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Theorem plymulx0 28377
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 3611 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F  e.  (Poly `  RR ) )
2 ax-resscn 9552 . . . . . . 7  |-  RR  C_  CC
3 1re 9598 . . . . . . 7  |-  1  e.  RR
4 plyid 22479 . . . . . . 7  |-  ( ( RR  C_  CC  /\  1  e.  RR )  ->  Xp  e.  (Poly `  RR ) )
52, 3, 4mp2an 672 . . . . . 6  |-  Xp  e.  (Poly `  RR )
65a1i 11 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  Xp  e.  (Poly `  RR ) )
7 simprl 756 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
8 simprr 757 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  RR )
97, 8readdcld 9626 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
107, 8remulcld 9627 . . . . 5  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
111, 6, 9, 10plymul 22488 . . . 4  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  e.  (Poly `  RR ) )
12 0re 9599 . . . 4  |-  0  e.  RR
13 eqid 2443 . . . . 5  |-  (coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( F  oF  x.  Xp ) )
1413coef2 22501 . . . 4  |-  ( ( ( F  oF  x.  Xp )  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
1511, 12, 14sylancl 662 . . 3  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) ) : NN0 --> RR )
1615feqmptd 5911 . 2  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  ( (coeff `  ( F  oF  x.  Xp ) ) `  n ) ) )
17 cnex 9576 . . . . . . . . 9  |-  CC  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  CC  e.  _V )
19 plyf 22468 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
201, 19syl 16 . . . . . . . 8  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  ->  F : CC --> CC )
21 plyf 22468 . . . . . . . . . 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 756 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  x  e.  CC )
25 simprr 757 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
y  e.  CC )
2624, 25mulcomd 9620 . . . . . . . 8  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
2718, 20, 23, 26caofcom 6557 . . . . . . 7  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( F  oF  x.  Xp )  =  ( Xp  oF  x.  F
) )
2827fveq2d 5860 . . . . . 6  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( Xp  oF  x.  F
) ) )
2928fveq1d 5858 . . . . 5  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
( (coeff `  ( F  oF  x.  Xp ) ) `  n )  =  ( (coeff `  ( Xp  oF  x.  F
) ) `  n
) )
3029adantr 465 . . . 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 465 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  F  e.  (Poly `  RR ) )
33 simpr 461 . . . . . 6  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
34 eqid 2443 . . . . . . 7  |-  (coeff `  Xp )  =  (coeff `  Xp
)
35 eqid 2443 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
3634, 35coemul 22521 . . . . . 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 1229 . . . . 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 11779 . . . . . . . . . 10  |-  ( i  e.  ( 0 ... n )  ->  i  e.  NN0 )
39 coeidp 22532 . . . . . . . . . 10  |-  ( i  e.  NN0  ->  ( (coeff `  Xp ) `  i )  =  if ( i  =  1 ,  1 ,  0 ) )
4038, 39syl 16 . . . . . . . . 9  |-  ( i  e.  ( 0 ... n )  ->  (
(coeff `  Xp
) `  i )  =  if ( i  =  1 ,  1 ,  0 ) )
4140oveq1d 6296 . . . . . . . 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 6364 . . . . . . . 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 2500 . . . . . . 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 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 ) ) ) ) )
4544sumeq2dv 13504 . . . . 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 4028 . . . . . . . . . 10  |-  ( i  e.  { 1 }  <-> 
i  =  1 )
4746bicomi 202 . . . . . . . . 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 22501 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  F ) : NN0 --> RR )
501, 12, 49sylancl 662 . . . . . . . . . . . 12  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  F ) : NN0 --> RR )
