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Theorem plymulx0 26960
Description: Coefficients of a polynomial multiplyed by  Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)
Assertion
Ref Expression
plymulx0  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
Distinct variable group:    n, F

Proof of Theorem plymulx0
Dummy variables  i  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 3490 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F  e.  (Poly `  RR ) )
2 ax-resscn 9351 . . . . . . . 8  |-  RR  C_  CC
3 1re 9397 . . . . . . . 8  |-  1  e.  RR
4 plyid 21689 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  1  e.  RR )  ->  Xp  e.  (Poly `  RR ) )
52, 3, 4mp2an 672 . . . . . . 7  |-  Xp  e.  (Poly `  RR )
65a1i 11 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  Xp  e.  (Poly `  RR ) )
7 simprl 755 . . . . . . 7  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
8 simprr 756 . . . . . . 7  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  RR )
97, 8readdcld 9425 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
107, 8remulcld 9426 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
111, 6, 9, 10plymul 21698 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  e.  (Poly `  RR ) )
12 0re 9398 . . . . 5  |-  0  e.  RR
13 eqid 2443 . . . . . 6  |-  (coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( F  oF  x.  Xp ) )
1413coef2 21711 . . . . 5  |-  ( ( ( F  oF  x.  Xp )  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
1511, 12, 14sylancl 662 . . . 4  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
16 ffn 5571 . . . 4  |-  ( (coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR  ->  (coeff `  ( F  oF  x.  Xp ) )  Fn  NN0 )
1715, 16syl 16 . . 3  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  Fn 
NN0 )
18 dffn5 5749 . . 3  |-  ( (coeff `  ( F  oF  x.  Xp ) )  Fn  NN0  <->  (coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  ( (coeff `  ( F  oF  x.  Xp ) ) `  n ) ) )
1917, 18sylib 196 . 2  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  ( (coeff `  ( F  oF  x.  Xp ) ) `  n ) ) )
20 cnex 9375 . . . . . . . . 9  |-  CC  e.  _V
2120a1i 11 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  CC  e.  _V )
22 plyf 21678 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
231, 22syl 16 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F : CC --> CC )
24 plyf 21678 . . . . . . . . . 10  |-  ( Xp  e.  (Poly `  RR )  ->  Xp : CC --> CC )
255, 24ax-mp 5 . . . . . . . . 9  |-  Xp : CC --> CC
2625a1i 11 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  Xp : CC --> CC )
27 simprl 755 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  x  e.  CC )
28 simprr 756 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
y  e.  CC )
2927, 28mulcomd 9419 . . . . . . . 8  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
3021, 23, 26, 29caofcom 6364 . . . . . . 7  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  =  ( Xp  oF  x.  F
) )
3130fveq2d 5707 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( Xp  oF  x.  F
) ) )
3231fveq1d 5705 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( (coeff `  ( F  oF  x.  Xp ) ) `  n )  =  ( (coeff `  ( Xp  oF  x.  F
) ) `  n
) )
3332adantr 465 . . . 4  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( F  oF  x.  Xp ) ) `
 n )  =  ( (coeff `  (
Xp  oF  x.  F ) ) `
 n ) )
345a1i 11 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  Xp  e.  (Poly `  RR )
)
351adantr 465 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  RR ) )
36 simpr 461 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
37 eqid 2443 . . . . . . 7  |-  (coeff `  Xp )  =  (coeff `  Xp
)
38 eqid 2443 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
3937, 38coemul 21731 . . . . . 6  |-  ( ( Xp  e.  (Poly `  RR )  /\  F  e.  (Poly `  RR )  /\  n  e.  NN0 )  ->  ( (coeff `  ( Xp  oF  x.  F ) ) `  n )  =  sum_ i  e.  ( 0 ... n ) ( ( (coeff `  Xp ) `  i )  x.  (
(coeff `  F ) `  ( n  -  i
) ) ) )
4034, 35, 36, 39syl3anc 1218 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( Xp  oF  x.  F ) ) `  n )  =  sum_ i  e.  ( 0 ... n ) ( ( (coeff `  Xp ) `  i )  x.  (
