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Theorem plyco 23274
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 23231 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
31, 2syl 17 . . . 4  |-  ( ph  ->  G : CC --> CC )
43ffvelrnda 6037 . . 3  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
53feqmptd 5932 . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  ( G `
 z ) ) )
6 plyco.1 . . . 4  |-  ( ph  ->  F  e.  (Poly `  S ) )
7 eqid 2471 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
8 eqid 2471 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
97, 8coeid 23271 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
x ^ k ) ) ) )
106, 9syl 17 . . 3  |-  ( ph  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( x ^ k
) ) ) )
11 oveq1 6315 . . . . 5  |-  ( x  =  ( G `  z )  ->  (
x ^ k )  =  ( ( G `
 z ) ^
k ) )
1211oveq2d 6324 . . . 4  |-  ( x  =  ( G `  z )  ->  (
( (coeff `  F
) `  k )  x.  ( x ^ k
) )  =  ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )
1312sumeq2sdv 13847 . . 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 6072 . 2  |-  ( ph  ->  ( F  o.  G
)  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
15 dgrcl 23266 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
166, 15syl 17 . . 3  |-  ( ph  ->  (deg `  F )  e.  NN0 )
17 oveq2 6316 . . . . . . . 8  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
1817sumeq1d 13844 . . . . . . 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 4483 . . . . . 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 2533 . . . . 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 323 . . . 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 6316 . . . . . . . 8  |-  ( x  =  d  ->  (
0 ... x )  =  ( 0 ... d
) )
2322sumeq1d 13844 . . . . . . 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 4483 . . . . . 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 2533 . . . . 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 323 . . . 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 6316 . . . . . . . 8  |-  ( x  =  ( d  +  1 )  ->  (
0 ... x )  =  ( 0 ... (
d  +  1 ) ) )
2827sumeq1d 13844 . . . . . . 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 4483 . . . . . 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 2533 . . . . 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 323 . . . 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 6316 . . . . . . . 8  |-  ( x  =  (deg `  F
)  ->  ( 0 ... x )  =  ( 0 ... (deg `  F ) ) )
3332sumeq1d 13844 . . . . . . 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 4483 . . . . . 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 2533 . . . . 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 323 . . . 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 10972 . . . . . . . . 9  |-  0  e.  ZZ
384exp0d 12448 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 0 )  =  1 )
3938oveq2d 6324 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
40 plybss 23227 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
416, 40syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S  C_  CC )
42 0cnd 9654 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  e.  CC )
4342snssd 4108 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { 0 }  C_  CC )
4441, 43unssd 3601 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
457coef 23263 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
466, 45syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
47 0nn0 10908 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
48 ffvelrn 6035 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  0  e.  NN0 )  -> 
( (coeff `  F
) `  0 )  e.  ( S  u.  {
0 } ) )
4946, 47, 48sylancl 675 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  ( S  u.  {
0 } ) )
5044, 49sseldd 3419 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  CC )
5150adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( (coeff `  F ) `  0
)  e.  CC )
5251mulid1d 9678 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
5339, 52eqtrd 2505 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
5453, 51eqeltrd 2549 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  e.  CC )
55 fveq2 5879 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
56 oveq2 6316 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
0 ) )
5755, 56oveq12d 6326 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( ( G `  z ) ^ 0 ) ) )
5857fsum1 13885 . . . . . . . . 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 676 . . . . . . . 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 2505 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( (coeff `  F ) `  0
) )
6160mpteq2dva 4482 . . . . . 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 4883 . . . . . 6  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
6361, 62syl6eqr 2523 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( CC  X.  {
( (coeff `  F
) `  0 ) } ) )
64 plyconst 23239 . . . . . . 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 673 . . . . . 6  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  ( S  u.  {
0 } ) ) )
66 plyun0 23230 . . . . . 6  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
6765, 66syl6eleq 2559 . . . . 5  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  S ) )
6863, 67eqeltrd 2549 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
)
69 simprr 774 . . . . . . . . 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 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( S  u.  { 0 } ) 
C_  CC )
71 peano2nn0 10934 . . . . . . . . . . . . . 14  |-  ( d  e.  NN0  ->  ( d  +  1 )  e. 