5150ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (coeff `  F ) : NN0 --> RR )
52 fznn0sub 11725 . . . . . . . . . . . 12  |-  ( i  e.  ( 0 ... n )  ->  (
n  -  i )  e.  NN0 )
5352adantl 466 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
n  -  i )  e.  NN0 )
5451, 53ffvelrnd 6017 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  RR )
5554recnd 9625 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
(coeff `  F ) `  ( n  -  i
) )  e.  CC )
5655mulid2d 9617 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
1  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  ( (coeff `  F ) `  ( n  -  i
) ) )
5755mul02d 9781 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  i  e.  ( 0 ... n
) )  ->  (
0  x.  ( (coeff `  F ) `  (
n  -  i ) ) )  =  0 )
5848, 56, 57ifbieq12d 3953 . . . . . . 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 13504 . . . . . 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 2458 . . . . . . 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 2458 . . . . . . 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 6289 . . . . . . . . . . 11  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
63 0z 10881 . . . . . . . . . . . 12  |-  0  e.  ZZ
64 fzsn 11734 . . . . . . . . . . . 12  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
6563, 64ax-mp 5 . . . . . . . . . . 11  |-  ( 0 ... 0 )  =  { 0 }
6662, 65syl6eq 2500 . . . . . . . . . 10  |-  ( n  =  0  ->  (
0 ... n )  =  { 0 } )
67 elsni 4039 . . . . . . . . . . . . 13  |-  ( i  e.  { 0 }  ->  i  =  0 )
6867adantl 466 . . . . . . . . . . . 12  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  i  =  0 )
69 ax-1ne0 9564 . . . . . . . . . . . . . 14  |-  1  =/=  0
7069nesymi 2716 . . . . . . . . . . . . 13  |-  -.  0  =  1
71 eqeq1 2447 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  (
i  =  1  <->  0  =  1 ) )
7270, 71mtbiri 303 . . . . . . . . . . . 12  |-  ( i  =  0  ->  -.  i  =  1 )
7368, 72syl 16 . . . . . . . . . . 11  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  -.  i  =  1 )
7447notbii 296 . . . . . . . . . . . 12  |-  ( -.  i  =  1  <->  -.  i  e.  { 1 } )
7574biimpi 194 . . . . . . . . . . 11  |-  ( -.  i  =  1  ->  -.  i  e.  { 1 } )
76 iffalse 3935 . . . . . . . . . . 11  |-  ( -.  i  e.  { 1 }  ->  if (
i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
7773, 75, 763syl 20 . . . . . . . . . 10  |-  ( ( n  =  0  /\  i  e.  { 0 } )  ->  if ( i  e.  {
1 } ,  ( (coeff `  F ) `  ( n  -  i
) ) ,  0 )  =  0 )
7866, 77sumeq12rdv 13508 . . . . . . . . 9  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  sum_ i  e.  {
0 } 0 )
79 snfi 7598 . . . . . . . . . . 11  |-  { 0 }  e.  Fin
8079olci 391 . . . . . . . . . 10  |-  ( { 0 }  C_  ( ZZ>=
`  0 )  \/ 
{ 0 }  e.  Fin )
81 sumz 13523 . . . . . . . . . 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 2500 . . . . . . . 8  |-  ( n  =  0  ->  sum_ i  e.  ( 0 ... n
) if ( i  e.  { 1 } ,  ( (coeff `  F ) `  (
n  -  i ) ) ,  0 )  =  0 )
8483adantl 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 )
85 simpll 753 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  F  e.  ( (Poly `  RR )  \  {
0p } ) )
8633adantr 465 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN0 )
87 simpr 461 . . . . . . . . . 10  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  -.  n  =  0
)
8887neqned 2646 . . . . . . . . 9  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  =/=  0 )
89 elnnne0 10815 . . . . . . . . 9  |-  ( n  e.  NN  <->  ( n  e.  NN0  /\  n  =/=  0 ) )
9086, 88, 89sylanbrc 664 . . . . . . . 8  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN0 )  /\  -.  n  =  0 )  ->  n  e.  NN )
91 1nn0 10817 . . . . . . . . . . . . 13  |-  1  e.  NN0
9291a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  NN0 )
93 simpr 461 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN )
9493nnnn0d 10858 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  n  e.  NN0 )