(coeff `  F ) `  ( n  -  i
) ) ) )
41 elfznn0 11493 . . . . . . . . . 10  |-  ( i  e.  ( 0 ... n )  ->  i  e.  NN0 )
42 coeidp 21742 . . . . . . . . . 10  |-  ( i  e.  NN0  ->  ( (coeff `  Xp ) `  i )  =  if ( i  =  1 ,  1 ,  0 ) )
4341, 42syl 16 . . . . . . . . 9  |-  ( i  e.  ( 0 ... n )  ->  (
(coeff `  Xp
) `  i )  =  if ( i  =  1 ,  1 ,  0 ) )
4443oveq1d 6118 . . . . . . . 8  |-  ( i  e.  ( 0 ... n )  ->  (
( (coeff `  Xp ) `  i
)  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  ( if ( i  =  1 ,  1 ,  0 )  x.  (
(coeff `  F ) `  ( n  -  i
) ) ) )
45 ovif 6180 . . . . . . . 8  |-  ( if ( i  =  1 ,  1 ,  0 )  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  if ( i  =  1 ,  ( 1  x.  ( (coeff `  F
) `  ( n  -  i ) ) ) ,  ( 0  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) )
4644, 45syl6eq 2491 . . . . . . 7  |-  ( i  e.  ( 0 ... n )  ->  (
( (coeff `  Xp ) `  i
)  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  if ( i  =  1 ,  ( 1  x.  ( (coeff `  F
) `  ( n  -  i ) ) ) ,  ( 0  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) ) )
4746adantl 466 . . . . . 6  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
( (coeff `  Xp ) `  i
)  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  if ( i  =  1 ,  ( 1  x.  ( (coeff `  F
) `  ( n  -  i ) ) ) ,  ( 0  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) ) )
4847sumeq2dv 13192 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  sum_ i  e.  ( 0 ... n ) ( ( (coeff `  Xp ) `  i )  x.  (
(coeff `  F ) `  ( n  -  i
) ) )  = 
sum_ i  e.  ( 0 ... n ) if ( i  =  1 ,  ( 1  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) ,  ( 0  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) ) )
49 elsn 3903 . . . . . . . . . 10  |-  ( i  e.  { 1 }  <-> 
i  =  1 )
5049bicomi 202 . . . . . . . . 9  |-  ( i  =  1  <->  i  e.  { 1 } )
5150a1i 11 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
i  =  1  <->  i  e.  { 1 } ) )
5238coef2 21711 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  F ) : NN0 --> RR )
531, 12, 52sylancl 662 . . . . . . . . . . . 12  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  F ) : NN0 --> RR )
5453ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (coeff `  F ) : NN0 --> RR )
55 fznn0sub 11499 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... n )  ->  (
n  -  i )  e.  NN0 )
5655adantl 466 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
n  -  i )  e.  NN0 )
5754, 56ffvelrnd 5856 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  RR )
582, 57sseldi 3366 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  CC )
5958mulid2d 9416 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
1  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  ( (coeff `  F ) `  ( n  -  i
) ) )
6058mul02d 9579 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
0  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  0 )
6151, 59, 60ifbieq12d 3828 . . . . . . 7  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  if ( i  =  1 ,  ( 1  x.  ( (coeff `  F
) `  ( n  -  i ) ) ) ,  ( 0  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) )  =  if ( i  e. 
{ 1 } , 
( (coeff `  F
) `  ( n  -  i ) ) ,  0 ) )
6261sumeq2dv 13192 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  sum_ i  e.  ( 0 ... n ) if ( i  =  1 ,  ( 1  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) ,  ( 0  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) )  = 
sum_ i  e.  ( 0 ... n ) if ( i  e. 
{ 1 } , 
( (coeff `  F
) `  ( n  -  i ) ) ,  0 ) )
63 eqeq2 2452 . . . . . . 7  |-  ( 0  =  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  (
n  -  1 ) ) )  ->  ( sum_ i  e.  ( 0 ... n ) if ( i  e.  {
1 } ,  ( (coeff `  F ) `  ( n  -  i
) ) ,  0 )  =  0  <->  sum_ i  e.  ( 0 ... n ) if ( i  e.  {
1 } ,  ( (coeff `  F ) `  ( n  -  i
) ) ,  0 )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  (
n  -  1 ) ) ) ) )
64 eqeq2 2452 . . . . . . 7  |-  ( ( (coeff `  F ) `  ( n  -  1 ) )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) )  -> 
( sum_ i  e.  ( 0 ... n ) if ( i  e. 
