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Theorem plyco 20113
Description: The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
plyco.1  |-  ( ph  ->  F  e.  (Poly `  S ) )
plyco.2  |-  ( ph  ->  G  e.  (Poly `  S ) )
plyco.3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
plyco.4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
Assertion
Ref Expression
plyco  |-  ( ph  ->  ( F  o.  G
)  e.  (Poly `  S ) )
Distinct variable groups:    x, y, F    x, G, y    ph, x, y    x, S, y

Proof of Theorem plyco
Dummy variables  k 
d  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyco.2 . . . . 5  |-  ( ph  ->  G  e.  (Poly `  S ) )
2 plyf 20070 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
31, 2syl 16 . . . 4  |-  ( ph  ->  G : CC --> CC )
43ffvelrnda 5829 . . 3  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
53feqmptd 5738 . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  ( G `
 z ) ) )
6 plyco.1 . . . 4  |-  ( ph  ->  F  e.  (Poly `  S ) )
7 eqid 2404 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
8 eqid 2404 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
97, 8coeid 20110 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
x ^ k ) ) ) )
106, 9syl 16 . . 3  |-  ( ph  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( x ^ k
) ) ) )
11 oveq1 6047 . . . . 5  |-  ( x  =  ( G `  z )  ->  (
x ^ k )  =  ( ( G `
 z ) ^
k ) )
1211oveq2d 6056 . . . 4  |-  ( x  =  ( G `  z )  ->  (
( (coeff `  F
) `  k )  x.  ( x ^ k
) )  =  ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )
1312sumeq2sdv 12453 . . 3  |-  ( x  =  ( G `  z )  ->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( x ^ k
) )  =  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
( G `  z
) ^ k ) ) )
144, 5, 10, 13fmptco 5860 . 2  |-  ( ph  ->  ( F  o.  G
)  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
15 dgrcl 20105 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
166, 15syl 16 . . 3  |-  ( ph  ->  (deg `  F )  e.  NN0 )
17 oveq2 6048 . . . . . . . 8  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
1817sumeq1d 12450 . . . . . . 7  |-  ( x  =  0  ->  sum_ k  e.  ( 0 ... x
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )
1918mpteq2dv 4256 . . . . . 6  |-  ( x  =  0  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
2019eleq1d 2470 . . . . 5  |-  ( x  =  0  ->  (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) )
2120imbi2d 308 . . . 4  |-  ( x  =  0  ->  (
( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... x
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) )  <->  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) ) )
22 oveq2 6048 . . . . . . . 8  |-  ( x  =  d  ->  (
0 ... x )  =  ( 0 ... d
) )
2322sumeq1d 12450 . . . . . . 7  |-  ( x  =  d  ->  sum_ k  e.  ( 0 ... x
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )
2423mpteq2dv 4256 . . . . . 6  |-  ( x  =  d  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
2524eleq1d 2470 . . . . 5  |-  ( x  =  d  ->  (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) )
2625imbi2d 308 . . . 4  |-  ( x  =  d  ->  (
( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... x
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) )  <->  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) ) )
27 oveq2 6048 . . . . . . . 8  |-  ( x  =  ( d  +  1 )  ->  (
0 ... x )  =  ( 0 ... (
d  +  1 ) ) )
2827sumeq1d 12450 . . . . . . 7  |-  ( x  =  ( d  +  1 )  ->  sum_ k  e.  ( 0 ... x
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )
2928mpteq2dv 4256 . . . . . 6  |-  ( x  =  ( d  +  1 )  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
3029eleq1d 2470 . . . . 5  |-  ( x  =  ( d  +  1 )  ->  (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) )
3130imbi2d 308 . . . 4  |-  ( x  =  ( d  +  1 )  ->  (
( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... x
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) )  <->  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) ) )
32 oveq2 6048 . . . . . . . 8  |-  ( x  =  (deg `  F
)  ->  ( 0 ... x )  =  ( 0 ... (deg `  F ) ) )
3332sumeq1d 12450 . . . . . . 7  |-  ( x  =  (deg `  F
)  ->  sum_ k  e.  ( 0 ... x
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )
3433mpteq2dv 4256 . . . . . 6  |-  ( x  =  (deg `  F
)  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
3534eleq1d 2470 . . . . 5  |-  ( x  =  (deg `  F
)  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
( G `  z
) ^ k ) ) )  e.  (Poly `  S ) ) )
3635imbi2d 308 . . . 4  |-  ( x  =  (deg `  F
)  ->  ( ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... x ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) )  <->  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
( G `  z
) ^ k ) ) )  e.  (Poly `  S ) ) ) )
37 0z 10249 . . . . . . . . 9  |-  0  e.  ZZ
384exp0d 11472 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 0 )  =  1 )
3938oveq2d 6056 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
40 plybss 20066 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
416, 40syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S  C_  CC )
42 0cn 9040 . . . . . . . . . . . . . . . . 17  |-  0  e.  CC
4342a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  e.  CC )
4443snssd 3903 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { 0 }  C_  CC )
