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Theorem coemulc 21681
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 3372 . . . . 5  |-  CC  C_  CC
2 plyconst 21633 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
31, 2mpan 665 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
4 plyssc 21627 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
54sseli 3349 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
6 plymulcl 21648 . . . 4  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  F  e.  (Poly `  CC )
)  ->  ( ( CC  X.  { A }
)  oF  x.  F )  e.  (Poly `  CC ) )
73, 5, 6syl2an 474 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (
( CC  X.  { A } )  oF  x.  F )  e.  (Poly `  CC )
)
8 eqid 2441 . . . 4  |-  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )  =  (coeff `  ( ( CC  X.  { A } )  oF  x.  F ) )
98coef3 21659 . . 3  |-  ( ( ( CC  X.  { A } )  oF  x.  F )  e.  (Poly `  CC )  ->  (coeff `  ( ( CC  X.  { A }
)  oF  x.  F ) ) : NN0 --> CC )
10 ffn 5556 . . 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 5594 . . . . 5  |-  ( A  e.  CC  ->  ( NN0  X.  { A }
) : NN0 --> { A } )
1312adantr 462 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  ( NN0  X.  { A }
) : NN0 --> { A } )
14 ffn 5556 . . . 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 2441 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
1716coef3 21659 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
1817adantl 463 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (coeff `  F ) : NN0 --> CC )
19 ffn 5556 . . . 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 10581 . . . 4  |-  NN0  e.  _V
2221a1i 11 . . 3  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  NN0  e.  _V )
23 inidm 3556 . . 3  |-  ( NN0 
i^i  NN0 )  =  NN0
2415, 20, 22, 22, 23offn 6330 . 2  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  (
( NN0  X.  { A } )  oF  x.  (coeff `  F
) )  Fn  NN0 )
253ad2antrr 720 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( CC  X.  { A } )  e.  (Poly `  CC ) )
26 eqid 2441 . . . . . . 7  |-  (coeff `  ( CC  X.  { A } ) )  =  (coeff `  ( CC  X.  { A } ) )
2726coefv0 21674 . . . . . 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 748 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  A  e.  CC )
30 0cn 9374 . . . . . 6  |-  0  e.  CC
31 fvconst2g 5928 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( ( CC  X.  { A } ) ` 
0 )  =  A )
3229, 30, 31sylancl 657 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( CC  X.  { A } ) ` 
0 )  =  A )
3328, 32eqtr3d 2475 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  ( CC  X.  { A }
) ) `  0
)  =  A )
34 simpr 458 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
3534nn0cnd 10634 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  CC )
3635subid1d 9704 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( n  -  0 )  =  n )
3736fveq2d 5692 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  0 ) )  =  ( (coeff `  F ) `  n
) )
3833, 37oveq12d 6108 . . 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 721 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  CC ) )
4026, 16coemul 21678 . . . . 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 1213 . . . 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 10891 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
4334, 42syl6eleq 2531 . . . . . 6  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  ->  n  e.  ( ZZ>= ` 
0 ) )
44 fzss2 11494 . . . . . 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 11458 . . . . . . . 8  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
4746adantl 463 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( 0 ... 0
) )  ->  k  =  0 )
48 fveq2 5688 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  ( CC  X.  { A } ) ) `  k )  =  ( (coeff `  ( CC  X.  { A } ) ) ` 
0 ) )
49 oveq2 6098 . . . . . . . . 9  |-  ( k  =  0  ->  (
n  -  k )  =  ( n  - 
0 ) )
5049fveq2d 5692 . . . . . . . 8  |-  ( k  =  0  ->  (
(coeff `  F ) `  ( n  -  k
) )  =  ( (coeff `  F ) `  ( n  -  0 ) ) )
5148, 50oveq12d 6108 . . . . . . 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 5840 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  e.  CC )
5429, 53mulcld 9402 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( A  x.  (
