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Theorem plyco 21712
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 21669 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
31, 2syl 16 . . . 4  |-  ( ph  ->  G : CC --> CC )
43ffvelrnda 5846 . . 3  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
53feqmptd 5747 . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  ( G `
 z ) ) )
6 plyco.1 . . . 4  |-  ( ph  ->  F  e.  (Poly `  S ) )
7 eqid 2443 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
8 eqid 2443 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
97, 8coeid 21709 . . . 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 6101 . . . . 5  |-  ( x  =  ( G `  z )  ->  (
x ^ k )  =  ( ( G `
 z ) ^
k ) )
1211oveq2d 6110 . . . 4  |-  ( x  =  ( G `  z )  ->  (
( (coeff `  F
) `  k )  x.  ( x ^ k
) )  =  ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )
1312sumeq2sdv 13184 . . 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 5879 . 2  |-  ( ph  ->  ( F  o.  G
)  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
15 dgrcl 21704 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
166, 15syl 16 . . 3  |-  ( ph  ->  (deg `  F )  e.  NN0 )
17 oveq2 6102 . . . . . . . 8  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
1817sumeq1d 13181 . . . . . . 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 4382 . . . . . 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 2509 . . . . 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 316 . . . 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 6102 . . . . . . . 8  |-  ( x  =  d  ->  (
0 ... x )  =  ( 0 ... d
) )
2322sumeq1d 13181 . . . . . . 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 4382 . . . . . 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 2509 . . . . 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 316 . . . 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 6102 . . . . . . . 8  |-  ( x  =  ( d  +  1 )  ->  (
0 ... x )  =  ( 0 ... (
d  +  1 ) ) )
2827sumeq1d 13181 . . . . . . 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 4382 . . . . . 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 2509 . . . . 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 316 . . . 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 6102 . . . . . . . 8  |-  ( x  =  (deg `  F
)  ->  ( 0 ... x )  =  ( 0 ... (deg `  F ) ) )
3332sumeq1d 13181 . . . . . . 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 4382 . . . . . 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 2509 . . . . 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 316 . . . 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 10660 . . . . . . . . 9  |-  0  e.  ZZ
384exp0d 12005 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 0 )  =  1 )
3938oveq2d 6110 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
40 plybss 21665 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
416, 40syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S  C_  CC )
42 0cnd 9382 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  e.  CC )
4342snssd 4021 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { 0 }  C_  CC )
4441, 43unssd 3535 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
457coef 21701 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
466, 45syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
47 0nn0 10597 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
48 ffvelrn 5844 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  0  e.  NN0 )  -> 
( (coeff `  F
) `  0 )  e.  ( S  u.  {
0 } ) )
4946, 47, 48sylancl 662 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  ( S  u.  {
0 } ) )
5044, 49sseldd 3360 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  CC )
5150adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( (coeff `  F ) `  0
)  e.  CC )
5251mulid1d 9406 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
5339, 52eqtrd 2475 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
5453, 51eqeltrd 2517 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  e.  CC )
55 fveq2 5694 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
56 oveq2 6102 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
0 ) )
5755, 56oveq12d 6112 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( ( G `  z ) ^ 0 ) ) )
5857fsum1 13221 . . . . . . . . 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 ) ) )
5937, 54, 58sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( ( (coeff `  F ) `  0
)  x.  ( ( G `  z ) ^ 0 ) ) )
6059, 53eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( (coeff `  F ) `  0
) )
6160mpteq2dva 4381 . . . . . 6  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
62 fconstmpt 4885 . . . . . 6  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
6361, 62syl6eqr 2493 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( CC  X.  {
( (coeff `  F
) `  0 ) } ) )
64 plyconst 21677 . . . . . . 7  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  ( (coeff `  F ) `  0
)  e.  ( S  u.  { 0 } ) )  ->  ( CC  X.  { ( (coeff `  F ) `  0
) } )  e.  (Poly `  ( S  u.  { 0 } ) ) )
6544, 49, 64syl2anc 661 . . . . . 6  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  ( S  u.  {
0 } ) ) )
66 plyun0 21668 . . . . . 6  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
6765, 66syl6eleq 2533 . . . . 5  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  S ) )
6863, 67eqeltrd 2517 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
)
69 simprr 756 . . . . . . . . 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 ) )
7044adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( S  u.  { 0 } ) 
C_  CC )
71 peano2nn0 10623 . . . . . . . . . . . . . 14  |-  ( d  e.  NN0  ->  ( d  +  1 )  e. 