4541, 44unssd 3483 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
467coef 20102 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
476, 46syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
48 0nn0 10192 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
49 ffvelrn 5827 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  0  e.  NN0 )  -> 
( (coeff `  F
) `  0 )  e.  ( S  u.  {
0 } ) )
5047, 48, 49sylancl 644 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  ( S  u.  {
0 } ) )
5145, 50sseldd 3309 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  CC )
5251adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( (coeff `  F ) `  0
)  e.  CC )
5352mulid1d 9061 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
5439, 53eqtrd 2436 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
5554, 52eqeltrd 2478 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  e.  CC )
56 fveq2 5687 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
57 oveq2 6048 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
0 ) )
5856, 57oveq12d 6058 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( ( G `  z ) ^ 0 ) ) )
5958fsum1 12490 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  ( ( (coeff `  F ) `  0
)  x.  ( ( G `  z ) ^ 0 ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( ( G `  z ) ^ 0 ) ) )
6037, 55, 59sylancr 645 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( ( (coeff `  F ) `  0
)  x.  ( ( G `  z ) ^ 0 ) ) )
6160, 54eqtrd 2436 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( (coeff `  F ) `  0
) )
6261mpteq2dva 4255 . . . . . 6  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
63 fconstmpt 4880 . . . . . 6  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
6462, 63syl6eqr 2454 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( CC  X.  {
( (coeff `  F
) `  0 ) } ) )
65 plyconst 20078 . . . . . . 7  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  ( (coeff `  F ) `  0
)  e.  ( S  u.  { 0 } ) )  ->  ( CC  X.  { ( (coeff `  F ) `  0
) } )  e.  (Poly `  ( S  u.  { 0 } ) ) )
6645, 50, 65syl2anc 643 . . . . . 6  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  ( S  u.  {
0 } ) ) )
67 plyun0 20069 . . . . . 6  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
6866, 67syl6eleq 2494 . . . . 5  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  S ) )
6964, 68eqeltrd 2478 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
)
70 simprr 734 . . . . . . . . 9  |-  ( (
ph  /\  ( d  e.  NN0  /\  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) ) )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) )
7145adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( S  u.  { 0 } ) 
C_  CC )
72 peano2nn0 10216 . . . . . . . . . . . . . 14  |-  ( d  e.  NN0  ->  ( d  +  1 )  e. 
NN0 )
73 ffvelrn 5827 . . . . . . . . . . . . . 14  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  ( d  +  1 )  e.  NN0 )  ->  ( (coeff `  F
) `  ( d  +  1 ) )  e.  ( S  u.  { 0 } ) )
7447, 72, 73syl2an 464 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (coeff `  F ) `  (
d  +  1 ) )  e.  ( S  u.  { 0 } ) )
75 plyconst 20078 . . . . . . . . . . . . 13  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  ( (coeff `  F ) `  (
d  +  1 ) )  e.  ( S  u.  { 0 } ) )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  ( S  u.  { 0 } ) ) )
7671, 74, 75syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  ( S  u.  { 0 } ) ) )
7776, 67syl6eleq 2494 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  S )
)
78 nn0p1nn 10215 . . . . . . . . . . . . 13  |-  ( d  e.  NN0  ->  ( d  +  1 )  e.  NN )
79 oveq2 6048 . . . . . . . . . . . . . . . . 17  |-  ( x  =  1  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
1 ) )
8079mpteq2dv 4256 . . . . . . . . . . . . . . . 16  |-  ( x  =  1  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) ) )
8180eleq1d 2470 . . . . . . . . . . . . . . 15  |-  ( x  =  1  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
1 ) )  e.  (Poly `  S )
) )
8281imbi2d 308 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  (
( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ x ) )  e.  (Poly `  S
) )  <->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  e.  (Poly `  S ) ) ) )
83 oveq2 6048 . . . . . . . . . . . . . . . . 17  |-  ( x  =  d  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
d ) )
8483mpteq2dv 4256 . . . . . . . . . . . . . . . 16  |-  ( x  =  d  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) ) )
8584eleq1d 2470 . . . . . . . . . . . . . . 15  |-  ( x  =  d  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) )  e.  (Poly `  S )
) )
8685imbi2d 308 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  (
( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ x ) )  e.  (Poly `  S
) )  <->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) ) ) )
87 oveq2 6048 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( d  +  1 )  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
8887mpteq2dv 4256 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( d  +  1 )  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )
8988eleq1d 2470 . . . . . . . . . . . . . . 15  |-  ( x  =  ( d  +  1 )  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) )
