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Theorem coemulc 19851
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 3283 . . . . 5  |-  CC  C_  CC
2 plyconst 19803 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
31, 2mpan 651 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
4 plyssc 19797 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
54sseli 3262 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
6 plymulcl 19818 . . . 4  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  F  e.  (Poly `  CC )
)  ->  ( ( CC  X.  { A }
)  o F  x.  F )  e.  (Poly `  CC ) )
73, 5, 6syl2an 463 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (
( CC  X.  { A } )  o F  x.  F )  e.  (Poly `  CC )
)
8 eqid 2366 . . . 4  |-  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )  =  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )
98coef3 19829 . . 3  |-  ( ( ( CC  X.  { A } )  o F  x.  F )  e.  (Poly `  CC )  ->  (coeff `  ( ( CC  X.  { A }
)  o F  x.  F ) ) : NN0 --> CC )
10 ffn 5495 . . 3  |-  ( (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) ) : NN0 --> CC  ->  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )  Fn  NN0 )
117, 9, 103syl 18 . 2  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  ( ( CC  X.  { A } )  o F  x.  F ) )  Fn  NN0 )
12 fconstg 5534 . . . . 5  |-  ( A  e.  CC  ->  ( NN0  X.  { A }
) : NN0 --> { A } )
1312adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  ( NN0  X.  { A }
) : NN0 --> { A } )
14 ffn 5495 . . . 4  |-  ( ( NN0  X.  { A } ) : NN0 --> { A }  ->  ( NN0  X.  { A }
)  Fn  NN0 )
1513, 14syl 15 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  ( NN0  X.  { A }
)  Fn  NN0 )
16 eqid 2366 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
1716coef3 19829 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1817adantl 452 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  F ) : NN0 --> CC )
19 ffn 5495 . . . 4  |-  ( (coeff `  F ) : NN0 --> CC 
->  (coeff `  F )  Fn  NN0 )
2018, 19syl 15 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  F )  Fn  NN0 )
21 nn0ex 10120 . . . 4  |-  NN0  e.  _V
2221a1i 10 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  NN0  e.  _V )
23 inidm 3466 . . 3  |-  ( NN0 
i^i  NN0 )  =  NN0
2415, 20, 22, 22, 23offn 6216 . 2  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (
( NN0  X.  { A } )  o F  x.  (coeff `  F
) )  Fn  NN0 )
253ad2antrr 706 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( CC  X.  { A } )  e.  (Poly `  CC ) )
26 eqid 2366 . . . . . . 7  |-  (coeff `  ( CC  X.  { A } ) )  =  (coeff `  ( CC  X.  { A } ) )
2726coefv0 19844 . . . . . 6  |-  ( ( CC  X.  { A } )  e.  (Poly `  CC )  ->  (
( CC  X.  { A } ) `  0
)  =  ( (coeff `  ( CC  X.  { A } ) ) ` 
0 ) )
2825, 27syl 15 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( CC  X.  { A } ) ` 
0 )  =  ( (coeff `  ( CC  X.  { A } ) ) `  0 ) )
29 simpll 730 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  A  e.  CC )
30 0cn 8978 . . . . . 6  |-  0  e.  CC
31 fvconst2g 5845 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( ( CC  X.  { A } ) ` 
0 )  =  A )
3229, 30, 31sylancl 643 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( CC  X.  { A } ) ` 
0 )  =  A )
3328, 32eqtr3d 2400 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  ( CC  X.  { A }
) ) `  0
)  =  A )
34 simpr 447 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
3534nn0cnd 10169 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  CC )
3635subid1d 9293 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( n  -  0 )  =  n )
3736fveq2d 5636 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  0 ) )  =  ( (coeff `  F ) `  n
) )
3833, 37oveq12d 5999 . . 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 707 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  CC ) )
4026, 16coemul 19848 . . . . 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 1183 . . . 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 10413 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
4334, 42syl6eleq 2456 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  ( ZZ>= ` 
0 ) )
44 fzss2 10984 . . . . . 6  |-  ( n  e.  ( ZZ>= `  0
)  ->  ( 0 ... 0 )  C_  ( 0 ... n
) )
4543, 44syl 15 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( 0 ... 0
)  C_  ( 0 ... n ) )
46 elfz1eq 10960 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
4746adantl 452 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( 0 ... 0
) )  ->  k  =  0 )
48 fveq2 5632 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  ( CC  X.  { A } ) ) `  k )  =  ( (coeff `  ( CC  X.  { A } ) ) ` 
0 ) )
49 oveq2 5989 . . . . . . . . 9  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
5049fveq2d 5636 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  F ) `  ( n  -  k
) )  =  ( (coeff `  F ) `  ( n  -  0 ) ) )
5148, 50oveq12d 5999 . . . . . . 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 15 . . . . . 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 ) ) ) )
53 ffvelrn 5770 . . . . . . . . . 10  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  n  e.  NN0 )  ->  (
(coeff `  F ) `  n )  e.  CC )
5418, 53sylan 457 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  e.  CC )
5529, 54mulcld 9002 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( A  x.  (
