  (
(coeff `  G ) `  ( k  +  M
) )  =  ( (coeff `  G ) `  ( ( j  -  M )  +  M
) ) )
135 oveq2 6316 . . . . . . . 8  |-  ( k  =  ( j  -  M )  ->  ( A ^ k )  =  ( A ^ (
j  -  M ) ) )
136134, 135oveq12d 6326 . . . . . . 7  |-  ( k  =  ( j  -  M )  ->  (
( (coeff `  G
) `  ( k  +  M ) )  x.  ( A ^ k
) )  =  ( ( (coeff `  G
) `  ( (
j  -  M )  +  M ) )  x.  ( A ^
( j  -  M
) ) ) )
13733, 126, 34, 132, 136fsumshft 13918 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( (deg
`  G )  -  M ) ) ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( A ^ k
) )  =  sum_ j  e.  ( (
0  +  M ) ... ( ( (deg
`  G )  -  M )  +  M
) ) ( ( (coeff `  G ) `  ( ( j  -  M )  +  M
) )  x.  ( A ^ ( j  -  M ) ) ) )
1383, 8sseldi 3416 . . . . . . . . . 10  |-  ( ph  ->  (deg `  G )  e.  CC )
139138, 108npcand 10009 . . . . . . . . 9  |-  ( ph  ->  ( ( (deg `  G )  -  M
)  +  M )  =  (deg `  G
) )
140109, 139oveq12d 6326 . . . . . . . 8  |-  ( ph  ->  ( ( 0  +  M ) ... (
( (deg `  G
)  -  M )  +  M ) )  =  ( M ... (deg `  G ) ) )
141140sumeq1d 13844 . . . . . . 7  |-  ( ph  -> 
sum_ j  e.  ( ( 0  +  M
) ... ( ( (deg
`  G )  -  M )  +  M
) ) ( ( (coeff `  G ) `  ( ( j  -  M )  +  M
) )  x.  ( A ^ ( j  -  M ) ) )  =  sum_ j  e.  ( M ... (deg `  G ) ) ( ( (coeff `  G
) `  ( (
j  -  M )  +  M ) )  x.  ( A ^
( j  -  M
) ) ) )
142 elfzelz 11826 . . . . . . . . . . . . . 14  |-  ( j  e.  ( M ... (deg `  G ) )  ->  j  e.  ZZ )
143142adantl 473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  j  e.  ZZ )
1443, 143sseldi 3416 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  j  e.  CC )
145108adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  M  e.  CC )
146144, 145npcand 10009 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( j  -  M )  +  M )  =  j )
147146fveq2d 5883 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( (coeff `  G ) `  (
( j  -  M
)  +  M ) )  =  ( (coeff `  G ) `  j
) )
148147oveq1d 6323 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( (coeff `  G ) `  (
( j  -  M
)  +  M ) )  x.  ( A ^ ( j  -  M ) ) )  =  ( ( (coeff `  G ) `  j
)  x.  ( A ^ ( j  -  M ) ) ) )
149129adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  A  e.  CC )
150 elaa2lem.an0 . . . . . . . . . . . . 13  |-  ( ph  ->  A  =/=  0 )
151150adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  A  =/=  0
)
15233adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  M  e.  ZZ )
153149, 151, 152, 143expsubd 12465 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( A ^
( j  -  M
) )  =  ( ( A ^ j
)  /  ( A ^ M ) ) )
154153oveq2d 6324 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( (coeff `  G ) `  j
)  x.  ( A ^ ( j  -  M ) ) )  =  ( ( (coeff `  G ) `  j
)  x.  ( ( A ^ j )  /  ( A ^ M ) ) ) )
15586adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  (coeff `  G
) : NN0 --> CC )
156 0red 9662 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  0  e.  RR )
15740adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  M  e.  RR )
158143zred 11063 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  j  e.  RR )
15932nn0ge0d 10952 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  <_  M )
160159adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  0  <_  M
)
161 elfzle1 11828 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( M ... (deg `  G ) )  ->  M  <_  j
)
162161adantl 473 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  M  <_  j
)
163156, 157, 158, 160, 162letrd 9809 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  0  <_  j
)
164143, 163jca 541 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( j  e.  ZZ  /\  0  <_ 
j ) )
165 elnn0z 10974 . . . . . . . . . . . . . 14  |-  ( j  e.  NN0  <->  ( j  e.  ZZ  /\  0  <_ 
j ) )
166164, 165sylibr 217 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  j  e.  NN0 )