NN0 )
72 ffvelrn 6035 . . . . . . . . . . . . . 14  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  ( d  +  1 )  e.  NN0 )  ->  ( (coeff `  F
) `  ( d  +  1 ) )  e.  ( S  u.  { 0 } ) )
7346, 71, 72syl2an 485 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (coeff `  F ) `  (
d  +  1 ) )  e.  ( S  u.  { 0 } ) )
74 plyconst 23239 . . . . . . . . . . . . 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 673 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  ( S  u.  { 0 } ) ) )
7675, 66syl6eleq 2559 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  S )
)
77 nn0p1nn 10933 . . . . . . . . . . . . 13  |-  ( d  e.  NN0  ->  ( d  +  1 )  e.  NN )
78 oveq2 6316 . . . . . . . . . . . . . . . . 17  |-  ( x  =  1  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
1 ) )
7978mpteq2dv 4483 . . . . . . . . . . . . . . . 16  |-  ( x  =  1  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) ) )
8079eleq1d 2533 . . . . . . . . . . . . . . 15  |-  ( x  =  1  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
1 ) )  e.  (Poly `  S )
) )
8180imbi2d 323 . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . . . 17  |-  ( x  =  d  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
d ) )
8382mpteq2dv 4483 . . . . . . . . . . . . . . . 16  |-  ( x  =  d  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) ) )
8483eleq1d 2533 . . . . . . . . . . . . . . 15  |-  ( x  =  d  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) )  e.  (Poly `  S )
) )
8584imbi2d 323 . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( d  +  1 )  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
8786mpteq2dv 4483 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( d  +  1 )  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )
8887eleq1d 2533 . . . . . . . . . . . . . . 15  |-  ( x  =  ( d  +  1 )  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) )
8988imbi2d 323 . . . . . . . . . . . . . 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 12449 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 1 )  =  ( G `  z
) )
9190mpteq2dva 4482 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  ( z  e.  CC  |->  ( G `  z ) ) )
9291, 5eqtr4d 2508 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  G )
9392, 1eqeltrd 2549 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  e.  (Poly `  S ) )
94 simprr 774 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  -> 
( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) )
951adantr 472 . . . . . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . . . 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 23251 . . . . . . . . . . . . . . . . . 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 626 . . . . . . . . . . . . . . . . 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 9638 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  e.  _V
103102a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  CC  e.  _V )
104 ovex 6336 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( G `  z ) ^ d )  e. 
_V
105104a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ d )  e.  _V )
1064adantlr 729 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  ( G `  z )  e.  CC )
107 eqidd 2472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  =  ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) ) )
1085adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  G  =  ( z  e.  CC  |->  ( G `  z ) ) )
109103, 105, 106, 107, 108offval2 6567 . . . . . . . . . . . . . . . . . . 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 10900 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( d  e.  NN  ->  d  e.  NN0 )
111110ad2antlr 741 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  d  e.  NN0 )
112106, 111expp1d 12455 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ ( d  +  1 ) )  =  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) )
113112mpteq2dva 4482 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) ) )
114109, 113eqtr4d 2508 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  oF  x.  G )  =  ( z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) )
115114eleq1d 2533 . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . 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 442 . . . . . . . . . . . . . . 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 28 . . . . . . . . . . . . . 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 10649 . . . . . . . . . . . . 13  |-  ( ( d  +  1 )  e.  NN  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) )
12077, 119syl 17 . . . . . . . . . . . 12  |-  ( d  e.  NN0  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) )  e.  (Poly `  S ) ) )
121120impcom 437 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
)
12296adantlr 729 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  +  y )  e.  S )
12398adantlr 729 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  x.  y )  e.  S
)
12476, 121, 122, 123plymul 23251 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  oF  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  e.  (Poly `  S ) )
125124adantrr 731 . . . . . . . . 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 729 . . . . . . . . 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 23250 . . . . . . . 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 626 . . . . . . 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 13831 . . . . . . . . . . 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 6336 . . . . . . . . . . 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 2472 . . . . . . . . . 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 5889 . . . . . . . . . . . 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 6336 . . . . . . . . . . . 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 4883 . . . . . . . . . . . 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 2472 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) ) )
142129, 136, 138, 140, 141offval2 6567 . . . . . . . . . 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 6567 . . . . . . . . 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 770 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  NN0 )
145 nn0uz 11217 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
146144, 145syl6eleq 2559 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  ( ZZ>= `  0 )
)
1477coef3 23265 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1486, 147syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  (coeff `  F ) : NN0 --> CC )
149148ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (coeff `  F ) : NN0 --> CC )
150 elfznn0 11913 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... ( d  +  1 ) )  ->  k  e.  NN0 )
151 ffvelrn 6035 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
(coeff `  F ) `  k )  e.  CC )
152149, 150, 151syl2an 485 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( (coeff `  F ) `  k
)  e.  CC )
1534adantlr 729 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  ( G `  z )  e.  CC )
154 expcl 12328 . . . . . . . . . . . . 13  |-  ( ( ( G `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( G `  z ) ^ k
)  e.  CC )
155153, 150, 154syl2an 485 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( G `
 z ) ^
k )  e.  CC )
156152, 155mulcld 9681 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  CC )
157 fveq2 5879 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  ( d  +  1 ) ) )
158 oveq2 6316 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
159157, 158oveq12d 6326 . . . . . . . . . . 11  |-  ( k  =  ( d  +  1 )  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) ) )
160146, 156, 159fsump1 13894 . . . . . . . . . 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 4482 . . . . . . . . 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 2508 . . . . . . . 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 2533 . . . . . . 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 222 . . . . . 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 442 . . . . 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 28 . . . 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 11053 . . 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 2549 1  |-  ( ph  ->  ( F  o.  G
)  e.  (Poly `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388    C_ wss 3390   {csn 3959    |-> cmpt 4454    X. cxp 4837    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   sum_csu 13829  Polycply 23217  coeffccoe 23219  degcdgr 23220
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
This theorem is referenced by:  dgrcolem1  23306  dgrcolem2  23307  taylply2  23402  ftalem7  24084
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