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Theorem plyco 19639
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 19596 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
31, 2syl 15 . . . 4  |-  ( ph  ->  G : CC --> CC )
4 ffvelrn 5679 . . . 4  |-  ( ( G : CC --> CC  /\  z  e.  CC )  ->  ( G `  z
)  e.  CC )
53, 4sylan 457 . . 3  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
63feqmptd 5591 . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  ( G `
 z ) ) )
7 plyco.1 . . . 4  |-  ( ph  ->  F  e.  (Poly `  S ) )
8 eqid 2296 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
9 eqid 2296 . . . . 5  |-  (deg `  F )  =  (deg
`  F )
108, 9coeid 19636 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
x ^ k ) ) ) )
117, 10syl 15 . . 3  |-  ( ph  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( x ^ k
) ) ) )
12 oveq1 5881 . . . . 5  |-  ( x  =  ( G `  z )  ->  (
x ^ k )  =  ( ( G `
 z ) ^
k ) )
1312oveq2d 5890 . . . 4  |-  ( x  =  ( G `  z )  ->  (
( (coeff `  F
) `  k )  x.  ( x ^ k
) )  =  ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )
1413sumeq2sdv 12193 . . 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 ) ) )
155, 6, 11, 14fmptco 5707 . 2  |-  ( ph  ->  ( F  o.  G
)  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
16 dgrcl 19631 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
177, 16syl 15 . . 3  |-  ( ph  ->  (deg `  F )  e.  NN0 )
18 oveq2 5882 . . . . . . . 8  |-  ( x  =  0  ->  (
0 ... x )  =  ( 0 ... 0
) )
1918sumeq1d 12190 . . . . . . 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 ) ) )
2019mpteq2dv 4123 . . . . . 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
) ) ) )
2120eleq1d 2362 . . . . 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 )
) )
2221imbi2d 307 . . . 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 )
) ) )
23 oveq2 5882 . . . . . . . 8  |-  ( x  =  d  ->  (
0 ... x )  =  ( 0 ... d
) )
2423sumeq1d 12190 . . . . . . 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 ) ) )
2524mpteq2dv 4123 . . . . . 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
) ) ) )
2625eleq1d 2362 . . . . 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 )
) )
2726imbi2d 307 . . . 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 )
) ) )
28 oveq2 5882 . . . . . . . 8  |-  ( x  =  ( d  +  1 )  ->  (
0 ... x )  =  ( 0 ... (
d  +  1 ) ) )
2928sumeq1d 12190 . . . . . . 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 ) ) )
3029mpteq2dv 4123 . . . . . 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
) ) ) )
3130eleq1d 2362 . . . . 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 )
) )
3231imbi2d 307 . . . 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 )
) ) )
33 oveq2 5882 . . . . . . . 8  |-  ( x  =  (deg `  F
)  ->  ( 0 ... x )  =  ( 0 ... (deg `  F ) ) )
3433sumeq1d 12190 . . . . . . 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
) ) )
3534mpteq2dv 4123 . . . . . 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
) ) ) )
3635eleq1d 2362 . . . . 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 ) ) )
3736imbi2d 307 . . . 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 ) ) ) )
38 0z 10051 . . . . . . . . 9  |-  0  e.  ZZ
395exp0d 11255 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 0 )  =  1 )
4039oveq2d 5890 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
41 plybss 19592 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
427, 41syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S  C_  CC )
43 0cn 8847 . . . . . . . . . . . . . . . . 17  |-  0  e.  CC
4443a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  e.  CC )
4544snssd 3776 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { 0 }  C_  CC )
4642, 45unssd 3364 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
478coef 19628 . . . . . . . . . . . . . . . 16  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
487, 47syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
49 0nn0 9996 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
50 ffvelrn 5679 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  0  e.  NN0 )  -> 
( (coeff `  F
) `  0 )  e.  ( S  u.  {
0 } ) )
5148, 49, 50sylancl 643 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  ( S  u.  {
0 } ) )
5246, 51sseldd 3194 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (coeff `  F
) `  0 )  e.  CC )
5352adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  ( (coeff `  F ) `  0
)  e.  CC )
5453mulid1d 8868 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
5540, 54eqtrd 2328 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
5655, 53eqeltrd 2370 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( (coeff `  F ) `  0 )  x.  ( ( G `  z ) ^ 0 ) )  e.  CC )
57 fveq2 5541 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
58 oveq2 5882 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
0 ) )
5957, 58oveq12d 5892 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( ( G `  z ) ^ 0 ) ) )
6059fsum1 12230 . . . . . . . . 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 ) ) )
6138, 56, 60sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( ( (coeff `  F ) `  0
)  x.  ( ( G `  z ) ^ 0 ) ) )
6261, 55eqtrd 2328 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  =  ( (coeff `  F ) `  0
) )
6362mpteq2dva 4122 . . . . . 6  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
