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Theorem plymulx0 29508
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 3544 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F  e.  (Poly `  RR ) )
2 ax-resscn 9614 . . . . . . 7  |-  RR  C_  CC
3 1re 9660 . . . . . . 7  |-  1  e.  RR
4 plyid 23242 . . . . . . 7  |-  ( ( RR  C_  CC  /\  1  e.  RR )  ->  Xp  e.  (Poly `  RR ) )
52, 3, 4mp2an 686 . . . . . 6  |-  Xp  e.  (Poly `  RR )
65a1i 11 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  Xp  e.  (Poly `  RR ) )
7 simprl 772 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
8 simprr 774 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  RR )
97, 8readdcld 9688 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
107, 8remulcld 9689 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
111, 6, 9, 10plymul 23251 . . . 4  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  e.  (Poly `  RR ) )
12 0re 9661 . . . 4  |-  0  e.  RR
13 eqid 2471 . . . . 5  |-  (coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( F  oF  x.  Xp ) )
1413coef2 23264 . . . 4  |-  ( ( ( F  oF  x.  Xp )  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
1511, 12, 14sylancl 675 . . 3  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
1615feqmptd 5932 . 2  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  ( (coeff `  ( F  oF  x.  Xp ) ) `  n ) ) )
17 cnex 9638 . . . . . . . . 9  |-  CC  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  CC  e.  _V )
19 plyf 23231 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
201, 19syl 17 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F : CC --> CC )
21 plyf 23231 . . . . . . . . . 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 772 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  x  e.  CC )
25 simprr 774 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
y  e.  CC )
2624, 25mulcomd 9682 . . . . . . . 8  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
2718, 20, 23, 26caofcom 6582 . . . . . . 7  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  =  ( Xp  oF  x.  F
) )
2827fveq2d 5883 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( Xp  oF  x.  F
) ) )
2928fveq1d 5881 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( (coeff `  ( F  oF  x.  Xp ) ) `  n )  =  ( (coeff `  ( Xp  oF  x.  F
) ) `  n
) )
3029adantr 472 . . . 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 472 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  RR ) )
33 simpr 468 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
34 eqid 2471 . . . . . . 7  |-  (coeff `  Xp )  =  (coeff `  Xp
)
35 eqid 2471 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
3634, 35coemul 23285 . . . . . 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 1292 . . . . 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 11913 . . . . . . . . . 10  |-  ( i  e.  ( 0 ... n )  ->  i  e.  NN0 )
39 coeidp 23296 . . . . . . . . . 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 6323 . . . . . . . 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 6392 . . . . . . . 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 2521 . . . . . . 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 473 . . . . . 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 13846 . . . . 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 3973 . . . . . . . . . 10  |-  ( i  e.  { 1 }  <-> 
i  =  1 )
4746bicomi 207 . . . . . . . . 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 23264 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  F ) : NN0 --> RR )
501, 12, 49sylancl 675 . . . . . . . . . . . 12  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  F ) : NN0 --> RR )
5150ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (coeff `  F ) : NN0 --> RR )
52 fznn0sub 11857 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... n )  ->  (
n  -  i )  e.  NN0 )
5352adantl 473 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
n  -  i )  e.  NN0 )
5451, 53ffvelrnd 6038 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  RR )
5554recnd 9687 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  CC )
5655mulid2d 9679 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
1  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  ( (coeff `  F ) `  ( n  -  i
) ) )
5755mul02d 9849 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
0  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  0 )
5848, 56, 57ifbieq12d 3899 . . . . . . 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 13846 . . . . . 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 2482 . . . . . . 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 2482 . . . . . . 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 6316 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
63 0z 10972 . . . . . . . . . . . 12  |-  0  e.  ZZ
64 fzsn 11866 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6563, 64ax-mp 5 . . . . . . . . . . 11  |-  ( 0 ... 0 )  =  { 0 }
6662, 65syl6eq 2521 . . . . . . . . . 10  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
67 elsni 3985 . . . . . . . . . . . . 13  |-  ( i  e.  { 0 }  ->  i  =  0 )
6867adantl 473 . . . . . . . . . . . 12  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  i  =  0 )
69 ax-1ne0 9626 . . . . . . . . . . . . . 14  |-  1  =/=  0
7069nesymi 2700 . . . . . . . . . . . . 13  |-  -.  0  =  1