{ 1 } , 
( (coeff `  F
) `  ( n  -  i ) ) ,  0 )  =  ( (coeff `  F
) `  ( n  -  1 ) )  <->  sum_ i  e.  ( 0 ... n ) if ( i  e.  {
1 } ,  ( (coeff `  F ) `  ( n  -  i
) ) ,  0 )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  (
n  -  1 ) ) ) ) )
65 oveq2 6111 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
66 0z 10669 . . . . . . . . . . . 12  |-  0  e.  ZZ
67 fzsn 11512 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6866, 67ax-mp 5 . . . . . . . . . . 11  |-  ( 0 ... 0 )  =  { 0 }
6965, 68syl6eq 2491 . . . . . . . . . 10  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
70 elsni 3914 . . . . . . . . . . . . 13  |-  ( i  e.  { 0 }  ->  i  =  0 )
7170adantl 466 . . . . . . . . . . . 12  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  i  =  0 )
72 ax-1ne0 9363 . . . . . . . . . . . . . 14  |-  1  =/=  0
7372nesymi 2660 . . . . . . . . . . . . 13  |-  -.  0  =  1
74 eqeq1 2449 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  (
i  =  1  <->  0  =  1 ) )
7573, 74mtbiri 303 . . . . . . . . . . . 12  |-  ( i  =  0  ->  -.  i  =  1 )
7671, 75syl 16 . . . . . . . . . . 11  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  -.  i  =  1 )
7750notbii 296 . . . . . . . . . . . 12  |-  ( -.  i  =  1  <->  -.  i  e.  { 1 } )
7877biimpi 194 . . . . . . . . . . 11  |-  ( -.  i  =  1  ->  -.  i  e.  { 1 } )
79 iffalse 3811 . . . . . . . . . . 11  |-  ( -.  i  e.  { 1 }  ->  if (
i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
8076, 78, 793syl 20 . . . . . . . . . 10  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  if ( i  e.  {
1 } ,  ( (coeff `  F ) `  ( n  -  i
) ) ,  0 )  =  0 )
8169, 80sumeq12rdv 13196 . . . . . . . . 9  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  sum_ i  e.  {
0 } 0 )
82 snfi 7402 . . . . . . . . . . 11  |-  { 0 }  e.  Fin
8382olci 391 . . . . . . . . . 10  |-  ( { 0 }  C_  ( ZZ>=
`  0 )  \/ 
{ 0 }  e.  Fin )
84 sumz 13211 . . . . . . . . . 10  |-  ( ( { 0 }  C_  ( ZZ>= `  0 )  \/  { 0 }  e.  Fin )  ->  sum_ i  e.  { 0 } 0  =  0 )
8583, 84ax-mp 5 . . . . . . . . 9  |-  sum_ i  e.  { 0 } 0  =  0
8681, 85syl6eq 2491 . . . . . . . 8  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
8786adantl 466 . . . . . . 7  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  n  =  0 )  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
88 simpll 753 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  F  e.  ( (Poly `  RR )  \  {
0p } ) )
8936adantr 465 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN0 )
90 simpr 461 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  -.  n  =  0
)
9190neneqad 2693 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  =/=  0 )
9289, 91jca 532 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  -> 
( n  e.  NN0  /\  n  =/=  0 ) )
93 elnnne0 10605 . . . . . . . . 9  |-  ( n  e.  NN  <->  ( n  e.  NN0  /\  n  =/=  0 ) )
9492, 93sylibr 212 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN )
95 1nn0 10607 . . . . . . . . . . . . 13  |-  1  e.  NN0
9695a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  NN0 )
97 simpr 461 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN )
9897nnnn0d 10648 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN0 )
99 nnge1 10360 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  1  <_  n )
10097, 99syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  <_  n
)
101 elfz2nn0 11492 . . . . . . . . . . . 12  |-  ( 1  e.  ( 0 ... n )  <->  ( 1  e.  NN0  /\  n  e.  NN0  /\  1  <_  n ) )
10296, 98, 100, 101syl3anbrc 1172 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  ( 0 ... n ) )
1033elexi 2994 . . . . . . . . . . . 12  |-  1  e.  _V
104103snss 4011 . . . . . . . . . . 11  |-  ( 1  e.  ( 0 ... n )  <->  { 1 }  C_  ( 0 ... n ) )
105102, 104sylib 196 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  { 1 } 
C_  ( 0 ... n ) )
10653ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  (coeff `  F
) : NN0 --> RR )
107 oveq2 6111 . . . . . . . . . . . . . . . 16  |-  ( i  =  1  ->  (
n  -  i )  =  ( n  - 
1 ) )
10849, 107sylbi 195 . . . . . . . . . . . . . . 15  |-  ( i  e.  { 1 }  ->  ( n  -  i )  =  ( n  -  1 ) )
109108adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  =  ( n  -  1 ) )
110 nnm1nn0 10633 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
111110ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  - 
1 )  e.  NN0 )
112109, 111eqeltrd 2517 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  e.  NN0 )
113106, 112ffvelrnd 5856 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  RR )
1142, 113sseldi 3366 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
115114ralrimiva 2811 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  A. i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
116 fzfi 11806 . . . . . . . . . . . 12  |-  ( 0 ... n )  e. 
