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Theorem coemulc 20178
Description: The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
coemulc  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )  =  ( ( NN0  X.  { A } )  o F  x.  (coeff `  F
) ) )

Proof of Theorem coemulc
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3369 . . . . 5  |-  CC  C_  CC
2 plyconst 20130 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
31, 2mpan 653 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
4 plyssc 20124 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
54sseli 3346 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
6 plymulcl 20145 . . . 4  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  F  e.  (Poly `  CC )
)  ->  ( ( CC  X.  { A }
)  o F  x.  F )  e.  (Poly `  CC ) )
73, 5, 6syl2an 465 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (
( CC  X.  { A } )  o F  x.  F )  e.  (Poly `  CC )
)
8 eqid 2438 . . . 4  |-  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )  =  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )
98coef3 20156 . . 3  |-  ( ( ( CC  X.  { A } )  o F  x.  F )  e.  (Poly `  CC )  ->  (coeff `  ( ( CC  X.  { A }
)  o F  x.  F ) ) : NN0 --> CC )
10 ffn 5594 . . 3  |-  ( (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) ) : NN0 --> CC  ->  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )  Fn  NN0 )
117, 9, 103syl 19 . 2  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )  Fn  NN0 )
12 fconstg 5633 . . . . 5  |-  ( A  e.  CC  ->  ( NN0  X.  { A }
) : NN0 --> { A } )
1312adantr 453 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  ( NN0  X.  { A }
) : NN0 --> { A } )
14 ffn 5594 . . . 4  |-  ( ( NN0  X.  { A } ) : NN0 --> { A }  ->  ( NN0  X.  { A }
)  Fn  NN0 )
1513, 14syl 16 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  ( NN0  X.  { A }
)  Fn  NN0 )
16 eqid 2438 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
1716coef3 20156 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1817adantl 454 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  F ) : NN0 --> CC )
19 ffn 5594 . . . 4  |-  ( (coeff `  F ) : NN0 --> CC 
->  (coeff `  F )  Fn  NN0 )
2018, 19syl 16 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  F )  Fn  NN0 )
21 nn0ex 10232 . . . 4  |-  NN0  e.  _V
2221a1i 11 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  NN0  e.  _V )
23 inidm 3552 . . 3  |-  ( NN0 
i^i  NN0 )  =  NN0
2415, 20, 22, 22, 23offn 6319 . 2  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (
( NN0  X.  { A } )  o F  x.  (coeff `  F
) )  Fn  NN0 )
253ad2antrr 708 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( CC  X.  { A } )  e.  (Poly `  CC ) )
26 eqid 2438 . . . . . . 7  |-  (coeff `  ( CC  X.  { A } ) )  =  (coeff `  ( CC  X.  { A } ) )
2726coefv0 20171 . . . . . 6  |-  ( ( CC  X.  { A } )  e.  (Poly `  CC )  ->  (
( CC  X.  { A } ) `  0
)  =  ( (coeff `  ( CC  X.  { A } ) ) ` 
0 ) )
2825, 27syl 16 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( CC  X.  { A } ) ` 
0 )  =  ( (coeff `  ( CC  X.  { A } ) ) `  0 ) )
29 simpll 732 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  A  e.  CC )
30 0cn 9089 . . . . . 6  |-  0  e.  CC
31 fvconst2g 5948 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( ( CC  X.  { A } ) ` 
0 )  =  A )
3229, 30, 31sylancl 645 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( CC  X.  { A } ) ` 
0 )  =  A )
3328, 32eqtr3d 2472 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  ( CC  X.  { A }
) ) `  0
)  =  A )
34 simpr 449 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
3534nn0cnd 10281 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  CC )
3635subid1d 9405 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( n  -  0 )  =  n )
3736fveq2d 5735 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  0 ) )  =  ( (coeff `  F ) `  n
) )
3833, 37oveq12d 6102 . . 3  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) )  =  ( A  x.  (
(coeff `  F ) `  n ) ) )
395ad2antlr 709 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  CC ) )
4026, 16coemul 20175 . . . . 5  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  F  e.  (Poly `  CC )  /\  n  e.  NN0 )  ->  ( (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) ) `  n )  =  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  ( CC  X.  { A } ) ) `  k )  x.  (