167155, 166ffvelrnd 6038 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( (coeff `  G ) `  j
)  e.  CC )
168149, 166expcld 12454 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( A ^
j )  e.  CC )
169129, 32expcld 12454 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A ^ M
)  e.  CC )
170169adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( A ^ M )  e.  CC )
171149, 151, 152expne0d 12460 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( A ^ M )  =/=  0
)
172167, 168, 170, 171divassd 10440 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( ( (coeff `  G ) `  j )  x.  ( A ^ j ) )  /  ( A ^ M ) )  =  ( ( (coeff `  G ) `  j
)  x.  ( ( A ^ j )  /  ( A ^ M ) ) ) )
173172eqcomd 2477 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( (coeff `  G ) `  j
)  x.  ( ( A ^ j )  /  ( A ^ M ) ) )  =  ( ( ( (coeff `  G ) `  j )  x.  ( A ^ j ) )  /  ( A ^ M ) ) )
174154, 173eqtr2d 2506 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( ( (coeff `  G ) `  j )  x.  ( A ^ j ) )  /  ( A ^ M ) )  =  ( ( (coeff `  G ) `  j
)  x.  ( A ^ ( j  -  M ) ) ) )
175148, 174eqtr4d 2508 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( (coeff `  G ) `  (
( j  -  M
)  +  M ) )  x.  ( A ^ ( j  -  M ) ) )  =  ( ( ( (coeff `  G ) `  j )  x.  ( A ^ j ) )  /  ( A ^ M ) ) )
176175sumeq2dv 13846 . . . . . . 7  |-  ( ph  -> 
sum_ j  e.  ( M ... (deg `  G ) ) ( ( (coeff `  G
) `  ( (
j  -  M )  +  M ) )  x.  ( A ^
( j  -  M
) ) )  = 
sum_ j  e.  ( M ... (deg `  G ) ) ( ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  /  ( A ^ M ) ) )
177141, 176eqtrd 2505 . . . . . 6  |-  ( ph  -> 
sum_ j  e.  ( ( 0  +  M
) ... ( ( (deg
`  G )  -  M )  +  M
) ) ( ( (coeff `  G ) `  ( ( j  -  M )  +  M
) )  x.  ( A ^ ( j  -  M ) ) )  =  sum_ j  e.  ( M ... (deg `  G ) ) ( ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  /  ( A ^ M ) ) )
17832, 11syl6eleq 2559 . . . . . . . 8  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
179 fzss1 11863 . . . . . . . 8  |-  ( M  e.  ( ZZ>= `  0
)  ->  ( M ... (deg `  G )
)  C_  ( 0 ... (deg `  G
) ) )
180178, 179syl 17 . . . . . . 7  |-  ( ph  ->  ( M ... (deg `  G ) )  C_  ( 0 ... (deg `  G ) ) )
181167, 168mulcld 9681 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  e.  CC )
182181, 170, 171divcld 10405 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( M ... (deg `  G ) ) )  ->  ( ( ( (coeff `  G ) `  j )  x.  ( A ^ j ) )  /  ( A ^ M ) )  e.  CC )
18333ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  M  e.  ZZ )
1848ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  (deg `  G
)  e.  ZZ )
185 eldifi 3544 . . . . . . . . . . . . . . . . . . 19  |-  ( j  e.  ( ( 0 ... (deg `  G
) )  \  ( M ... (deg `  G
) ) )  -> 
j  e.  ( 0 ... (deg `  G
) ) )
186 elfznn0 11913 . . . . . . . . . . . . . . . . . . . 20  |-  ( j  e.  ( 0 ... (deg `  G )
)  ->  j  e.  NN0 )
187186nn0zd 11061 . . . . . . . . . . . . . . . . . . 19  |-  ( j  e.  ( 0 ... (deg `  G )
)  ->  j  e.  ZZ )
188185, 187syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( j  e.  ( ( 0 ... (deg `  G
) )  \  ( M ... (deg `  G
) ) )  -> 
j  e.  ZZ )
189188ad2antlr 741 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  j  e.  ZZ )
190183, 184, 1893jca 1210 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  ( M  e.  ZZ  /\  (deg `  G )  e.  ZZ  /\  j  e.  ZZ ) )
191 simpr 468 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  -.  j  <  M )
19240ad2antrr 740 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  M  e.  RR )
193189zred 11063 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  j  e.  RR )
194192, 193lenltd 9798 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  ( M  <_ 
j  <->  -.  j  <  M ) )
195191, 194mpbird 240 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  M  <_  j
)