9089imbi2d 308 . . . . . . . . . . . . . 14  |-  ( x  =  ( d  +  1 )  ->  (
( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ x ) )  e.  (Poly `  S
) )  <->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) )  e.  (Poly `  S ) ) ) )
914exp1d 11473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 1 )  =  ( G `  z
) )
9291mpteq2dva 4255 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  ( z  e.  CC  |->  ( G `  z ) ) )
9392, 5eqtr4d 2439 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  G )
9493, 1eqeltrd 2478 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  e.  (Poly `  S ) )
95 simprr 734 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  -> 
( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) )
961adantr 452 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  ->  G  e.  (Poly `  S
) )
97 plyco.3 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
9897adantlr 696 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
99 plyco.4 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
10099adantlr 696 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
10195, 96, 98, 100plymul 20090 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  -> 
( ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) )  o F  x.  G )  e.  (Poly `  S
) )
102101expr 599 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  e.  (Poly `  S )  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  o F  x.  G )  e.  (Poly `  S )
) )
103 cnex 9027 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  e.  _V
104103a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  CC  e.  _V )
105 ovex 6065 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( G `  z ) ^ d )  e. 
_V
106105a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ d )  e.  _V )
1074adantlr 696 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  ( G `  z )  e.  CC )
108 eqidd 2405 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  =  ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) ) )
1095adantr 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  G  =  ( z  e.  CC  |->  ( G `  z ) ) )
110104, 106, 107, 108, 109offval2 6281 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  o F  x.  G )  =  ( z  e.  CC  |->  ( ( ( G `  z ) ^ d
)  x.  ( G `
 z ) ) ) )
111 nnnn0 10184 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( d  e.  NN  ->  d  e.  NN0 )
112111ad2antlr 708 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  d  e.  NN0 )
113107, 112expp1d 11479 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ ( d  +  1 ) )  =  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) )
114113mpteq2dva 4255 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) ) )
115110, 114eqtr4d 2439 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  o F  x.  G )  =  ( z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) )
116115eleq1d 2470 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  o F  x.  G )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) )  e.  (Poly `  S ) ) )
117102, 116sylibd 206 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  e.  (Poly `  S )  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) )  e.  (Poly `  S ) ) )
118117expcom 425 . . . . . . . . . . . . . . 15  |-  ( d  e.  NN  ->  ( ph  ->  ( ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
)  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) ) )
119118a2d 24 . . . . . . . . . . . . . 14  |-  ( d  e.  NN  ->  (
( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) )  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) ) )
12082, 86, 90, 90, 94, 119nnind 9974 . . . . . . . . . . . . 13  |-  ( ( d  +  1 )  e.  NN  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) )
12178, 120syl 16 . . . . . . . . . . . 12  |-  ( d  e.  NN0  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) )  e.  (Poly `  S ) ) )
122121impcom 420 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
)
12397adantlr 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  +  y )  e.  S )
12499adantlr 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  x.  y )  e.  S
)
12577, 122, 123, 124plymul 20090 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  e.  (Poly `  S ) )
126125adantrr 698 . . . . . . . . 9  |-  ( (
ph  /\  ( d  e.  NN0  /\  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) ) )  ->  ( ( CC 
X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  e.  (Poly `  S ) )
12797adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  +  y )  e.  S )
12870, 126, 127plyadd 20089 . . . . . . . 8  |-  ( (
ph  /\  ( d  e.  NN0  /\  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) ) )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) )  e.  (Poly `  S ) )
129128expr 599 . . . . . . 7  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) )  e.  (Poly `  S ) ) )
130103a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  CC  e.  _V )
131 sumex 12436 . . . . . . . . . . 11  |-  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  _V
132131a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  _V )
133 ovex 6065 . . . . . . . . . . 11  |-  ( ( (coeff `  F ) `  ( d  +  1 ) )  x.  (
( G `  z
) ^ ( d  +  1 ) ) )  e.  _V
134133a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) )  e. 
