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Theorem coemulc 22379
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 } )  oF  x.  F ) )  =  ( ( NN0  X.  { A } )  oF  x.  (coeff `  F
) ) )

Proof of Theorem coemulc
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3516 . . . . 5  |-  CC  C_  CC
2 plyconst 22331 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
31, 2mpan 670 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
4 plyssc 22325 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
54sseli 3493 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
6 plymulcl 22346 . . . 4  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  F  e.  (Poly `  CC )
)  ->  ( ( CC  X.  { A }
)  oF  x.  F )  e.  (Poly `  CC ) )
73, 5, 6syl2an 477 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (
( CC  X.  { A } )  oF  x.  F )  e.  (Poly `  CC )
)
8 eqid 2460 . . . 4  |-  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )  =  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )
98coef3 22357 . . 3  |-  ( ( ( CC  X.  { A } )  oF  x.  F )  e.  (Poly `  CC )  ->  (coeff `  ( ( CC  X.  { A }
)  oF  x.  F ) ) : NN0 --> CC )
10 ffn 5722 . . 3  |-  ( (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) ) : NN0 --> CC  ->  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )  Fn  NN0 )
117, 9, 103syl 20 . 2  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )  Fn  NN0 )
12 fconstg 5763 . . . . 5  |-  ( A  e.  CC  ->  ( NN0  X.  { A }
) : NN0 --> { A } )
1312adantr 465 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  ( NN0  X.  { A }
) : NN0 --> { A } )
14 ffn 5722 . . . 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 2460 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
1716coef3 22357 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1817adantl 466 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  F ) : NN0 --> CC )
19 ffn 5722 . . . 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 10790 . . . 4  |-  NN0  e.  _V
2221a1i 11 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  NN0  e.  _V )
23 inidm 3700 . . 3  |-  ( NN0 
i^i  NN0 )  =  NN0
2415, 20, 22, 22, 23offn 6526 . 2  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (
( NN0  X.  { A } )  oF  x.  (coeff `  F
) )  Fn  NN0 )
253ad2antrr 725 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( CC  X.  { A } )  e.  (Poly `  CC ) )
26 eqid 2460 . . . . . . 7  |-  (coeff `  ( CC  X.  { A } ) )  =  (coeff `  ( CC  X.  { A } ) )
2726coefv0 22372 . . . . . 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 753 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  A  e.  CC )
30 0cn 9577 . . . . . 6  |-  0  e.  CC
31 fvconst2g 6105 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( ( CC  X.  { A } ) ` 
0 )  =  A )
3229, 30, 31sylancl 662 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( CC  X.  { A } ) ` 
0 )  =  A )
3328, 32eqtr3d 2503 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  ( CC  X.  { A }
) ) `  0
)  =  A )
34 simpr 461 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
3534nn0cnd 10843 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  CC )
3635subid1d 9908 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( n  -  0 )  =  n )
3736fveq2d 5861 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  0 ) )  =  ( (coeff `  F ) `  n
) )
3833, 37oveq12d 6293 . . 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 726 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  CC ) )
4026, 16coemul 22376 . . . . 5  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  F  e.  (Poly `  CC )  /\  n  e.  NN0 )  ->  ( (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) ) `  n )  =  sum_ k  e.  ( 0 ... n ) ( ( (coeff `  ( CC  X.  { A } ) ) `  k )  x.  (
(coeff `  F ) `  ( n  -  k
) ) ) )
4125, 39, 34, 40syl3anc 1223 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  (
( CC  X.  { A } )  oF  x.  F ) ) `
 n )  = 
sum_ k  e.  ( 0 ... n ) ( ( (coeff `  ( CC  X.  { A } ) ) `  k )  x.  (
(coeff `  F ) `  ( n  -  k
) ) ) )
42 nn0uz 11105 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
4334, 42syl6eleq 2558 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  ( ZZ>= ` 
0 ) )
44 fzss2 11712 . . . . . 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 11686 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
4746adantl 466 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( 0 ... 0
) )  ->  k  =  0 )
48 fveq2 5857 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  ( CC  X.  { A } ) ) `  k )  =  ( (coeff `  ( CC  X.  { A } ) ) ` 
0 ) )
49 oveq2 6283 . . . . . . . . 9  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
5049fveq2d 5861 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  F ) `  ( n  -  k
) )  =  ( (coeff `  F ) `  ( n  -  0 ) ) )
5148, 50oveq12d 6293 . . . . . . 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 6012 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  e.  CC )