196 elfzle2 11829 . . . . . . . . . . . . . . . . . 18  |-  ( j  e.  ( 0 ... (deg `  G )
)  ->  j  <_  (deg
`  G ) )
197185, 196syl 17 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( ( 0 ... (deg `  G
) )  \  ( M ... (deg `  G
) ) )  -> 
j  <_  (deg `  G
) )
198197ad2antlr 741 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  j  <_  (deg `  G ) )
199190, 195, 198jca32 544 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  ( ( M  e.  ZZ  /\  (deg `  G )  e.  ZZ  /\  j  e.  ZZ )  /\  ( M  <_ 
j  /\  j  <_  (deg
`  G ) ) ) )
200 elfz2 11817 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( M ... (deg `  G ) )  <-> 
( ( M  e.  ZZ  /\  (deg `  G )  e.  ZZ  /\  j  e.  ZZ )  /\  ( M  <_ 
j  /\  j  <_  (deg
`  G ) ) ) )
201199, 200sylibr 217 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  j  e.  ( M ... (deg `  G ) ) )
202 eldifn 3545 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( ( 0 ... (deg `  G
) )  \  ( M ... (deg `  G
) ) )  ->  -.  j  e.  ( M ... (deg `  G
) ) )
203202ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  j  <  M )  ->  -.  j  e.  ( M ... (deg `  G ) ) )
204201, 203condan 811 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
j  <  M )
205204adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  -> 
j  <  M )
2069a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  ->  M  = inf ( {
n  e.  NN0  | 
( (coeff `  G
) `  n )  =/=  0 } ,  RR ,  <  ) )
20712a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  ->  { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 }  C_  ( ZZ>=
`  0 ) )
208185, 186syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( j  e.  ( ( 0 ... (deg `  G
) )  \  ( M ... (deg `  G
) ) )  -> 
j  e.  NN0 )
209208adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( j  e.  ( ( 0 ... (deg `  G ) )  \ 
( M ... (deg `  G ) ) )  /\  -.  ( (coeff `  G ) `  j
)  =  0 )  ->  j  e.  NN0 )
210 neqne 2651 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  ( (coeff `  G
) `  j )  =  0  ->  (
(coeff `  G ) `  j )  =/=  0
)
211210adantl 473 . . . . . . . . . . . . . . . . . 18  |-  ( ( j  e.  ( ( 0 ... (deg `  G ) )  \ 
( M ... (deg `  G ) ) )  /\  -.  ( (coeff `  G ) `  j
)  =  0 )  ->  ( (coeff `  G ) `  j
)  =/=  0 )
212209, 211jca 541 . . . . . . . . . . . . . . . . 17  |-  ( ( j  e.  ( ( 0 ... (deg `  G ) )  \ 
( M ... (deg `  G ) ) )  /\  -.  ( (coeff `  G ) `  j
)  =  0 )  ->  ( j  e. 
NN0  /\  ( (coeff `  G ) `  j
)  =/=  0 ) )
213 fveq2 5879 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  j  ->  (
(coeff `  G ) `  n )  =  ( (coeff `  G ) `  j ) )
214213neeq1d 2702 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  j  ->  (
( (coeff `  G
) `  n )  =/=  0  <->  ( (coeff `  G ) `  j
)  =/=  0 ) )
215214elrab 3184 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  { n  e. 
NN0  |  ( (coeff `  G ) `  n
)  =/=  0 }  <-> 
( j  e.  NN0  /\  ( (coeff `  G
) `  j )  =/=  0 ) )
216212, 215sylibr 217 . . . . . . . . . . . . . . . 16  |-  ( ( j  e.  ( ( 0 ... (deg `  G ) )  \ 
( M ... (deg `  G ) ) )  /\  -.  ( (coeff `  G ) `  j
)  =  0 )  ->  j  e.  {
n  e.  NN0  | 
( (coeff `  G
) `  n )  =/=  0 } )
217216adantll 728 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  -> 
j  e.  { n  e.  NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } )
218 infssuzle 11267 . . . . . . . . . . . . . . 15  |-  ( ( { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 }  C_  ( ZZ>=
`  0 )  /\  j  e.  { n  e.  NN0  |  ( (coeff `  G ) `  n
)  =/=  0 } )  -> inf ( {
n  e.  NN0  | 
( (coeff `  G
) `  n )  =/=  0 } ,  RR ,  <  )  <_  j
)
219207, 217, 218syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  -> inf ( { n  e.  NN0  |  ( (coeff `  G
) `  n )  =/=  0 } ,  RR ,  <  )  <_  j
)
220206, 219eqbrtrd 4416 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  ->  M  <_  j )