64 fconstmpt 4748 . . . . . 6  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
6563, 64syl6eqr 2346 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  =  ( CC  X.  {
( (coeff `  F
) `  0 ) } ) )
66 plyconst 19604 . . . . . . 7  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  ( (coeff `  F ) `  0
)  e.  ( S  u.  { 0 } ) )  ->  ( CC  X.  { ( (coeff `  F ) `  0
) } )  e.  (Poly `  ( S  u.  { 0 } ) ) )
6746, 51, 66syl2anc 642 . . . . . 6  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  ( S  u.  {
0 } ) ) )
68 plyun0 19595 . . . . . 6  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
6967, 68syl6eleq 2386 . . . . 5  |-  ( ph  ->  ( CC  X.  {
( (coeff `  F
) `  0 ) } )  e.  (Poly `  S ) )
7065, 69eqeltrd 2370 . . . 4  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
)
71 simprr 733 . . . . . . . . 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 ) )
7246adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( S  u.  { 0 } ) 
C_  CC )
73 peano2nn0 10020 . . . . . . . . . . . . . 14  |-  ( d  e.  NN0  ->  ( d  +  1 )  e. 
NN0 )
74 ffvelrn 5679 . . . . . . . . . . . . . 14  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  ( d  +  1 )  e.  NN0 )  ->  ( (coeff `  F
) `  ( d  +  1 ) )  e.  ( S  u.  { 0 } ) )
7548, 73, 74syl2an 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (coeff `  F ) `  (
d  +  1 ) )  e.  ( S  u.  { 0 } ) )
76 plyconst 19604 . . . . . . . . . . . . 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 } ) ) )
7772, 75, 76syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  ( S  u.  { 0 } ) ) )
7877, 68syl6eleq 2386 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  e.  (Poly `  S )
)
79 nn0p1nn 10019 . . . . . . . . . . . . 13  |-  ( d  e.  NN0  ->  ( d  +  1 )  e.  NN )
80 oveq2 5882 . . . . . . . . . . . . . . . . 17  |-  ( x  =  1  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
1 ) )
8180mpteq2dv 4123 . . . . . . . . . . . . . . . 16  |-  ( x  =  1  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) ) )
8281eleq1d 2362 . . . . . . . . . . . . . . 15  |-  ( x  =  1  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
1 ) )  e.  (Poly `  S )
) )
8382imbi2d 307 . . . . . . . . . . . . . 14  |-  ( x  =  1  ->  (
( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ x ) )  e.  (Poly `  S
) )  <->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  e.  (Poly `  S ) ) ) )
84 oveq2 5882 . . . . . . . . . . . . . . . . 17  |-  ( x  =  d  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
d ) )
8584mpteq2dv 4123 . . . . . . . . . . . . . . . 16  |-  ( x  =  d  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) ) )
8685eleq1d 2362 . . . . . . . . . . . . . . 15  |-  ( x  =  d  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) )  e.  (Poly `  S )
) )
8786imbi2d 307 . . . . . . . . . . . . . 14  |-  ( x  =  d  ->  (
( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ x ) )  e.  (Poly `  S
) )  <->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) ) ) )
88 oveq2 5882 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( d  +  1 )  ->  (
( G `  z
) ^ x )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
8988mpteq2dv 4123 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( d  +  1 )  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ x ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )
9089eleq1d 2362 . . . . . . . . . . . . . . 15  |-  ( x  =  ( d  +  1 )  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ x
) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) )
9190imbi2d 307 . . . . . . . . . . . . . 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 ) ) ) )
925exp1d 11256 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z ) ^ 1 )  =  ( G `  z
) )
9392mpteq2dva 4122 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  ( z  e.  CC  |->  ( G `  z ) ) )
9493, 6eqtr4d 2331 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  =  G )
9594, 1eqeltrd 2370 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ 1 ) )  e.  (Poly `  S ) )
96 simprr 733 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  -> 
( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) )
971adantr 451 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  ->  G  e.  (Poly `  S
) )
98 plyco.3 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
9998adantlr 695 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  e.  (Poly `  S ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
100 plyco.4 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
101100adantlr 695 . . . . . . . . . . . . . . . . . . 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 )
10296, 97, 99, 101plymul 19616 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( d  e.  NN  /\  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
) ) )  -> 
( ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) )  o F  x.  G )  e.  (Poly `  S
) )
103102expr 598 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  e.  (Poly `  S )  ->  (
( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  o F  x.  G )  e.  (Poly `  S )
) )
104 cnex 8834 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  e.  _V
105104a1i 10 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  CC  e.  _V )
106 ovex 5899 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( G `  z ) ^ d )  e. 