9593nnge1d 10584 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  <_  n
)
96 elfz2nn0 11777 . . . . . . . . . . . 12  |-  ( 1  e.  ( 0 ... n )  <->  ( 1  e.  NN0  /\  n  e.  NN0  /\  1  <_  n ) )
9792, 94, 95, 96syl3anbrc 1181 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  1  e.  ( 0 ... n ) )
9897snssd 4160 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  { 1 } 
C_  ( 0 ... n ) )
9950ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  (coeff `  F
) : NN0 --> RR )
100 oveq2 6289 . . . . . . . . . . . . . . . 16  |-  ( i  =  1  ->  (
n  -  i )  =  ( n  - 
1 ) )
10146, 100sylbi 195 . . . . . . . . . . . . . . 15  |-  ( i  e.  { 1 }  ->  ( n  -  i )  =  ( n  -  1 ) )
102101adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  =  ( n  -  1 ) )
103 nnm1nn0 10843 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
104103ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  - 
1 )  e.  NN0 )
105102, 104eqeltrd 2531 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( n  -  i )  e.  NN0 )
10699, 105ffvelrnd 6017 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  RR )
107106recnd 9625 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( (Poly `  RR )  \  { 0p }
)  /\  n  e.  NN )  /\  i  e.  { 1 } )  ->  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
108107ralrimiva 2857 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  A. i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  e.  CC )
109 fzfi 12061 . . . . . . . . . . . 12  |-  ( 0 ... n )  e. 
Fin
110109olci 391 . . . . . . . . . . 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 13527 . . . . . . . . . 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 1228 . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  (coeff `  F
) : NN0 --> RR )
115103adantl 466 . . . . . . . . . . . 12  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( n  - 
1 )  e.  NN0 )
116114, 115ffvelrnd 6017 . . . . . . . . . . 11  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  RR )
117116recnd 9625 . . . . . . . . . 10  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  e.  CC )
118100fveq2d 5860 . . . . . . . . . . 11  |-  ( i  =  1  ->  (
(coeff `  F ) `  ( n  -  i
) )  =  ( (coeff `  F ) `  ( n  -  1 ) ) )
119118sumsn 13542 . . . . . . . . . 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 663 . . . . . . . . 9  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN )  ->  sum_ i  e.  {
1 }  ( (coeff `  F ) `  (
n  -  i ) )  =  ( (coeff `  F ) `  (
n  -  1 ) ) )
121113, 120eqtr3d 2486 . . . . . . . 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 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 ) ) )
12360, 61, 84, 122ifbothda 3961 . . . . . 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 2484 . . . . 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 2488 . . . 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 2484 . . 3  |-  ( ( F  e.  ( (Poly `  RR )  \  {
0p } )  /\  n  e.  NN0 )  ->  ( (coeff `  ( F  oF  x.  Xp ) ) `
 n )  =  if ( n  =  0 ,  0 ,  ( (coeff `  F
) `  ( n  -  1 ) ) ) )
127126mpteq2dva 4523 . 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 2484 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 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    \ cdif 3458    C_ wss 3461   ifcif 3926   {csn 4014   class class class wbr 4437    |-> cmpt 4495   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   Fincfn 7518   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    x. cmul 9500    <_ cle 9632    - cmin 9810   NNcn 10542   NN0cn0 10801   ZZcz 10870   ZZ>=cuz 11090   ...cfz 11681   sum_csu 13487   0pc0p 21949  Polycply 22454   Xpcidp 22455  coeffccoe 22456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-fz 11682  df-fzo 11804  df-fl 11908  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-rlim 13291  df-sum 13488  df-0p 21950  df-ply 22458  df-idp 22459  df-coe 22460  df-dgr 22461
This theorem is referenced by:  plymulx  28378
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