22140ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  ->  M  e.  RR )
222188zred 11063 . . . . . . . . . . . . . . 15  |-  ( j  e.  ( ( 0 ... (deg `  G
) )  \  ( M ... (deg `  G
) ) )  -> 
j  e.  RR )
223222ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  -> 
j  e.  RR )
224221, 223lenltd 9798 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  -> 
( M  <_  j  <->  -.  j  <  M ) )
225220, 224mpbid 215 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  /\  -.  ( (coeff `  G
) `  j )  =  0 )  ->  -.  j  <  M )
226205, 225condan 811 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
( (coeff `  G
) `  j )  =  0 )
227226oveq1d 6323 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  =  ( 0  x.  ( A ^ j
) ) )
228129adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  ->  A  e.  CC )
229208adantl 473 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
j  e.  NN0 )
230228, 229expcld 12454 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
( A ^ j
)  e.  CC )
231230mul02d 9849 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
( 0  x.  ( A ^ j ) )  =  0 )
232227, 231eqtrd 2505 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  =  0 )
233232oveq1d 6323 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
( ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  /  ( A ^ M ) )  =  ( 0  /  ( A ^ M ) ) )
234129, 150, 33expne0d 12460 . . . . . . . . . 10  |-  ( ph  ->  ( A ^ M
)  =/=  0 )
235169, 234div0d 10404 . . . . . . . . 9  |-  ( ph  ->  ( 0  /  ( A ^ M ) )  =  0 )
236235adantr 472 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
( 0  /  ( A ^ M ) )  =  0 )
237233, 236eqtrd 2505 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( ( 0 ... (deg `  G )
)  \  ( M ... (deg `  G )
) ) )  -> 
( ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  /  ( A ^ M ) )  =  0 )
238 fzfid 12224 . . . . . . 7  |-  ( ph  ->  ( 0 ... (deg `  G ) )  e. 
Fin )
239180, 182, 237, 238fsumss 13868 . . . . . 6  |-  ( ph  -> 
sum_ j  e.  ( M ... (deg `  G ) ) ( ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  /  ( A ^ M ) )  = 
sum_ j  e.  ( 0 ... (deg `  G ) ) ( ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  /  ( A ^ M ) ) )
240137, 177, 2393eqtrd 2509 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( (deg
`  G )  -  M ) ) ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( A ^ k
) )  =  sum_ j  e.  ( 0 ... (deg `  G
) ) ( ( ( (coeff `  G
) `  j )  x.  ( A ^ j
) )  /  ( A ^ M ) ) )
24189, 55syldan 478 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( (coeff `  G ) `  (
k  +  M ) )  e.  ZZ )
24256fvmpt2 5972 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  ( (coeff `  G ) `  ( k  +  M
) )  e.  ZZ )  ->  ( I `  k )  =  ( (coeff `  G ) `  ( k  +  M
) ) )
24389, 241, 242syl2anc 673 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( I `  k )  =  ( (coeff `  G ) `  ( k  +  M
) ) )
244243adantlr 729 . . . . . . . 8  |-  ( ( ( ph  /\  z  =  A )  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( I `  k )  =  ( (coeff `  G ) `  ( k  +  M
) ) )
245 oveq1 6315 . . . . . . . . 9  |-  ( z  =  A  ->  (
z ^ k )  =  ( A ^
k ) )
246245ad2antlr 741 . . . . . . . 8  |-  ( ( ( ph  /\  z  =  A )  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( z ^
k )  =  ( A ^ k ) )
247244, 246oveq12d 6326 . . . . . . 7  |-  ( ( ( ph  /\  z  =  A )  /\  k  e.  ( 0 ... (
(deg `  G )  -  M ) ) )  ->  ( ( I `
 k )  x.  ( z ^ k
) )  =  ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( A ^ k
) ) )
248247sumeq2dv 13846 . . . . . 6  |-  ( (
ph  /\  z  =  A )  ->  sum_ k  e.  ( 0 ... (
(deg `  G )  -  M ) ) ( ( I `  k
)  x.  ( z ^ k ) )  =  sum_ k  e.  ( 0 ... ( (deg
`  G )  -  M ) ) ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( A ^ k
) ) )
249 fzfid 12224 . . . . . . 7  |-  ( ph  ->  ( 0 ... (
(deg `  G )  -  M ) )  e. 