(coeff `  F ) `  ( n  -  k
) ) ) )
4125, 39, 34, 40syl3anc 1185 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  (
( CC  X.  { A } )  o F  x.  F ) ) `
 n )  = 
sum_ k  e.  ( 0 ... n ) ( ( (coeff `  ( CC  X.  { A } ) ) `  k )  x.  (
(coeff `  F ) `  ( n  -  k
) ) ) )
42 nn0uz 10525 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
4334, 42syl6eleq 2528 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  ( ZZ>= ` 
0 ) )
44 fzss2 11097 . . . . . 6  |-  ( n  e.  ( ZZ>= `  0
)  ->  ( 0 ... 0 )  C_  ( 0 ... n
) )
4543, 44syl 16 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( 0 ... 0
)  C_  ( 0 ... n ) )
46 elfz1eq 11073 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
4746adantl 454 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( 0 ... 0
) )  ->  k  =  0 )
48 fveq2 5731 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  ( CC  X.  { A } ) ) `  k )  =  ( (coeff `  ( CC  X.  { A } ) ) ` 
0 ) )
49 oveq2 6092 . . . . . . . . 9  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
5049fveq2d 5735 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  F ) `  ( n  -  k
) )  =  ( (coeff `  F ) `  ( n  -  0 ) ) )
5148, 50oveq12d 6102 . . . . . . 7  |-  ( k  =  0  ->  (
( (coeff `  ( CC  X.  { A }
) ) `  k
)  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  =  ( ( (coeff `  ( CC  X.  { A }
) ) `  0
)  x.  ( (coeff `  F ) `  (
n  -  0 ) ) ) )
5247, 51syl 16 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( 0 ... 0
) )  ->  (
( (coeff `  ( CC  X.  { A }
) ) `  k
)  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  =  ( ( (coeff `  ( CC  X.  { A }
) ) `  0
)  x.  ( (coeff `  F ) `  (
n  -  0 ) ) ) )
5318ffvelrnda 5873 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  e.  CC )
5429, 53mulcld 9113 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( A  x.  (
(coeff `  F ) `  n ) )  e.  CC )
5538, 54eqeltrd 2512 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) )  e.  CC )
5655adantr 453 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( 0 ... 0
) )  ->  (
( (coeff `  ( CC  X.  { A }
) ) `  0
)  x.  ( (coeff `  F ) `  (
n  -  0 ) ) )  e.  CC )
5752, 56eqeltrd 2512 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( 0 ... 0
) )  ->  (
( (coeff `  ( CC  X.  { A }
) ) `  k
)  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  e.  CC )
58 eldifn 3472 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  -.  k  e.  ( 0 ... 0 ) )
5958adantl 454 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  -.  k  e.  ( 0 ... 0
) )
60 eldifi 3471 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  ( 0 ... n
) )
61 elfznn0 11088 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
6260, 61syl 16 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  NN0 )
63 eqid 2438 . . . . . . . . . . . . . 14  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
6426, 63dgrub 20158 . . . . . . . . . . . . 13  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0  /\  ( (coeff `  ( CC  X.  { A } ) ) `  k )  =/=  0
)  ->  k  <_  (deg
`  ( CC  X.  { A } ) ) )
65643expia 1156 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0 )  ->  (
( (coeff `  ( CC  X.  { A }
) ) `  k
)  =/=  0  -> 
k  <_  (deg `  ( CC  X.  { A }
) ) ) )
6625, 62, 65syl2an 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (
(coeff `  ( CC  X.  { A } ) ) `  k )  =/=  0  ->  k  <_  (deg `  ( CC  X.  { A } ) ) ) )
67 0dgr 20169 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { A } ) )  =  0 )
6867ad3antrrr 712 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  (deg `  ( CC  X.  { A }
) )  =  0 )
6968breq2d 4227 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( k  <_  (deg `  ( CC  X.  { A } ) )  <->  k  <_  0
) )
7062adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  k  e.  NN0 )
71 nn0le0eq0 10255 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( k  <_  0  <->  k  = 
0 ) )
7270, 71syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( k  <_  0  <->  k  =  0 ) )
7369, 72bitrd 246 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( k  <_  (deg `  ( CC  X.  { A } ) )  <->  k  =  0 ) )
7466, 73sylibd 207 . . . . . . . . . 10  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (
(coeff `  ( CC  X.  { A } ) ) `  k )  =/=  0  ->  k  =  0 ) )
75 id 21 . . . . . . . . . . 11  |-  ( k  =  0  ->  k  =  0 )
76 0z 10298 . . . . . . . . . . . 12  |-  0  e.  ZZ
77 elfz3 11072 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
7876, 77ax-mp 5 . . . . . . . . . . 11  |-  0  e.  ( 0 ... 0
)
7975, 78syl6eqel 2526 . . . . . . . . . 10  |-  ( k  =  0  ->  k  e.  ( 0 ... 0
) )
8074, 79syl6 32 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (
(coeff `  ( CC  X.  { A } ) ) `  k )  =/=  0  ->  k  e.  ( 0 ... 0
) ) )
8180necon1bd 2674 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( -.  k  e.  ( 0 ... 0 )  -> 
( (coeff `  ( CC  X.  { A }
) ) `  k
)  =  0 ) )
8259, 81mpd 15 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (coeff `  ( CC  X.  { A } ) ) `  k )  =  0 )
8382oveq1d 6099 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (
(coeff `  ( CC  X.  { A } ) ) `  k )  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  =  ( 0  x.  ( (coeff `  F ) `  (
n  -  k ) ) ) )
8418adantr 453 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
(coeff `  F ) : NN0 --> CC )
85 fznn0sub 11090 . . . . . . . . 9  |-  ( k  e.  ( 0 ... n )  ->  (
n  -  k )  e.  NN0 )
8660, 85syl 16 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  (
n  -  k )  e.  NN0 )
87 ffvelrn 5871 . . . . . . . 8  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  (
n  -  k )  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  k ) )  e.  CC )
8884, 86, 87syl2an 465 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (coeff `  F ) `  (
n  -  k ) )  e.  CC )
8988mul02d 9269 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( 0  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  =  0 )
9083, 89eqtrd 2470 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (
(coeff `  ( CC  X.  { A } ) ) `  k )  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  =  0 )
91 fzfid 11317 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( 0 ... n
)  e.  Fin )
9245, 57, 90, 91fsumss 12524 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  ( CC  X.  { A }
) ) `  k
)  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  =  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  ( CC  X.  { A }
) ) `  k
)  x.  ( (coeff `  F ) `  (
n  -  k ) ) ) )
9351fsum1 12540 . . . . 5  |-  ( ( 0  e.  ZZ  /\  ( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  ( CC  X.  { A } ) ) `  k )  x.  (
(coeff `  F ) `  ( n  -  k
) ) )  =  ( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) ) )
9476, 55, 93sylancr 646 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  ( CC  X.  { A }
) ) `  k
)  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  =  ( ( (coeff `  ( CC  X.  { A }
) ) `  0
)  x.  ( (coeff `  F ) `  (
n  -  0 ) ) ) )
9541, 92, 943eqtr2d 2476 . . 3  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  (
( CC  X.  { A } )  o F  x.  F ) ) `
 n )  =  ( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) ) )
96 simpl 445 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  A  e.  CC )
97 eqidd 2439 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  =  ( (coeff `  F ) `  n
) )
9822, 96, 20, 97ofc1 6330 . . 3  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( ( NN0 
X.  { A }
)  o F  x.  (coeff `  F ) ) `
 n )  =  ( A  x.  (
(coeff `  F ) `  n ) ) )
9938, 95, 983eqtr4d 2480 . 2  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  (
( CC  X.  { A } )  o F  x.  F ) ) `
 n )  =  ( ( ( NN0 
X.  { A }
)  o F  x.  (coeff `  F ) ) `
 n ) )
10011, 24, 99eqfnfvd 5833 1  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )  =  ( ( NN0  X.  { A } )  o F  x.  (coeff `  F
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958    \ cdif 3319    C_ wss 3322   {csn 3816   class class class wbr 4215    X. cxp 4879    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306   CCcc 8993   0cc0 8995    x. cmul 9000    <_ cle 9126    - cmin 9296   NN0cn0 10226   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048   sum_csu 12484  Polycply 20108  coeffccoe 20110  degcdgr 20111
This theorem is referenced by:  coe0  20179  coesub  20180  mpaaeu  27346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-0p 19565  df-ply 20112  df-coe 20114  df-dgr 20115
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