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Theorem plymulx0 26862
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 3475 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F  e.  (Poly `  RR ) )
2 ax-resscn 9335 . . . . . . . 8  |-  RR  C_  CC
3 1re 9381 . . . . . . . 8  |-  1  e.  RR
4 plyid 21620 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  1  e.  RR )  ->  Xp  e.  (Poly `  RR ) )
52, 3, 4mp2an 667 . . . . . . 7  |-  Xp  e.  (Poly `  RR )
65a1i 11 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  Xp  e.  (Poly `  RR ) )
7 simprl 750 . . . . . . 7  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
8 simprr 751 . . . . . . 7  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  RR )
97, 8readdcld 9409 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
107, 8remulcld 9410 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
111, 6, 9, 10plymul 21629 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  e.  (Poly `  RR ) )
12 0re 9382 . . . . 5  |-  0  e.  RR
13 eqid 2441 . . . . . 6  |-  (coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( F  oF  x.  Xp ) )
1413coef2 21642 . . . . 5  |-  ( ( ( F  oF  x.  Xp )  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
1511, 12, 14sylancl 657 . . . 4  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
16 ffn 5556 . . . 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 5734 . . 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 9359 . . . . . . . . 9  |-  CC  e.  _V
2120a1i 11 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  CC  e.  _V )
22 plyf 21609 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
231, 22syl 16 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F : CC --> CC )
24 plyf 21609 . . . . . . . . . 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 750 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  x  e.  CC )
28 simprr 751 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
y  e.  CC )
2927, 28mulcomd 9403 . . . . . . . 8  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
3021, 23, 26, 29caofcom 6351 . . . . . . 7  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  =  ( Xp  oF  x.  F
) )
3130fveq2d 5692 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( Xp  oF  x.  F
) ) )
3231fveq1d 5690 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( (coeff `  ( F  oF  x.  Xp ) ) `  n )  =  ( (coeff `  ( Xp  oF  x.  F
) ) `  n
) )
3332adantr 462 . . . 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 462 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  RR ) )
36 simpr 458 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
37 eqid 2441 . . . . . . 7  |-  (coeff `  Xp )  =  (coeff `  Xp
)
38 eqid 2441 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
3937, 38coemul 21662 . . . . . 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 1213 . . . . 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 11477 . . . . . . . . . 10  |-  ( i  e.  ( 0 ... n )  ->  i  e.  NN0 )
42 coeidp 21673 . . . . . . . . . 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 6105 . . . . . . . 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 6167 . . . . . . . 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 2489 . . . . . . 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 463 . . . . . 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 13176 . . . . 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 3888 . . . . . . . . . 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 21642 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  F ) : NN0 --> RR )
531, 12, 52sylancl 657 . . . . . . . . . . . 12  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  F ) : NN0 --> RR )
5453ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (coeff `  F ) : NN0 --> RR )
55 fznn0sub 11483 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... n )  ->  (
n  -  i )  e.  NN0 )
5655adantl 463 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
n  -  i )  e.  NN0 )
5754, 56ffvelrnd 5841 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  RR )
582, 57sseldi 3351 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  CC )
5958mulid2d 9400 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
1  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  ( (coeff `  F ) `  ( n  -  i
) ) )
6058mul02d 9563 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
0  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  0 )
6151, 59, 60ifbieq12d 3813 . . . . . . 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 13176 . . . . . 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 2450 . . . . . . 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 2450 . . . . . . 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 6098 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
66 0z 10653 . . . . . . . . . . . 12  |-  0  e.  ZZ
67 fzsn 11496 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6866, 67ax-mp 5 . . . . . . . . . . 11  |-  ( 0 ... 0 )  =  { 0 }
6965, 68syl6eq 2489 . . . . . . . . . 10  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
70 elsni 3899 . . . . . . . . . . . . 13  |-  ( i  e.  { 0 }  ->  i  =  0 )
7170adantl 463 . . . . . . . . . . . 12  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  i  =  0 )
