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Theorem plymulx0 29432
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 3587 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F  e.  (Poly `  RR ) )
2 ax-resscn 9597 . . . . . . 7  |-  RR  C_  CC
3 1re 9643 . . . . . . 7  |-  1  e.  RR
4 plyid 23150 . . . . . . 7  |-  ( ( RR  C_  CC  /\  1  e.  RR )  ->  Xp  e.  (Poly `  RR ) )
52, 3, 4mp2an 676 . . . . . 6  |-  Xp  e.  (Poly `  RR )
65a1i 11 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  Xp  e.  (Poly `  RR ) )
7 simprl 762 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
8 simprr 764 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  RR )
97, 8readdcld 9671 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
107, 8remulcld 9672 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
111, 6, 9, 10plymul 23159 . . . 4  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  e.  (Poly `  RR ) )
12 0re 9644 . . . 4  |-  0  e.  RR
13 eqid 2422 . . . . 5  |-  (coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( F  oF  x.  Xp ) )
1413coef2 23172 . . . 4  |-  ( ( ( F  oF  x.  Xp )  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
1511, 12, 14sylancl 666 . . 3  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
1615feqmptd 5931 . 2  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  ( (coeff `  ( F  oF  x.  Xp ) ) `  n ) ) )
17 cnex 9621 . . . . . . . . 9  |-  CC  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  CC  e.  _V )
19 plyf 23139 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
201, 19syl 17 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F : CC --> CC )
21 plyf 23139 . . . . . . . . . 10  |-  ( Xp  e.  (Poly `  RR )  ->  Xp : CC --> CC )
225, 21ax-mp 5 . . . . . . . . 9  |-  Xp : CC --> CC
2322a1i 11 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  Xp : CC --> CC )
24 simprl 762 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  x  e.  CC )
25 simprr 764 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
y  e.  CC )
2624, 25mulcomd 9665 . . . . . . . 8  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
2718, 20, 23, 26caofcom 6574 . . . . . . 7  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  =  ( Xp  oF  x.  F
) )
2827fveq2d 5882 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( Xp  oF  x.  F
) ) )
2928fveq1d 5880 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( (coeff `  ( F  oF  x.  Xp ) ) `  n )  =  ( (coeff `  ( Xp  oF  x.  F
) ) `  n
) )
3029adantr 466 . . . 4  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( F  oF  x.  Xp ) ) `
 n )  =  ( (coeff `  (
Xp  oF  x.  F ) ) `
 n ) )
315a1i 11 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  Xp  e.  (Poly `  RR )
)
321adantr 466 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  RR ) )
33 simpr 462 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
34 eqid 2422 . . . . . . 7  |-  (coeff `  Xp )  =  (coeff `  Xp
)
35 eqid 2422 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
3634, 35coemul 23193 . . . . . 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
) ) ) )
3731, 32, 33, 36syl3anc 1264 . . . . 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
) ) ) )
38 elfznn0 11888 . . . . . . . . . 10  |-  ( i  e.  ( 0 ... n )  ->  i  e.  NN0 )
39 coeidp 23204 . . . . . . . . . 10  |-  ( i  e.  NN0  ->  ( (coeff `  Xp ) `  i )  =  if ( i  =  1 ,  1 ,  0 ) )
4038, 39syl 17 . . . . . . . . 9  |-  ( i  e.  ( 0 ... n )  ->  (
(coeff `  Xp
) `  i )  =  if ( i  =  1 ,  1 ,  0 ) )
4140oveq1d 6317 . . . . . . . 8  |-  ( i  e.  ( 0 ... n )  ->  (
( (coeff `  Xp ) `  i
)  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  ( if ( i  =  1 ,  1 ,  0 )  x.  (
(coeff `  F ) `  ( n  -  i
) ) ) )
42 ovif 6384 . . . . . . . 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 ) ) ) )
4341, 42syl6eq 2479 . . . . . . 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 ) ) ) ) )
4443adantl 467 . . . . . 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 ) ) ) ) )
4544sumeq2dv 13757 . . . . 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 ) ) ) ) )
46 elsn 4010 . . . . . . . . . 10  |-  ( i  e.  { 1 }  <-> 
i  =  1 )
4746bicomi 205 . . . . . . . . 9  |-  ( i  =  1  <->  i  e.  { 1 } )
4847a1i 11 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
