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Theorem coemulc 21725
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 3378 . . . . 5  |-  CC  C_  CC
2 plyconst 21677 . . . . 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 21671 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
54sseli 3355 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
6 plymulcl 21692 . . . 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 2443 . . . 4  |-  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )  =  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )
98coef3 21703 . . 3  |-  ( ( ( CC  X.  { A } )  oF  x.  F )  e.  (Poly `  CC )  ->  (coeff `  ( ( CC  X.  { A }
)  oF  x.  F ) ) : NN0 --> CC )
10 ffn 5562 . . 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 5600 . . . . 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 5562 . . . 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 2443 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
1716coef3 21703 . . . . 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 5562 . . . 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 10588 . . . 4  |-  NN0  e.  _V
2221a1i 11 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  NN0  e.  _V )
23 inidm 3562 . . 3  |-  ( NN0 
i^i  NN0 )  =  NN0
2415, 20, 22, 22, 23offn 6334 . 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 2443 . . . . . . 7  |-  (coeff `  ( CC  X.  { A } ) )  =  (coeff `  ( CC  X.  { A } ) )
2726coefv0 21718 . . . . . 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 9381 . . . . . 6  |-  0  e.  CC
31 fvconst2g 5934 . . . . . 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 2477 . . . 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 10641 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  CC )
3635subid1d 9711 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( n  -  0 )  =  n )
3736fveq2d 5698 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  0 ) )  =  ( (coeff `  F ) `  n
) )
3833, 37oveq12d 6112 . . 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 21722 . . . . 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 1218 . . . 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 10898 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
4334, 42syl6eleq 2533 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  ( ZZ>= ` 
0 ) )
44 fzss2 11501 . . . . . 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 11465 . . . . . . . 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 5694 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  ( CC  X.  { A } ) ) `  k )  =  ( (coeff `  ( CC  X.  { A } ) ) ` 
0 ) )
49 oveq2 6102 . . . . . . . . 9  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
5049fveq2d 5698 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  F ) `  ( n  -  k
) )  =  ( (coeff `  F ) `  ( n  -  0 ) ) )
5148, 50oveq12d 6112 . . . . . . 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 5846 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  e.  CC )
5429, 53mulcld 9409 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( A  x.  (
(coeff `  F ) `  n ) )  e.  CC )
5538, 54eqeltrd 2517 . . . . . . 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 2517 . . . . 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 3482 . . . . . . . . 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 3481 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  ( 0 ... n
) )
61 elfznn0 11484 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
6260, 61syl 16 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  NN0 )
63 eqid 2443 . . . . . . . . . . . . . 14  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
6426, 63dgrub 21705 . . . . . . . . . . . . 13  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0  /\  ( (coeff `  ( CC  X.  { A } ) ) `  k )  =/=  0
)  ->  k  <_  (deg
`  ( CC  X.  { A } ) ) )
65643expia 1189 . . . . . . . . . . . 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 21716 . . . . . . . . . . . . . 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 4307 . . . . . . . . . . . 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 10611 . . . . . . . . . . . . 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 10660 . . . . . . . . . . . 12  |-  0  e.  ZZ
77 elfz3 11464 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
7876, 77ax-mp 5 . . . . . . . . . . 11  |-  0  e.  ( 0 ... 0
)
7975, 78syl6eqel 2531 . . . . . . . . . 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 2682 . . . . . . . 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 6109 . . . . . 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 11490 . . . . . . . . 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 5844 . . . . . . . 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 9570 . . . . . 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 2475 . . . . 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 11798 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( 0 ... n
)  e.  Fin )
9245, 57, 90, 91fsumss 13205 . . . 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 13221 . . . . 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 2481 . . 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 2444 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  =  ( (coeff `  F ) `  n
) )
9822, 96, 20, 97ofc1 6346 . . 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 2485 . 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 5803 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 1369    e. wcel 1756    =/= wne 2609   _Vcvv 2975    \ cdif 3328    C_ wss 3331   {csn 3880   class class class wbr 4295    X. cxp 4841    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6094    oFcof 6321   CCcc 9283   0cc0 9285    x. cmul 9290    <_ cle 9422    - cmin 9598   NN0cn0 10582   ZZcz 10649   ZZ>=cuz 10864   ...cfz 11440   sum_csu 13166  Polycply 21655  coeffccoe 21657  degcdgr 21658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-fz 11441  df-fzo 11552  df-fl 11645  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-rlim 12970  df-sum 13167  df-0p 21151  df-ply 21659  df-coe 21661  df-dgr 21662
This theorem is referenced by:  coe0  21726  coesub  21727  mpaaeu  29510
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