Fin )
250249, 132fsumcl 13876 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( (deg
`  G )  -  M ) ) ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( A ^ k
) )  e.  CC )
2512, 248, 129, 250fvmptd 5969 . . . . 5  |-  ( ph  ->  ( F `  A
)  =  sum_ k  e.  ( 0 ... (
(deg `  G )  -  M ) ) ( ( (coeff `  G
) `  ( k  +  M ) )  x.  ( A ^ k
) ) )
25217, 16coeid2 23272 . . . . . . . 8  |-  ( ( G  e.  (Poly `  ZZ )  /\  A  e.  CC )  ->  ( G `  A )  =  sum_ j  e.  ( 0 ... (deg `  G ) ) ( ( (coeff `  G
) `  j )  x.  ( A ^ j
) ) )
2535, 129, 252syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( G `  A
)  =  sum_ j  e.  ( 0 ... (deg `  G ) ) ( ( (coeff `  G
) `  j )  x.  ( A ^ j
) ) )
254253oveq1d 6323 . . . . . 6  |-  ( ph  ->  ( ( G `  A )  /  ( A ^ M ) )  =  ( sum_ j  e.  ( 0 ... (deg `  G ) ) ( ( (coeff `  G
) `  j )  x.  ( A ^ j
) )  /  ( A ^ M ) ) )
25586adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... (deg `  G ) ) )  ->  (coeff `  G
) : NN0 --> CC )
256186adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... (deg `  G ) ) )  ->  j  e.  NN0 )
257255, 256ffvelrnd 6038 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... (deg `  G ) ) )  ->  ( (coeff `  G ) `  j
)  e.  CC )
258129adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 0 ... (deg `  G ) ) )  ->  A  e.  CC )
259258, 256expcld 12454 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0 ... (deg `  G ) ) )  ->  ( A ^
j )  e.  CC )
260257, 259mulcld 9681 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0 ... (deg `  G ) ) )  ->  ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  e.  CC )
261238, 169, 260, 234fsumdivc 13924 . . . . . 6  |-  ( ph  ->  ( sum_ j  e.  ( 0 ... (deg `  G ) ) ( ( (coeff `  G
) `  j )  x.  ( A ^ j
) )  /  ( A ^ M ) )  =  sum_ j  e.  ( 0 ... (deg `  G ) ) ( ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  /  ( A ^ M ) ) )
262254, 261eqtrd 2505 . . . . 5  |-  ( ph  ->  ( ( G `  A )  /  ( A ^ M ) )  =  sum_ j  e.  ( 0 ... (deg `  G ) ) ( ( ( (coeff `  G ) `  j
)  x.  ( A ^ j ) )  /  ( A ^ M ) ) )
263240, 251, 2623eqtr4d 2515 . . . 4  |-  ( ph  ->  ( F `  A
)  =  ( ( G `  A )  /  ( A ^ M ) ) )
264 elaa2lem.ga . . . . 5  |-  ( ph  ->  ( G `  A
)  =  0 )
265264oveq1d 6323 . . . 4  |-  ( ph  ->  ( ( G `  A )  /  ( A ^ M ) )  =  ( 0  / 
( A ^ M
) ) )
266263, 265, 2353eqtrd 2509 . . 3  |-  ( ph  ->  ( F `  A
)  =  0 )
267125, 266jca 541 . 2  |-  ( ph  ->  ( ( (coeff `  F ) `  0
)  =/=  0  /\  ( F `  A
)  =  0 ) )
268 fveq2 5879 . . . . . 6  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
269268fveq1d 5881 . . . . 5  |-  ( f  =  F  ->  (
(coeff `  f ) `  0 )  =  ( (coeff `  F
) `  0 )
)
270269neeq1d 2702 . . . 4  |-  ( f  =  F  ->  (
( (coeff `  f
) `  0 )  =/=  0  <->  ( (coeff `  F ) `  0
)  =/=  0 ) )
271 fveq1 5878 . . . . 5  |-  ( f  =  F  ->  (
f `  A )  =  ( F `  A ) )
272271eqeq1d 2473 . . . 4  |-  ( f  =  F  ->  (
( f `  A
)  =  0  <->  ( F `  A )  =  0 ) )
273270, 272anbi12d 725 . . 3  |-  ( f  =  F  ->  (
( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 )  <-> 
( ( (coeff `  F ) `  0
)  =/=  0  /\  ( F `  A
)  =  0 ) ) )
274273rspcev 3136 . 2  |-  ( ( F  e.  (Poly `  ZZ )  /\  (
( (coeff `  F
) `  0 )  =/=  0  /\  ( F `  A )  =  0 ) )  ->  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) )
27560, 267, 274syl2anc 673 1  |-  ( ph  ->  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   {crab 2760    \ cdif 3387    C_ wss 3390   (/)c0 3722   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308  infcinf 7973   CCcc 9555   RRcr 9556   0cc0 9557    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   sum_csu 13829   0pc0p 22706  Polycply 23217  coeffccoe 23219  degcdgr 23220   AAcaa 23346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224  df-aa 23347
This theorem is referenced by:  elaa2  38211
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