72 ax-1ne0 9347 . . . . . . . . . . . . . 14  |-  1  =/=  0
7372nesymi 2646 . . . . . . . . . . . . 13  |-  -.  0  =  1
74 eqeq1 2447 . . . . . . . . . . . . 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 3796 . . . . . . . . . . 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 13180 . . . . . . . . 9  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  sum_ i  e.  {
0 } 0 )
82 snfi 7386 . . . . . . . . . . 11  |-  { 0 }  e.  Fin
8382olci 391 . . . . . . . . . 10  |-  ( { 0 }  C_  ( ZZ>=
`  0 )  \/ 
{ 0 }  e.  Fin )
84 sumz 13195 . . . . . . . . . 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 2489 . . . . . . . 8  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
8786adantl 463 . . . . . . 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 748 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  F  e.  ( (Poly `  RR )  \  {
0p } ) )
8936adantr 462 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN0 )
90 simpr 458 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  -.  n  =  0
)
9190neneqad 2679 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  =/=  0 )
9289, 91jca 529 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  -> 
( n  e.  NN0  /\  n  =/=  0 ) )
93 elnnne0 10589 . . . . . . . . 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 10591 . . . . . . . . . . . . 13  |-  1  e.  NN0
9695a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  NN0 )
97 simpr 458 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN )
9897nnnn0d 10632 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN0 )
99 nnge1 10344 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  1  <_  n )
10097, 99syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  <_  n
)
101 elfz2nn0 11476 . . . . . . . . . . . 12  |-  ( 1  e.  ( 0 ... n )  <->  ( 1  e.  NN0  /\  n  e.  NN0  /\  1  <_  n ) )
10296, 98, 100, 101syl3anbrc 1167 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  ( 0 ... n ) )
1033elexi 2980 . . . . . . . . . . . 12  |-  1  e.  _V
104103snss 3996 . . . . . . . . . . 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 720 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  (coeff `  F
) : NN0 --> RR )
107 oveq2 6098 . . . . . . . . . . . . . . . 16  |-  ( i  =  1  ->  (
n  -  i )  =  ( n  - 
1 ) )
10849, 107sylbi 195 . . . . . . . . . . . . . . 15  |-  ( i  e.  { 1 }  ->  ( n  -  i )  =  ( n  -  1 ) )
109108adantl 463 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  =  ( n  -  1 ) )
110 nnm1nn0 10617 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
111110ad2antlr 721 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  - 
1 )  e.  NN0 )
112109, 111eqeltrd 2515 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  e.  NN0 )
113106, 112ffvelrnd 5841 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  RR )
1142, 113sseldi 3351 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
115114ralrimiva 2797 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  A. i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
116 fzfi 11790 . . . . . . . . . . . 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 13199 . . . . . . . . . 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 1212 . . . . . . . . 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 462 . . . . . . . . . . . 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 5841 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  RR )
1242, 123sseldi 3351 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  CC )
125107fveq2d 5692 . . . . . . . . . . 11  |-  ( i  =  1  ->  (
(coeff `  F ) `  ( n  -  i
) )  =  ( (coeff `  F ) `  ( n  -  1 ) ) )
126125sumsn 13213 . . . . . . . . . 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 658 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  sum_ i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  =  ( (coeff `  F ) `  (
n  -  1 ) ) )
128120, 127eqtr3d 2475 . . . . . . . 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 656 . . . . . . 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 3821 . . . . . 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 2473 . . . . 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 2477 . . . 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 2473 . . 3  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( F  oF  x.  Xp ) ) `
 n )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F
) `  ( n  -  1 ) ) ) )
134133mpteq2dva 4375 . 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 2473 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970    \ cdif 3322    C_ wss 3325   ifcif 3788   {csn 3874   class class class wbr 4289    e. cmpt 4347    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   Fincfn 7306   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    <_ cle 9415    - cmin 9591   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   sum_csu 13159   0pc0p 21047  Polycply 21595   Xpcidp 21596  coeffccoe 21597
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 2263  df-mo 2264  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 21048  df-ply 21599  df-idp 21600  df-coe 21601  df-dgr 21602
This theorem is referenced by:  plymulx  26863
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