_V
107106a1i 10 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ d )  e.  _V )
1085adantlr 695 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  ( G `  z )  e.  CC )
109 eqidd 2297 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  =  ( z  e.  CC  |->  ( ( G `
 z ) ^
d ) ) )
1106adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  d  e.  NN )  ->  G  =  ( z  e.  CC  |->  ( G `  z ) ) )
111105, 107, 108, 109, 110offval2 6111 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  o F  x.  G )  =  ( z  e.  CC  |->  ( ( ( G `  z ) ^ d
)  x.  ( G `
 z ) ) ) )
112 nnnn0 9988 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( d  e.  NN  ->  d  e.  NN0 )
113112ad2antlr 707 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  d  e.  NN0 )
114108, 113expp1d 11262 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN )  /\  z  e.  CC )  ->  (
( G `  z
) ^ ( d  +  1 ) )  =  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) )
115114mpteq2dva 4122 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  d  e.  NN )  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( ( G `  z ) ^ d )  x.  ( G `  z
) ) ) )
116111, 115eqtr4d 2331 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  o F  x.  G )  =  ( z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) )
117116eleq1d 2362 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( ( z  e.  CC  |->  ( ( G `  z ) ^ d
) )  o F  x.  G )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) )  e.  (Poly `  S ) ) )
118103, 117sylibd 205 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( z  e.  CC  |->  ( ( G `  z
) ^ d ) )  e.  (Poly `  S )  ->  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) )  e.  (Poly `  S ) ) )
119118expcom 424 . . . . . . . . . . . . . . 15  |-  ( d  e.  NN  ->  ( ph  ->  ( ( z  e.  CC  |->  ( ( G `  z ) ^ d ) )  e.  (Poly `  S
)  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) ) )
120119a2d 23 . . . . . . . . . . . . . 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 )
) ) )
12183, 87, 91, 91, 95, 120nnind 9780 . . . . . . . . . . . . 13  |-  ( ( d  +  1 )  e.  NN  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
) )
12279, 121syl 15 . . . . . . . . . . . 12  |-  ( d  e.  NN0  ->  ( ph  ->  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) )  e.  (Poly `  S ) ) )
123122impcom 419 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  e.  (Poly `  S )
)
12498adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  +  y )  e.  S )
125100adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  x.  y )  e.  S
)
12678, 123, 124, 125plymul 19616 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  e.  (Poly `  S ) )
127126adantrr 697 . . . . . . . . 9  |-  ( (
ph  /\  ( d  e.  NN0  /\  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) ) )  ->  ( ( CC 
X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  e.  (Poly `  S ) )
12898adantlr 695 . . . . . . . . 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 )
12971, 127, 128plyadd 19615 . . . . . . . 8  |-  ( (
ph  /\  ( d  e.  NN0  /\  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  e.  (Poly `  S ) ) )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) )  e.  (Poly `  S ) )
130129expr 598 . . . . . . 7  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )  ->  ( ( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) )  e.  (Poly `  S ) ) )
131104a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  CC  e.  _V )
132 sumex 12176 . . . . . . . . . . 11  |-  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  _V
133132a1i 10 . . . . . . . . . 10  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... d
) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  _V )
134 ovex 5899 . . . . . . . . . . 11  |-  ( ( (coeff `  F ) `  ( d  +  1 ) )  x.  (
( G `  z
) ^ ( d  +  1 ) ) )  e.  _V
135134a1i 10 . . . . . . . . . 10  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) )  e. 
_V )
136 eqidd 2297 . . . . . . . . . 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 ) ) ) )
137 fvex 5555 . . . . . . . . . . . 12  |-  ( (coeff `  F ) `  (
d  +  1 ) )  e.  _V
138137a1i 10 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
(coeff `  F ) `  ( d  +  1 ) )  e.  _V )
139 ovex 5899 . . . . . . . . . . . 12  |-  ( ( G `  z ) ^ ( d  +  1 ) )  e. 