NN0 )
72 ffvelrn 5844 . . . . . . . . . . . . . 14  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  ( d  +  1 )  e.  NN0 )  ->  ( (coeff `  F
) `  ( d  +  1 ) )  e.  ( S  u.  { 0 } ) )
7346, 71, 72syl2an 477 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (coeff `  F ) `  (
d  +  1 ) )  e.  ( S  u.  { 0 } ) )
74 plyconst 21677 . . . . . . . . . . . . 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 } ) ) )
7570, 73, 74syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  ( S  u.  { 0 } ) ) )
7675, 66syl6eleq 2533 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  S )
)
77 nn0p1nn 10622 . . . . . . . . . . . . 13  |-  ( d  e.  NN0  ->  ( d  +  1 )  e.  NN )
78 oveq2 6102 . . . . . . . . . . . . . . . . 17  |-  ( x  =  1  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
1 ) )
7978mpteq2dv 4382 . . . . . . . . . . . . . . . 16  |-  ( x  =  1  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) ) )
8079eleq1d 2509 . . . . . . . . . . . . . . 15  |-  ( x  =  1  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
1 ) )  e.  (Poly `  S )
) )
8180imbi2d 316 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  (
( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ x ) )  e.  (Poly `  S
) )  <->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  e.  (Poly `  S ) ) ) )
82 oveq2 6102 . . . . . . . . . . . . . . . . 17  |-  ( x  =  d  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
d ) )
8382mpteq2dv 4382 . . . . . . . . . . . . . . . 16  |-  ( x  =  d  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) ) )
8483eleq1d 2509 . . . . . . . . . . . . . . 15  |-  ( x  =  d  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) )  e.  (Poly `  S )
) )
8584imbi2d 316 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  (
( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ x ) )  e.  (Poly `  S
) )  <->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) ) ) )
86 oveq2 6102 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( d  +  1 )  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
8786mpteq2dv 4382 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( d  +  1 )  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )
8887eleq1d 2509 . . . . . . . . . . . . . . 15  |-  ( x  =  ( d  +  1 )  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) )
8988imbi2d 316 . . . . . . . . . . . . . 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 ) ) ) )
904exp1d 12006 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 1 )  =  ( G `  z
) )
9190mpteq2dva 4381 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  ( z  e.  CC  |->  ( G `  z ) ) )
9291, 5eqtr4d 2478 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  G )
9392, 1eqeltrd 2517 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  e.  (Poly `  S ) )
94 simprr 756 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  -> 
( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) )
951adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  ->  G  e.  (Poly `  S
) )
96 plyco.3 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
9796adantlr 714 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
98 plyco.4 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
9998adantlr 714 . . . . . . . . . . . . . . . . . . 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 )
10094, 95, 97, 99plymul 21689 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  -> 
( ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) )  oF  x.  G )  e.  (Poly `  S
) )
101100expr 615 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  e.  (Poly `  S )  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  oF  x.  G )  e.  (Poly `  S )
) )
102 cnex 9366 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  e.  _V
103102a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  CC  e.  _V )
104 ovex 6119 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( G `  z ) ^ d )  e. 
_V
105104a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ d )  e.  _V )
1064adantlr 714 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  ( G `  z )  e.  CC )
107 eqidd 2444 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  =  ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) ) )
1085adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  G  =  ( z  e.  CC  |->  ( G `  z ) ) )
109103, 105, 106, 107, 108offval2 6339 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  oF  x.  G )  =  ( z  e.  CC  |->  ( ( ( G `  z ) ^ d
)  x.  ( G `
 z ) ) ) )
110 nnnn0 10589 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( d  e.  NN  ->  d  e.  NN0 )
111110ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  d  e.  NN0 )
112106, 111expp1d 12012 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ ( d  +  1 ) )  =  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) )
113112mpteq2dva 4381 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) ) )
114109, 113eqtr4d 2478 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  oF  x.  G )  =  ( z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) )
115114eleq1d 2509 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  oF  x.  G )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) )  e.  (Poly `  S ) ) )
116101, 115sylibd 214 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  e.  (Poly `  S )  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) )  e.  (Poly `  S ) ) )
117116expcom 435 . . . . . . . . . . . . . . 15  |-  ( d  e.  NN  ->  ( ph  ->  ( ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
)  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) ) )
118117a2d 26 . . . . . . . . . . . . . 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 )
) ) )
11981, 85, 89, 89, 93, 118nnind 10343 . . . . . . . . . . . . 13  |-  ( ( d  +  1 )  e.  NN  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) )
12077, 119syl 16 . . . . . . . . . . . 12  |-  ( d  e.  NN0  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) )  e.  (Poly `  S ) ) )
121120impcom 430 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
)
12296adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  +  y )  e.  S )
12398adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  x.  y )  e.  S
)
12476, 121, 122, 123plymul 21689 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  oF  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  e.  (Poly `  S ) )
125124adantrr 716 . . . . . . . . 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 ) ) } )  oF  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  e.  (Poly `  S ) )
12696adantlr 714 . . . . . . . . 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 )
12769, 125, 126plyadd 21688 . . . . . . . 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 ) ) )  oF  +  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  oF  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) )  e.  (Poly `  S ) )
128127expr 615 . . . . . . 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 ) ) )  oF  +  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  oF  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) )  e.  (Poly `  S ) ) )
129102a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  CC  e.  _V )
130 sumex 13168 . . . . . . . . . . 11  |-  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  _V
131130a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  _V )
132 ovex 6119 . . . . . . . . . . 11  |-  ( ( (coeff `  F ) `  ( d  +  1 ) )  x.  (
( G `  z
) ^ ( d  +  1 ) ) )  e.  _V
133132a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) )  e. 