(coeff `  F ) `  n ) )  e.  CC )
5638, 55eqeltrd 2440 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) )  e.  CC )
5756adantr 451 . . . . . 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 )
5852, 57eqeltrd 2440 . . . . 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 )
59 eldifn 3386 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  -.  k  e.  ( 0 ... 0 ) )
6059adantl 452 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  -.  k  e.  ( 0 ... 0
) )
61 eldifi 3385 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  ( 0 ... n
) )
62 elfznn0 10975 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
6361, 62syl 15 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  NN0 )
64 eqid 2366 . . . . . . . . . . . . . 14  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
6526, 64dgrub 19831 . . . . . . . . . . . . 13  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0  /\  ( (coeff `  ( CC  X.  { A } ) ) `  k )  =/=  0
)  ->  k  <_  (deg
`  ( CC  X.  { A } ) ) )
66653expia 1154 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0 )  ->  (
( (coeff `  ( CC  X.  { A }
) ) `  k
)  =/=  0  -> 
k  <_  (deg `  ( CC  X.  { A }
) ) ) )
6725, 63, 66syl2an 463 . . . . . . . . . . 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 } ) ) ) )
68 0dgr 19842 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { A } ) )  =  0 )
6968ad3antrrr 710 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  (deg `  ( CC  X.  { A }
) )  =  0 )
7069breq2d 4137 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( k  <_  (deg `  ( CC  X.  { A } ) )  <->  k  <_  0
) )
7163adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  k  e.  NN0 )
72 nn0le0eq0 10143 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( k  <_  0  <->  k  = 
0 ) )
7371, 72syl 15 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( k  <_  0  <->  k  =  0 ) )
7470, 73bitrd 244 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( k  <_  (deg `  ( CC  X.  { A } ) )  <->  k  =  0 ) )
7567, 74sylibd 205 . . . . . . . . . 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 ) )
76 id 19 . . . . . . . . . . 11  |-  ( k  =  0  ->  k  =  0 )
77 0z 10186 . . . . . . . . . . . 12  |-  0  e.  ZZ
78 elfz3 10959 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
7977, 78ax-mp 8 . . . . . . . . . . 11  |-  0  e.  ( 0 ... 0
)
8076, 79syl6eqel 2454 . . . . . . . . . 10  |-  ( k  =  0  ->  k  e.  ( 0 ... 0
) )
8175, 80syl6 29 . . . . . . . . 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
) ) )
8281necon1bd 2597 . . . . . . . 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 ) )
8360, 82mpd 14 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (coeff `  ( CC  X.  { A } ) ) `  k )  =  0 )
8483oveq1d 5996 . . . . . 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 ) ) ) )
8518adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
(coeff `  F ) : NN0 --> CC )
86 fznn0sub 10977 . . . . . . . . 9  |-  ( k  e.  ( 0 ... n )  ->  (
n  -  k )  e.  NN0 )
8761, 86syl 15 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  (
n  -  k )  e.  NN0 )
88 ffvelrn 5770 . . . . . . . 8  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  (
n  -  k )  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  k ) )  e.  CC )
8985, 87, 88syl2an 463 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (coeff `  F ) `  (
n  -  k ) )  e.  CC )
9089mul02d 9157 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( 0  x.  ( (coeff `  F ) `  (
n  -  k ) ) )  =  0 )
9184, 90eqtrd 2398 . . . . 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 )
92 fzfid 11199 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( 0 ... n
)  e.  Fin )
9345, 58, 91, 92fsumss 12406 . . . 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 ) ) ) )
9451fsum1 12422 . . . . 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 ) ) ) )
9577, 56, 94sylancr 644 . . . 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 ) ) ) )
9641, 93, 953eqtr2d 2404 . . 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 ) ) ) )
97 simpl 443 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  A  e.  CC )
98 eqidd 2367 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  =  ( (coeff `  F ) `  n
) )
9922, 97, 20, 98ofc1 6227 . . 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 ) ) )
10038, 96, 993eqtr4d 2408 . 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 ) )
10111, 24, 100eqfnfvd 5732 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 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   _Vcvv 2873    \ cdif 3235    C_ wss 3238   {csn 3729   class class class wbr 4125    X. cxp 4790    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981    o Fcof 6203   CCcc 8882   0cc0 8884    x. cmul 8889    <_ cle 9015    - cmin 9184   NN0cn0 10114   ZZcz 10175   ZZ>=cuz 10381   ...cfz 10935   sum_csu 12366  Polycply 19781  coeffccoe 19783  degcdgr 19784
This theorem is referenced by:  coe0  19852  coesub  19853  mpaaeu  26946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fz 10936  df-fzo 11026  df-fl 11089  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170  df-sum 12367  df-0p 19240  df-ply 19785  df-coe 19787  df-dgr 19788
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