i  =  1  <->  i  e.  { 1 } ) )
4935coef2 23172 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  F ) : NN0 --> RR )
501, 12, 49sylancl 666 . . . . . . . . . . . 12  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  F ) : NN0 --> RR )
5150ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (coeff `  F ) : NN0 --> RR )
52 fznn0sub 11832 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... n )  ->  (
n  -  i )  e.  NN0 )
5352adantl 467 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
n  -  i )  e.  NN0 )
5451, 53ffvelrnd 6035 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  RR )
5554recnd 9670 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  CC )
5655mulid2d 9662 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
1  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  ( (coeff `  F ) `  ( n  -  i
) ) )
5755mul02d 9832 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
0  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  0 )
5848, 56, 57ifbieq12d 3936 . . . . . . 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 ) )
5958sumeq2dv 13757 . . . . . 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 ) )
60 eqeq2 2437 . . . . . . 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 ) ) ) ) )
61 eqeq2 2437 . . . . . . 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 ) ) ) ) )
62 oveq2 6310 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
63 0z 10949 . . . . . . . . . . . 12  |-  0  e.  ZZ
64 fzsn 11841 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6563, 64ax-mp 5 . . . . . . . . . . 11  |-  ( 0 ... 0 )  =  { 0 }
6662, 65syl6eq 2479 . . . . . . . . . 10  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
67 elsni 4021 . . . . . . . . . . . . 13  |-  ( i  e.  { 0 }  ->  i  =  0 )
6867adantl 467 . . . . . . . . . . . 12  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  i  =  0 )
69 ax-1ne0 9609 . . . . . . . . . . . . . 14  |-  1  =/=  0
7069nesymi 2697 . . . . . . . . . . . . 13  |-  -.  0  =  1
71 eqeq1 2426 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  (
i  =  1  <->  0  =  1 ) )
7270, 71mtbiri 304 . . . . . . . . . . . 12  |-  ( i  =  0  ->  -.  i  =  1 )
7368, 72syl 17 . . . . . . . . . . 11  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  -.  i  =  1 )
7447notbii 297 . . . . . . . . . . . 12  |-  ( -.  i  =  1  <->  -.  i  e.  { 1 } )
7574biimpi 197 . . . . . . . . . . 11  |-  ( -.  i  =  1  ->  -.  i  e.  { 1 } )
76 iffalse 3918 . . . . . . . . . . 11  |-  ( -.  i  e.  { 1 }  ->  if (
i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
7773, 75, 763syl 18 . . . . . . . . . 10  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  if ( i  e.  {
1 } ,  ( (coeff `  F ) `  ( n  -  i
) ) ,  0 )  =  0 )
7866, 77sumeq12rdv 13761 . . . . . . . . 9  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  sum_ i  e.  {
0 } 0 )
79 snfi 7654 . . . . . . . . . . 11  |-  { 0 }  e.  Fin
8079olci 392 . . . . . . . . . 10  |-  ( { 0 }  C_  ( ZZ>=
`  0 )  \/ 
{ 0 }  e.  Fin )
81 sumz 13776 . . . . . . . . . 10  |-  ( ( { 0 }  C_  ( ZZ>= `  0 )  \/  { 0 }  e.  Fin )  ->  sum_ i  e.  { 0 } 0  =  0 )
8280, 81ax-mp 5 . . . . . . . . 9  |-  sum_ i  e.  { 0 } 0  =  0
8378, 82syl6eq 2479 . . . . . . . 8  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
8483adantl 467 . . . . . . 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 )
85 simpll 758 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  F  e.  ( (Poly `  RR )  \  {
0p } ) )
8633adantr 466 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN0 )
87 simpr 462 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  -.  n  =  0
)
8887neqned 2627 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  =/=  0 )
89 elnnne0 10884 . . . . . . . . 9  |-  ( n  e.  NN  <->  ( n  e.  NN0  /\  n  =/=  0 ) )
9086, 88, 89sylanbrc 668 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN )
91 1nn0 10886 . . . . . . . . . . . . 13  |-  1  e.  NN0
9291a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  NN0 )
93 simpr 462 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN )
9493nnnn0d 10926 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN0 )
9593nnge1d 10653 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  <_  n
)