_V )
134 eqidd 2444 . . . . . . . . . 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 ) ) ) )
135 fvex 5704 . . . . . . . . . . . 12  |-  ( (coeff `  F ) `  (
d  +  1 ) )  e.  _V
136135a1i 11 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
(coeff `  F ) `  ( d  +  1 ) )  e.  _V )
137 ovex 6119 . . . . . . . . . . . 12  |-  ( ( G `  z ) ^ ( d  +  1 ) )  e. 
_V
138137a1i 11 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
( G `  z
) ^ ( d  +  1 ) )  e.  _V )
139 fconstmpt 4885 . . . . . . . . . . . 12  |-  ( CC 
X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  ( d  +  1 ) ) )
140139a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  ( d  +  1 ) ) ) )
141 eqidd 2444 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) ) )
142129, 136, 138, 140, 141offval2 6339 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  oF  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  =  ( z  e.  CC  |->  ( ( (coeff `  F ) `  ( d  +  1 ) )  x.  (
( G `  z
) ^ ( d  +  1 ) ) ) ) )
143129, 131, 133, 134, 142offval2 6339 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  oF  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  oF  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 ) ) ) ) ) )
144 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  NN0 )
145 nn0uz 10898 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
146144, 145syl6eleq 2533 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  ( ZZ>= `  0 )
)
1477coef3 21703 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1486, 147syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (coeff `  F ) : NN0 --> CC )
149148ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (coeff `  F ) : NN0 --> CC )
150 elfznn0 11484 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... ( d  +  1 ) )  ->  k  e.  NN0 )
151 ffvelrn 5844 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
(coeff `  F ) `  k )  e.  CC )
152149, 150, 151syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( (coeff `  F ) `  k
)  e.  CC )
1534adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  ( G `  z )  e.  CC )
154 expcl 11886 . . . . . . . . . . . . 13  |-  ( ( ( G `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( G `  z ) ^ k
)  e.  CC )
155153, 150, 154syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( G `
 z ) ^
k )  e.  CC )
156152, 155mulcld 9409 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  CC )
157 fveq2 5694 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  ( d  +  1 ) ) )
158 oveq2 6102 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
159157, 158oveq12d 6112 . . . . . . . . . . 11  |-  ( k  =  ( d  +  1 )  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) ) )
160146, 156, 159fsump1 13226 . . . . . . . . . 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 ) ) ) ) )
161160mpteq2dva 4381 . . . . . . . . 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 ) ) ) ) ) )
162143, 161eqtr4d 2478 . . . . . . . 8  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  oF  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  oF  x.  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
163162eleq1d 2509 . . . . . . 7  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  oF  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  oF  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 )
) )
164128, 163sylibd 214 . . . . . 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 )
) )
165164expcom 435 . . . . 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 )
) ) )
166165a2d 26 . . . 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 ) ) ) )
16721, 26, 31, 36, 68, 166nn0ind 10741 . . 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 ) ) )
16816, 167mpcom 36 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
( G `  z
) ^ k ) ) )  e.  (Poly `  S ) )
16914, 168eqeltrd 2517 1  |-  ( ph  ->  ( F  o.  G
)  e.  (Poly `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2975    u. cun 3329    C_ wss 3331   {csn 3880    e. cmpt 4353    X. cxp 4841    o. ccom 4847   -->wf 5417   ` cfv 5421  (class class class)co 6094    oFcof 6321   CCcc 9283   0cc0 9285   1c1 9286    + caddc 9288    x. cmul 9290   NNcn 10325   NN0cn0 10582   ZZcz 10649   ZZ>=cuz 10864   ...cfz 11440   ^cexp 11868   sum_csu 13166  Polycply 21655  coeffccoe 21657  degcdgr 21658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-fz 11441  df-fzo 11552  df-fl 11645  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-rlim 12970  df-sum 13167  df-0p 21151  df-ply 21659  df-coe 21661  df-dgr 21662
This theorem is referenced by:  dgrcolem1  21743  dgrcolem2  21744  taylply2  21836  ftalem7  22419
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