71 eqeq1 2475 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  (
i  =  1  <->  0  =  1 ) )
7270, 71mtbiri 310 . . . . . . . . . . . 12  |-  ( i  =  0  ->  -.  i  =  1 )
7368, 72syl 17 . . . . . . . . . . 11  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  -.  i  =  1 )
7447notbii 303 . . . . . . . . . . . 12  |-  ( -.  i  =  1  <->  -.  i  e.  { 1 } )
7574biimpi 199 . . . . . . . . . . 11  |-  ( -.  i  =  1  ->  -.  i  e.  { 1 } )
76 iffalse 3881 . . . . . . . . . . 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 13850 . . . . . . . . 9  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  sum_ i  e.  {
0 } 0 )
79 snfi 7668 . . . . . . . . . . 11  |-  { 0 }  e.  Fin
8079olci 398 . . . . . . . . . 10  |-  ( { 0 }  C_  ( ZZ>=
`  0 )  \/ 
{ 0 }  e.  Fin )
81 sumz 13865 . . . . . . . . . 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 2521 . . . . . . . 8  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
8483adantl 473 . . . . . . 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 768 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  F  e.  ( (Poly `  RR )  \  {
0p } ) )
8633adantr 472 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN0 )
87 simpr 468 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  -.  n  =  0
)
8887neqned 2650 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  =/=  0 )
89 elnnne0 10907 . . . . . . . . 9  |-  ( n  e.  NN  <->  ( n  e.  NN0  /\  n  =/=  0 ) )
9086, 88, 89sylanbrc 677 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN )
91 1nn0 10909 . . . . . . . . . . . . 13  |-  1  e.  NN0
9291a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  NN0 )
93 simpr 468 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN )
9493nnnn0d 10949 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN0 )
9593nnge1d 10674 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  <_  n
)
96 elfz2nn0 11911 . . . . . . . . . . . 12  |-  ( 1  e.  ( 0 ... n )  <->  ( 1  e.  NN0  /\  n  e.  NN0  /\  1  <_  n ) )
9792, 94, 95, 96syl3anbrc 1214 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  ( 0 ... n ) )
9897snssd 4108 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  { 1 } 
C_  ( 0 ... n ) )
9950ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  (coeff `  F
) : NN0 --> RR )
100 oveq2 6316 . . . . . . . . . . . . . . . 16  |-  ( i  =  1  ->  (
n  -  i )  =  ( n  - 
1 ) )
10146, 100sylbi 200 . . . . . . . . . . . . . . 15  |-  ( i  e.  { 1 }  ->  ( n  -  i )  =  ( n  -  1 ) )
102101adantl 473 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  =  ( n  -  1 ) )
103 nnm1nn0 10935 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
104103ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  - 
1 )  e.  NN0 )
105102, 104eqeltrd 2549 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  e.  NN0 )
10699, 105ffvelrnd 6038 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  RR )
107106recnd 9687 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
108107ralrimiva 2809 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  A. i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
109 fzfi 12223 . . . . . . . . . . . 12  |-  ( 0 ... n )  e. 
Fin
110109olci 398 . . . . . . . . . . 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 13869 . . . . . . . . . 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 1291 . . . . . . . . 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 472 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  (coeff `  F
) : NN0 --> RR )
115103adantl 473 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( n  - 
1 )  e.  NN0 )
116114, 115ffvelrnd 6038 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  RR )
117116recnd 9687 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  CC )
118100fveq2d 5883 . . . . . . . . . . 11  |-  ( i  =  1  ->  (
(coeff `  F ) `  ( n  -  i
) )  =  ( (coeff `  F ) `  ( n  -  1 ) ) )
119118sumsn 13884 . . . . . . . . . 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 676 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  sum_ i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  =  ( (coeff `  F ) `  (
n  -  1 ) ) )
121113, 120eqtr3d 2507 . . . . . . . 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 673 . . . . . . 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 3907 . . . . . 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 2505 . . . . 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 2509 . . . 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 2505 . . 3  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( F  oF  x.  Xp ) ) `
 n )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F
) `  ( n  -  1 ) ) ) )
127126mpteq2dva 4482 . 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 2505 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 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    \ cdif 3387    C_ wss 3390   ifcif 3872   {csn 3959   class class class wbr 4395    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    <_ cle 9694    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   sum_csu 13829   0pc0p 22706  Polycply 23217   Xpcidp 23218  coeffccoe 23219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-idp 23222  df-coe 23223  df-dgr 23224
This theorem is referenced by:  plymulx  29509
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