(coeff `  F ) `  n ) )  e.  CC )
5538, 54eqeltrd 2515 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( (coeff `  ( CC  X.  { A } ) ) ` 
0 )  x.  (
(coeff `  F ) `  ( n  -  0 ) ) )  e.  CC )
5655adantr 462 . . . . . 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 2515 . . . . 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 3476 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  -.  k  e.  ( 0 ... 0 ) )
5958adantl 463 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  -.  k  e.  ( 0 ... 0
) )
60 eldifi 3475 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  ( 0 ... n
) )
61 elfznn0 11477 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
6260, 61syl 16 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... n )  \ 
( 0 ... 0
) )  ->  k  e.  NN0 )
63 eqid 2441 . . . . . . . . . . . . . 14  |-  (deg `  ( CC  X.  { A } ) )  =  (deg `  ( CC  X.  { A } ) )
6426, 63dgrub 21661 . . . . . . . . . . . . 13  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0  /\  ( (coeff `  ( CC  X.  { A } ) ) `  k )  =/=  0
)  ->  k  <_  (deg
`  ( CC  X.  { A } ) ) )
65643expia 1184 . . . . . . . . . . . 12  |-  ( ( ( CC  X.  { A } )  e.  (Poly `  CC )  /\  k  e.  NN0 )  ->  (
( (coeff `  ( CC  X.  { A }
) ) `  k
)  =/=  0  -> 
k  <_  (deg `  ( CC  X.  { A }
) ) ) )
6625, 62, 65syl2an 474 . . . . . . . . . . 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 21672 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { A } ) )  =  0 )
6867ad3antrrr 724 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  (deg `  ( CC  X.  { A }
) )  =  0 )
6968breq2d 4301 . . . . . . . . . . . 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 463 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  k  e.  NN0 )
71 nn0le0eq0 10604 . . . . . . . . . . . . 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 10653 . . . . . . . . . . . 12  |-  0  e.  ZZ
77 elfz3 11457 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
7876, 77ax-mp 5 . . . . . . . . . . 11  |-  0  e.  ( 0 ... 0
)
7975, 78syl6eqel 2529 . . . . . . . . . 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 2677 . . . . . . . 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 6105 . . . . . 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 462 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
(coeff `  F ) : NN0 --> CC )
85 fznn0sub 11483 . . . . . . . . 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 5838 . . . . . . . 8  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  (
n  -  k )  e.  NN0 )  -> 
( (coeff `  F
) `  ( n  -  k ) )  e.  CC )
8884, 86, 87syl2an 474 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  F  e.  (Poly `  S )
)  /\  n  e.  NN0 )  /\  k  e.  ( ( 0 ... n )  \  (
0 ... 0 ) ) )  ->  ( (coeff `  F ) `  (
n  -  k ) )  e.  CC )
8988mul02d 9563 . . . . . 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 2473 . . . . 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 11791 . . . . 5  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( 0 ... n
)  e.  Fin )
9245, 57, 90, 91fsumss 13198 . . . 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 13214 . . . . 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 658 . . . 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 2479 . . 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 454 . . . 4  |-  ( ( A  e.  CC  /\  F  e.  (Poly `  S
) )  ->  A  e.  CC )
97 eqidd 2442 . . . 4  |-  ( ( ( A  e.  CC  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( (coeff `  F
) `  n )  =  ( (coeff `  F ) `  n
) )
9822, 96, 20, 97ofc1 6342 . . 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 2483 . 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 5797 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 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    \ cdif 3322    C_ wss 3325   {csn 3874   class class class wbr 4289    X. cxp 4834    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   0cc0 9278    x. cmul 9283    <_ cle 9415    - cmin 9591   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   sum_csu 13159  Polycply 21611  coeffccoe 21613  degcdgr 21614
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-coe 21617  df-dgr 21618
This theorem is referenced by:  coe0  21682  coesub  21683  mpaaeu  29432
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