_V )
135 eqidd 2405 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) ) )
136 fvex 5701 . . . . . . . . . . . 12  |-  ( (coeff `  F ) `  (
d  +  1 ) )  e.  _V
137136a1i 11 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
(coeff `  F ) `  ( d  +  1 ) )  e.  _V )
138 ovex 6065 . . . . . . . . . . . 12  |-  ( ( G `  z ) ^ ( d  +  1 ) )  e. 
_V
139138a1i 11 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
( G `  z
) ^ ( d  +  1 ) )  e.  _V )
140 fconstmpt 4880 . . . . . . . . . . . 12  |-  ( CC 
X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  ( d  +  1 ) ) )
141140a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  ( d  +  1 ) ) ) )
142 eqidd 2405 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) ) )
143130, 137, 139, 141, 142offval2 6281 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  =  ( z  e.  CC  |->  ( ( (coeff `  F ) `  ( d  +  1 ) )  x.  (
( G `  z
) ^ ( d  +  1 ) ) ) ) )
144130, 132, 134, 135, 143offval2 6281 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  o F  x.  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) ) )  =  ( z  e.  CC  |->  ( sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  +  ( ( (coeff `  F ) `  (
d  +  1 ) )  x.  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) ) )
145 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  NN0 )
146 nn0uz 10476 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
147145, 146syl6eleq 2494 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  ( ZZ>= `  0 )
)
1487coef3 20104 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1496, 148syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (coeff `  F ) : NN0 --> CC )
150149ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (coeff `  F ) : NN0 --> CC )
151 elfznn0 11039 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... ( d  +  1 ) )  ->  k  e.  NN0 )
152 ffvelrn 5827 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
(coeff `  F ) `  k )  e.  CC )
153150, 151, 152syl2an 464 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( (coeff `  F ) `  k
)  e.  CC )
1544adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  ( G `  z )  e.  CC )
155 expcl 11354 . . . . . . . . . . . . 13  |-  ( ( ( G `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( G `  z ) ^ k
)  e.  CC )
156154, 151, 155syl2an 464 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( G `
 z ) ^
k )  e.  CC )
157153, 156mulcld 9064 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  CC )
158 fveq2 5687 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  ( d  +  1 ) ) )
159 oveq2 6048 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
160158, 159oveq12d 6058 . . . . . . . . . . 11  |-  ( k  =  ( d  +  1 )  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) ) )
161147, 157, 160fsump1 12495 . . . . . . . . . 10  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... (
d  +  1 ) ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  +  ( ( (coeff `  F ) `  (
d  +  1 ) )  x.  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) )
162161mpteq2dva 4255 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  =  ( z  e.  CC  |->  ( sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  +  ( ( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) ) ) ) )
163144, 162eqtr4d 2439 . . . . . . . 8  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  o F  x.  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
164163eleq1d 2470 . . . . . . 7  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  o F  x.  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) ) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) )
165129, 164sylibd 206 . . . . . 6  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) )
166165expcom 425 . . . . 5  |-  ( d  e.  NN0  ->  ( ph  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S )  ->  (
z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) ) )
167166a2d 24 . . . 4  |-  ( d  e.  NN0  ->  ( (
ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) )  -> 
( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (
d  +  1 ) ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) ) ) )
16821, 26, 31, 36, 69, 167nn0ind 10322 . . 3  |-  ( (deg
`  F )  e. 
NN0  ->  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
( G `  z
) ^ k ) ) )  e.  (Poly `  S ) ) )
16916, 168mpcom 34 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
( G `  z
) ^ k ) ) )  e.  (Poly `  S ) )
17014, 169eqeltrd 2478 1  |-  ( ph  ->  ( F  o.  G
)  e.  (Poly `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    C_ wss 3280   {csn 3774    e. cmpt 4226    X. cxp 4835    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   sum_csu 12434  Polycply 20056  coeffccoe 20058  degcdgr 20059
This theorem is referenced by:  dgrcolem1  20144  dgrcolem2  20145  taylply2  20237  ftalem7  20814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063
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