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Theorem plyco 22615
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 22572 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
31, 2syl 16 . . . 4  |-  ( ph  ->  G : CC --> CC )
43ffvelrnda 6016 . . 3  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
53feqmptd 5911 . . 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 22612 . . . 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 6288 . . . . 5  |-  ( x  =  ( G `  z )  ->  (
x ^ k )  =  ( ( G `
 z ) ^
k ) )
1211oveq2d 6297 . . . 4  |-  ( x  =  ( G `  z )  ->  (
( (coeff `  F
) `  k )  x.  ( x ^ k
) )  =  ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )
1312sumeq2sdv 13507 . . 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 6049 . 2  |-  ( ph  ->  ( F  o.  G
)  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
15 dgrcl 22607 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
166, 15syl 16 . . 3  |-  ( ph  ->  (deg `  F )  e.  NN0 )
17 oveq2 6289 . . . . . . . 8  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
1817sumeq1d 13504 . . . . . . 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 4524 . . . . . 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 2512 . . . . 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 6289 . . . . . . . 8  |-  ( x  =  d  ->  (
0 ... x )  =  ( 0 ... d
) )
2322sumeq1d 13504 . . . . . . 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 4524 . . . . . 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 2512 . . . . 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 6289 . . . . . . . 8  |-  ( x  =  ( d  +  1 )  ->  (
0 ... x )  =  ( 0 ... (
d  +  1 ) ) )
2827sumeq1d 13504 . . . . . . 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 4524 . . . . . 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 2512 . . . . 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 6289 . . . . . . . 8  |-  ( x  =  (deg `  F
)  ->  ( 0 ... x )  =  ( 0 ... (deg `  F ) ) )
3332sumeq1d 13504 . . . . . . 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 4524 . . . . . 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 2512 . . . . 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 10882 . . . . . . . . 9  |-  0  e.  ZZ
384exp0d 12285 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 0 )  =  1 )
3938oveq2d 6297 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
40 plybss 22568 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
416, 40syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S  C_  CC )
42 0cnd 9592 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  e.  CC )
4342snssd 4160 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { 0 }  C_  CC )
4441, 43unssd 3665 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
457coef 22604 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
466, 45syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
47 0nn0 10817 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
48 ffvelrn 6014 . . . . . . . . . . . . . . 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 3490 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  CC )
5150adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( (coeff `  F ) `  0
)  e.  CC )
5251mulid1d 9616 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
5339, 52eqtrd 2484 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
5453, 51eqeltrd 2531 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  e.  CC )
55 fveq2 5856 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
56 oveq2 6289 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
0 ) )
5755, 56oveq12d 6299 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( ( G `  z ) ^ 0 ) ) )
5857fsum1 13545 . . . . . . . . 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 2484 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( (coeff `  F ) `  0
) )
6160mpteq2dva 4523 . . . . . 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 5033 . . . . . 6  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
6361, 62syl6eqr 2502 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( CC  X.  {
( (coeff `  F
) `  0 ) } ) )
64 plyconst 22580 . . . . . . 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 22571 . . . . . 6  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
6765, 66syl6eleq 2541 . . . . 5  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  S ) )
6863, 67eqeltrd 2531 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
)
69 simprr 757 . . . . . . . . 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 10843 . . . . . . . . . . . . . 14  |-  ( d  e.  NN0  ->  ( d  +  1 )  e. 
NN0 )
72 ffvelrn 6014 . . . . . . . . . . . . . 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 22580 . . . . . . . . . . . . 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 2541 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  S )
)
77 nn0p1nn 10842 . . . . . . . . . . . . 13  |-  ( d  e.  NN0  ->  ( d  +  1 )  e.  NN )
78 oveq2 6289 . . . . . . . . . . . . . . . . 17  |-  ( x  =  1  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
1 ) )
7978mpteq2dv 4524 . . . . . . . . . . . . . . . 16  |-  ( x  =  1  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) ) )
8079eleq1d 2512 . . . . . . . . . . . . . . 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 6289 . . . . . . . . . . . . . . . . 17  |-  ( x  =  d  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
d ) )
8382mpteq2dv 4524 . . . . . . . . . . . . . . . 16  |-  ( x  =  d  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) ) )
8483eleq1d 2512 . . . . . . . . . . . . . . 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 6289 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( d  +  1 )  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
8786mpteq2dv 4524 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( d  +  1 )  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )
8887eleq1d 2512 . . . . . . . . . . . . . . 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 12286 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 1 )  =  ( G `  z
) )
9190mpteq2dva 4523 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  ( z  e.  CC  |->  ( G `  z ) ) )
9291, 5eqtr4d 2487 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  G )
9392, 1eqeltrd 2531 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  e.  (Poly `  S ) )
94 simprr 757 . . . . . . . . . . . . . . . . . . 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 22592 . . . . . . . . . . . . . . . . . 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 9576 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  e.  _V
103102a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  CC  e.  _V )
104 ovex 6309 . . . . . . . . . . . . . . . . . . . . 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 6541 . . . . . . . . . . . . . . . . . . 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 10809 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( d  e.  NN  ->  d  e.  NN0 )
111110ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  d  e.  NN0 )
112106, 111expp1d 12292 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ ( d  +  1 ) )  =  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) )
113112mpteq2dva 4523 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) ) )
114109, 113eqtr4d 2487 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  oF  x.  G )  =  ( z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) )
115114eleq1d 2512 . . . . . . . . . . . . . . . . 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 10561 . . . . . . . . . . . . 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 22592 . . . . . . . . . 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 22591 . . . . . . . 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 13491 . . . . . . . . . . 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 6309 . . . . . . . . . . 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 5866 . . . . . . . . . . . 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 6309 . . . . . . . . . . . 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 5033 . . . . . . . . . . . 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 6541 . . . . . . . . . 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 6541 . . . . . . . . 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 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  NN0 )
145 nn0uz 11125 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
146144, 145syl6eleq 2541 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  ( ZZ>= `  0 )
)
1477coef3 22606 . . . . . . . . . . . . . . 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 11781 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... ( d  +  1 ) )  ->  k  e.  NN0 )
151 ffvelrn 6014 . . . . . . . . . . . . 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 12165 . . . . . . . . . . . . 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 9619 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  CC )
157 fveq2 5856 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  ( d  +  1 ) ) )
158 oveq2 6289 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
159157, 158oveq12d 6299 . . . . . . . . . . 11  |-  ( k  =  ( d  +  1 )  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) ) )
160146, 156, 159fsump1 13552 . . . . . . . . . 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 4523 . . . . . . . . 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 2487 . . . . . . . 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 2512 . . . . . . 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 10966 . . 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 2531 1  |-  ( ph  ->  ( F  o.  G
)  e.  (Poly `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095    u. cun 3459    C_ wss 3461   {csn 4014    |-> cmpt 4495    X. cxp 4987    o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   NNcn 10543   NN0cn0 10802   ZZcz 10871   ZZ>=cuz 11091   ...cfz 11682   ^cexp 12147   sum_csu 13489  Polycply 22558  coeffccoe 22560  degcdgr 22561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11092  df-rp 11231  df-fz 11683  df-fzo 11806  df-fl 11910  df-seq 12089  df-exp 12148  df-hash 12387  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-clim 13292  df-rlim 13293  df-sum 13490  df-0p 22054  df-ply 22562  df-coe 22564  df-dgr 22565
This theorem is referenced by:  dgrcolem1  22646  dgrcolem2  22647  taylply2  22739  ftalem7  23328
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