_V
140139a1i 10 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (
( G `  z
) ^ ( d  +  1 ) )  e.  _V )
141 fconstmpt 4748 . . . . . . . . . . . 12  |-  ( CC 
X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  ( d  +  1 ) ) )
142141a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  ( d  +  1 ) ) ) )
143 eqidd 2297 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( G `
 z ) ^
( d  +  1 ) ) )  =  ( z  e.  CC  |->  ( ( G `  z ) ^ (
d  +  1 ) ) ) )
144131, 138, 140, 142, 143offval2 6111 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( ( CC  X.  { ( (coeff `  F ) `  (
d  +  1 ) ) } )  o F  x.  ( z  e.  CC  |->  ( ( G `  z ) ^ ( d  +  1 ) ) ) )  =  ( z  e.  CC  |->  ( ( (coeff `  F ) `  ( d  +  1 ) )  x.  (
( G `  z
) ^ ( d  +  1 ) ) ) ) )
145131, 133, 135, 136, 144offval2 6111 . . . . . . . . 9  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  o F  x.  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) ) )  =  ( z  e.  CC  |->  ( sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  +  ( ( (coeff `  F ) `  (
d  +  1 ) )  x.  ( ( G `  z ) ^ ( d  +  1 ) ) ) ) ) )
146 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  NN0 )
147 nn0uz 10278 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
148146, 147syl6eleq 2386 . . . . . . . . . . 11  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  d  e.  ( ZZ>= `  0 )
)
1498coef3 19630 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1507, 149syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  (coeff `  F ) : NN0 --> CC )
151150ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  (coeff `  F ) : NN0 --> CC )
152 elfznn0 10838 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... ( d  +  1 ) )  ->  k  e.  NN0 )
153 ffvelrn 5679 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
(coeff `  F ) `  k )  e.  CC )
154151, 152, 153syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( (coeff `  F ) `  k
)  e.  CC )
1555adantlr 695 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  ->  ( G `  z )  e.  CC )
156 expcl 11137 . . . . . . . . . . . . 13  |-  ( ( ( G `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( G `  z ) ^ k
)  e.  CC )
157155, 152, 156syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( G `
 z ) ^
k )  e.  CC )
158154, 157mulcld 8871 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  z  e.  CC )  /\  k  e.  (
0 ... ( d  +  1 ) ) )  ->  ( ( (coeff `  F ) `  k
)  x.  ( ( G `  z ) ^ k ) )  e.  CC )
159 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  ( d  +  1 ) ) )
160 oveq2 5882 . . . . . . . . . . . 12  |-  ( k  =  ( d  +  1 )  ->  (
( G `  z
) ^ k )  =  ( ( G `
 z ) ^
( d  +  1 ) ) )
161159, 160oveq12d 5892 . . . . . . . . . . 11  |-  ( k  =  ( d  +  1 )  ->  (
( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) )  =  ( ( (coeff `  F
) `  ( d  +  1 ) )  x.  ( ( G `
 z ) ^
( d  +  1 ) ) ) )
162148, 158, 161fsump1 12235 . . . . . . . . . 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 ) ) ) ) )
163162mpteq2dva 4122 . . . . . . . . 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 ) ) ) ) ) )
164145, 163eqtr4d 2331 . . . . . . . 8  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  o F  x.  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) ) )
165164eleq1d 2362 . . . . . . 7  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( (
( z  e.  CC  |->  sum_ k  e.  ( 0 ... d ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  o F  +  ( ( CC  X.  { ( (coeff `  F ) `  ( d  +  1 ) ) } )  o F  x.  (
z  e.  CC  |->  ( ( G `  z
) ^ ( d  +  1 ) ) ) ) )  e.  (Poly `  S )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( d  +  1 ) ) ( ( (coeff `  F
) `  k )  x.  ( ( G `  z ) ^ k
) ) )  e.  (Poly `  S )
) )
166130, 165sylibd 205 . . . . . 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 )
) )
167166expcom 424 . . . . 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 )
) ) )
168167a2d 23 . . . 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 ) ) ) )
16922, 27, 32, 37, 70, 168nn0ind 10124 . . 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 ) ) )
17017, 169mpcom 32 . 2  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
( G `  z
) ^ k ) ) )  e.  (Poly `  S ) )
17115, 170eqeltrd 2370 1  |-  ( ph  ->  ( F  o.  G
)  e.  (Poly `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    C_ wss 3165   {csn 3653    e. cmpt 4093    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120   sum_csu 12174  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  dgrcolem1  19670  dgrcolem2  19671  taylply2  19763  ftalem7  20332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
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