96 elfz2nn0 11886 . . . . . . . . . . . 12  |-  ( 1  e.  ( 0 ... n )  <->  ( 1  e.  NN0  /\  n  e.  NN0  /\  1  <_  n ) )
9792, 94, 95, 96syl3anbrc 1189 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  ( 0 ... n ) )
9897snssd 4142 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  { 1 } 
C_  ( 0 ... n ) )
9950ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  (coeff `  F
) : NN0 --> RR )
100 oveq2 6310 . . . . . . . . . . . . . . . 16  |-  ( i  =  1  ->  (
n  -  i )  =  ( n  - 
1 ) )
10146, 100sylbi 198 . . . . . . . . . . . . . . 15  |-  ( i  e.  { 1 }  ->  ( n  -  i )  =  ( n  -  1 ) )
102101adantl 467 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  =  ( n  -  1 ) )
103 nnm1nn0 10912 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
104103ad2antlr 731 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  - 
1 )  e.  NN0 )
105102, 104eqeltrd 2510 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  e.  NN0 )
10699, 105ffvelrnd 6035 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  RR )
107106recnd 9670 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
108107ralrimiva 2839 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  A. i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
109 fzfi 12185 . . . . . . . . . . . 12  |-  ( 0 ... n )  e. 
Fin
110109olci 392 . . . . . . . . . . 11  |-  ( ( 0 ... n ) 
C_  ( ZZ>= `  0
)  \/  ( 0 ... n )  e. 
Fin )
111110a1i 11 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( ( 0 ... n )  C_  ( ZZ>= `  0 )  \/  ( 0 ... n
)  e.  Fin )
)
112 sumss2 13780 . . . . . . . . . 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 ) )
11398, 108, 111, 112syl21anc 1263 . . . . . . . . 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 ) )
11450adantr 466 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  (coeff `  F
) : NN0 --> RR )
115103adantl 467 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( n  - 
1 )  e.  NN0 )
116114, 115ffvelrnd 6035 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  RR )
117116recnd 9670 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  CC )
118100fveq2d 5882 . . . . . . . . . . 11  |-  ( i  =  1  ->  (
(coeff `  F ) `  ( n  -  i
) )  =  ( (coeff `  F ) `  ( n  -  1 ) ) )
119118sumsn 13795 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( (coeff `  F ) `  ( n  -  1 ) )  e.  CC )  ->  sum_ i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  =  ( (coeff `  F ) `  (
n  -  1 ) ) )
1203, 117, 119sylancr 667 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  sum_ i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  =  ( (coeff `  F ) `  (
n  -  1 ) ) )
121113, 120eqtr3d 2465 . . . . . . . 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 ) ) )
12285, 90, 121syl2anc 665 . . . . . . 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 ) ) )
12360, 61, 84, 122ifbothda 3944 . . . . . 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 ) ) ) )
12459, 123eqtrd 2463 . . . . 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 ) ) ) )
12537, 45, 1243eqtrd 2467 . . . 4  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( Xp  oF  x.  F ) ) `  n )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  (
n  -  1 ) ) ) )
12630, 125eqtrd 2463 . . 3  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( F  oF  x.  Xp ) ) `
 n )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F
) `  ( n  -  1 ) ) ) )
127126mpteq2dva 4507 . 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 ) ) ) ) )
12816, 127eqtrd 2463 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   _Vcvv 3081    \ cdif 3433    C_ wss 3436   ifcif 3909   {csn 3996   class class class wbr 4420    |-> cmpt 4479   -->wf 5594   ` cfv 5598  (class class class)co 6302    oFcof 6540   Fincfn 7574   CCcc 9538   RRcr 9539   0cc0 9540   1c1 9541    x. cmul 9545    <_ cle 9677    - cmin 9861   NNcn 10610   NN0cn0 10870   ZZcz 10938   ZZ>=cuz 11160   ...cfz 11785   sum_csu 13740   0pc0p 22614  Polycply 23125   Xpcidp 23126  coeffccoe 23127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-fz 11786  df-fzo 11917  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-rlim 13541  df-sum 13741  df-0p 22615  df-ply 23129  df-idp 23130  df-coe 23131  df-dgr 23132
This theorem is referenced by:  plymulx  29433
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