5429, 53mulcld 9605 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( A  x.  (
(coeff `  F ) `  n ) )  e.  CC )
5538, 54eqeltrd 2548 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) )  e.  CC )
5655adantr 465 . . . . . 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 2548 . . . . 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 3620 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  -.  k  e.  ( 0 ... 0 ) )
5958adantl 466 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  -.  k  e.  ( 0 ... 0
) )
60 eldifi 3619 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  ( 0 ... n
) )
61 elfznn0 11759 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
6260, 61syl 16 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  NN0 )
63 eqid 2460 . . . . . . . . . . . . . 14  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
6426, 63dgrub 22359 . . . . . . . . . . . . 13  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0  /\  ( (coeff `  ( CC  X.  { A } ) ) `  k )  =/=  0
)  ->  k  <_  (deg
`  ( CC  X.  { A } ) ) )
65643expia 1193 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0 )  ->  (
( (coeff `  ( CC  X.  { A }
) ) `  k
)  =/=  0  -> 
k  <_  (deg `  ( CC  X.  { A }
) ) ) )
6625, 62, 65syl2an 477 . . . . . . . . . . 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 22370 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { A } ) )  =  0 )
6867ad3antrrr 729 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  (deg `  ( CC  X.  { A }
) )  =  0 )
6968breq2d 4452 . . . . . . . . . . . 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 466 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  k  e.  NN0 )
71 nn0le0eq0 10813 . . . . . . . . . . . . 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 253 . . . . . . . . . . 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 214 . . . . . . . . . 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 22 . . . . . . . . . . 11  |-  ( k  =  0  ->  k  =  0 )
76 0z 10864 . . . . . . . . . . . 12  |-  0  e.  ZZ
77 elfz3 11685 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
7876, 77ax-mp 5 . . . . . . . . . . 11  |-  0  e.  ( 0 ... 0
)
7975, 78syl6eqel 2556 . . . . . . . . . 10  |-  ( k  =  0  ->  k  e.  ( 0 ... 0
) )
8074, 79syl6 33 . . . . . . . . 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 2678 . . . . . . . 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 6290 . . . . . 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 465 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
(coeff `  F ) : NN0 --> CC )
85 fznn0sub 11705 . . . . . . . . 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 6010 . . . . . . . 8  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  (
n  -  k )  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  k ) )  e.  CC )
8884, 86, 87syl2an 477 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (coeff `  F ) `  (
n  -  k ) )  e.  CC )
8988mul02d 9766 . . . . . 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 2501 . . . . 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 12039 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( 0 ... n
)  e.  Fin )
9245, 57, 90, 91fsumss 13496 . . . 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 13513 . . . . 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 663 . . . 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 2507 . . 3  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  (
( CC  X.  { A } )  oF  x.  F ) ) `
 n )  =  ( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) ) )
96 simpl 457 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  A  e.  CC )
97 eqidd 2461 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  =  ( (coeff `  F ) `  n
) )
9822, 96, 20, 97ofc1 6538 . . 3  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( ( NN0 
X.  { A }
)  oF  x.  (coeff `  F )
) `  n )  =  ( A  x.  ( (coeff `  F ) `  n ) ) )
9938, 95, 983eqtr4d 2511 . 2  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  (
( CC  X.  { A } )  oF  x.  F ) ) `
 n )  =  ( ( ( NN0 
X.  { A }
)  oF  x.  (coeff `  F )
) `  n )
)
10011, 24, 99eqfnfvd 5969 1  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )  =  ( ( NN0  X.  { A } )  oF  x.  (coeff `  F
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106    \ cdif 3466    C_ wss 3469   {csn 4020   class class class wbr 4440    X. cxp 4990    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    oFcof 6513   CCcc 9479   0cc0 9481    x. cmul 9486    <_ cle 9618    - cmin 9794   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661   sum_csu 13457  Polycply 22309  coeffccoe 22311  degcdgr 22312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261  df-sum 13458  df-0p 21805  df-ply 22313  df-coe 22315  df-dgr 22316
This theorem is referenced by:  coe0  22380  coesub  22381  mpaaeu  30693
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