Fin
117116olci 391 . . . . . . . . . . 11  |-  ( ( 0 ... n ) 
C_  ( ZZ>= `  0
)  \/  ( 0 ... n )  e. 
Fin )
118117a1i 11 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( ( 0 ... n )  C_  ( ZZ>= `  0 )  \/  ( 0 ... n
)  e.  Fin )
)
119 sumss2 13215 . . . . . . . . . 10  |-  ( ( ( { 1 } 
C_  ( 0 ... n )  /\  A. i  e.  { 1 }  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )  /\  ( ( 0 ... n )  C_  ( ZZ>= `  0 )  \/  ( 0 ... n
)  e.  Fin )
)  ->  sum_ i  e. 
{ 1 }  (
(coeff `  F ) `  ( n  -  i
) )  =  sum_ i  e.  ( 0 ... n ) if ( i  e.  {
1 } ,  ( (coeff `  F ) `  ( n  -  i
) ) ,  0 ) )
120105, 115, 118, 119syl21anc 1217 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  sum_ i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  =  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 ) )
12153adantr 465 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  (coeff `  F
) : NN0 --> RR )
12297, 110syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( n  - 
1 )  e.  NN0 )
123121, 122ffvelrnd 5856 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  RR )
1242, 123sseldi 3366 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  CC )
125107fveq2d 5707 . . . . . . . . . . 11  |-  ( i  =  1  ->  (
(coeff `  F ) `  ( n  -  i
) )  =  ( (coeff `  F ) `  ( n  -  1 ) ) )
126125sumsn 13229 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( (coeff `  F ) `  ( n  -  1 ) )  e.  CC )  ->  sum_ i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  =  ( (coeff `  F ) `  (
n  -  1 ) ) )
1273, 124, 126sylancr 663 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  sum_ i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  =  ( (coeff `  F ) `  (
n  -  1 ) ) )
128120, 127eqtr3d 2477 . . . . . . . 8  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  sum_ i  e.  ( 0 ... n ) if ( i  e. 
{ 1 } , 
( (coeff `  F
) `  ( n  -  i ) ) ,  0 )  =  ( (coeff `  F
) `  ( n  -  1 ) ) )
12988, 94, 128syl2anc 661 . . . . . . 7  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  sum_ i  e.  ( 0 ... n ) if ( i  e.  {
1 } ,  ( (coeff `  F ) `  ( n  -  i
) ) ,  0 )  =  ( (coeff `  F ) `  (
n  -  1 ) ) )
13063, 64, 87, 129ifbothda 3836 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  sum_ i  e.  ( 0 ... n ) if ( i  e. 
{ 1 } , 
( (coeff `  F
) `  ( n  -  i ) ) ,  0 )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F
) `  ( n  -  1 ) ) ) )
13162, 130eqtrd 2475 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  sum_ i  e.  ( 0 ... n ) if ( i  =  1 ,  ( 1  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) ,  ( 0  x.  ( (coeff `  F ) `  (
n  -  i ) ) ) )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F
) `  ( n  -  1 ) ) ) )
13240, 48, 1313eqtrd 2479 . . . 4  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( Xp  oF  x.  F ) ) `  n )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  (
n  -  1 ) ) ) )
13333, 132eqtrd 2475 . . 3  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( F  oF  x.  Xp ) ) `
 n )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F
) `  ( n  -  1 ) ) ) )
134133mpteq2dva 4390 . 2  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( n  e.  NN0  |->  ( (coeff `  ( F  oF  x.  Xp ) ) `  n ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
13519, 134eqtrd 2475 1  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   _Vcvv 2984    \ cdif 3337    C_ wss 3340   ifcif 3803   {csn 3889   class class class wbr 4304    e. cmpt 4362    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103    oFcof 6330   Fincfn 7322   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295    x. cmul 9299    <_ cle 9431    - cmin 9607   NNcn 10334   NN0cn0 10591   ZZcz 10658   ZZ>=cuz 10873   ...cfz 11449   sum_csu 13175   0pc0p 21159  Polycply 21664   Xpcidp 21665  coeffccoe 21666
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-rlim 12979  df-sum 13176  df-0p 21160  df-ply 21668  df-idp 21669  df-coe 21670  df-dgr 21671
This theorem is referenced by:  plymulx  26961
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