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Theorem signsplypnf 29441
Description: The quotient of a polynomial  F by a monic monomial of same degree  G converges to the highest coefficient of  F. (Contributed by Thierry Arnoux, 18-Sep-2018.)
Hypotheses
Ref Expression
signsply0.d  |-  D  =  (deg `  F )
signsply0.c  |-  C  =  (coeff `  F )
signsply0.b  |-  B  =  ( C `  D
)
signsplypnf.g  |-  G  =  ( x  e.  RR+  |->  ( x ^ D
) )
Assertion
Ref Expression
signsplypnf  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  G )  ~~> r  B )
Distinct variable groups:    x, C    x, D    x, F    x, G
Allowed substitution hint:    B( x)

Proof of Theorem signsplypnf
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 plyf 23144 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
2 ffn 5744 . . . . 5  |-  ( F : CC --> CC  ->  F  Fn  CC )
31, 2syl 17 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  F  Fn  CC )
4 ovex 6331 . . . . . 6  |-  ( x ^ D )  e. 
_V
54rgenw 2787 . . . . 5  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
6 signsplypnf.g . . . . . 6  |-  G  =  ( x  e.  RR+  |->  ( x ^ D
) )
76fnmpt 5720 . . . . 5  |-  ( A. x  e.  RR+  ( x ^ D )  e. 
_V  ->  G  Fn  RR+ )
85, 7mp1i 13 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  G  Fn  RR+ )
9 cnex 9622 . . . . 5  |-  CC  e.  _V
109a1i 11 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  CC  e.  _V )
11 reex 9632 . . . . . 6  |-  RR  e.  _V
12 rpssre 11314 . . . . . 6  |-  RR+  C_  RR
1311, 12ssexi 4567 . . . . 5  |-  RR+  e.  _V
1413a1i 11 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  RR+  e.  _V )
15 ax-resscn 9598 . . . . . 6  |-  RR  C_  CC
1612, 15sstri 3474 . . . . 5  |-  RR+  C_  CC
17 dfss1 3668 . . . . 5  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
1816, 17mpbi 212 . . . 4  |-  ( CC 
i^i  RR+ )  =  RR+
19 signsply0.c . . . . 5  |-  C  =  (coeff `  F )
20 signsply0.d . . . . 5  |-  D  =  (deg `  F )
2119, 20coeid2 23185 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  CC )  ->  ( F `  x )  =  sum_ k  e.  ( 0 ... D ) ( ( C `  k )  x.  (
x ^ k ) ) )
226fvmpt2 5971 . . . . . 6  |-  ( ( x  e.  RR+  /\  (
x ^ D )  e.  _V )  -> 
( G `  x
)  =  ( x ^ D ) )
234, 22mpan2 676 . . . . 5  |-  ( x  e.  RR+  ->  ( G `
 x )  =  ( x ^ D
) )
2423adantl 468 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( G `  x )  =  ( x ^ D ) )
253, 8, 10, 14, 18, 21, 24offval 6550 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  G )  =  ( x  e.  RR+  |->  ( sum_ k  e.  ( 0 ... D
) ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) ) ) )
26 fzfid 12187 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
0 ... D )  e. 
Fin )
2716a1i 11 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  RR+  C_  CC )
2827sselda 3465 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  x  e.  CC )
29 dgrcl 23179 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
3020, 29syl5eqel 2515 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  D  e.  NN0 )
3130adantr 467 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  D  e.  NN0 )
3228, 31expcld 12417 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  e.  CC )
3319coef3 23178 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> CC )
3433ad2antrr 731 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  C : NN0 --> CC )
35 elfznn0 11889 . . . . . . . . 9  |-  ( k  e.  ( 0 ... D )  ->  k  e.  NN0 )
3635adantl 468 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  k  e.  NN0 )
3734, 36ffvelrnd 6036 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  ( C `  k )  e.  CC )
3828adantr 467 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  x  e.  CC )
3938, 36expcld 12417 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ k )  e.  CC )
4037, 39mulcld 9665 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
( C `  k
)  x.  ( x ^ k ) )  e.  CC )
41 rpne0 11319 . . . . . . . 8  |-  ( x  e.  RR+  ->  x  =/=  0 )
4241adantl 468 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  x  =/=  0 )
4330nn0zd 11040 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  D  e.  ZZ )
4443adantr 467 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  D  e.  ZZ )
4528, 42, 44expne0d 12423 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  =/=  0 )
4626, 32, 40, 45fsumdivc 13840 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( sum_ k  e.  ( 0 ... D ) ( ( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) )  = 
sum_ k  e.  ( 0 ... D ) ( ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) ) )
47 fzodisj 11954 . . . . . . . 8  |-  ( ( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  (/)
48 fzosn 11985 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  ( D..^ ( D  +  1 ) )  =  { D } )
4948ineq2d 3665 . . . . . . . 8  |-  ( D  e.  ZZ  ->  (
( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  ( ( 0..^ D )  i^i  { D }
) )
5047, 49syl5reqr 2479 . . . . . . 7  |-  ( D  e.  ZZ  ->  (
( 0..^ D )  i^i  { D }
)  =  (/) )
5144, 50syl 17 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( 0..^ D )  i^i  { D }
)  =  (/) )
52 fzval3 11984 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  (
0 ... D )  =  ( 0..^ ( D  +  1 ) ) )
5343, 52syl 17 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( 0..^ ( D  + 
1 ) ) )
54 nn0uz 11195 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
5530, 54syl6eleq 2521 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  D  e.  (
ZZ>= `  0 ) )
56 fzosplitsn 12018 . . . . . . . . 9  |-  ( D  e.  ( ZZ>= `  0
)  ->  ( 0..^ ( D  +  1 ) )  =  ( ( 0..^ D )  u.  { D }
) )
5755, 56syl 17 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  ( 0..^ ( D  +  1 ) )  =  ( ( 0..^ D )  u. 
{ D } ) )
5853, 57eqtrd 2464 . . . . . . 7  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( ( 0..^ D )  u.  { D }
) )
5958adantr 467 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
0 ... D )  =  ( ( 0..^ D )  u.  { D } ) )
6032adantr 467 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  e.  CC )
6142adantr 467 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  x  =/=  0 )
6244adantr 467 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  D  e.  ZZ )
6338, 61, 62expne0d 12423 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  =/=  0 )
6440, 60, 63divcld 10385 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  e.  CC )
6551, 59, 26, 64fsumsplit 13799 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  ( 0 ... D
) ( ( ( C `  k )  x.  ( x ^
k ) )  / 
( x ^ D
) )  =  (
sum_ k  e.  ( 0..^ D ) ( ( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  +  sum_ k  e.  { D }  ( (
( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) ) ) )
6646, 65eqtrd 2464 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( sum_ k  e.  ( 0 ... D ) ( ( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) )  =  ( sum_ k  e.  ( 0..^ D ) ( ( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  +  sum_ k  e.  { D }  ( (
( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) ) ) )
6766mpteq2dva 4508 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  ( sum_ k  e.  ( 0 ... D
) ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) ) )  =  ( x  e.  RR+  |->  ( sum_ k  e.  ( 0..^ D ) ( ( ( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) )  + 
sum_ k  e.  { D }  ( (
( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) ) ) ) )
6825, 67eqtrd 2464 . 2  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  G )  =  ( x  e.  RR+  |->  ( sum_ k  e.  ( 0..^ D ) ( ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) )  +  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) ) ) )
69 sumex 13747 . . . . 5  |-  sum_ k  e.  ( 0..^ D ) ( ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) )  e.  _V
7069a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  ( 0..^ D ) ( ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) )  e.  _V )
71 sumex 13747 . . . . 5  |-  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  e.  _V
7271a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  e.  _V )
7312a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  RR+  C_  RR )
74 fzofi 12188 . . . . . . 7  |-  ( 0..^ D )  e.  Fin
7574a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( 0..^ D )  e.  Fin )
76 ovex 6331 . . . . . . 7  |-  ( ( ( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) )  e. 
_V
7776a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  (
x  e.  RR+  /\  k  e.  ( 0..^ D ) ) )  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  e.  _V )
7833ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  C : NN0
--> CC )
79 elfzonn0 11962 . . . . . . . . . . 11  |-  ( k  e.  ( 0..^ D )  ->  k  e.  NN0 )
8079ad2antlr 732 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e. 
NN0 )
8178, 80ffvelrnd 6036 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( C `
 k )  e.  CC )
8228adantlr 720 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  e.  CC )
8382, 80expcld 12417 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ k )  e.  CC )
8432adantlr 720 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  e.  CC )
8541adantl 468 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  =/=  0 )
8644adantlr 720 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  ZZ )
8782, 85, 86expne0d 12423 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  =/=  0 )
8881, 83, 84, 87divassd 10420 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( ( ( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) )  =  ( ( C `  k )  x.  (
( x ^ k
)  /  ( x ^ D ) ) ) )
8988mpteq2dva 4508 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( ( C `  k )  x.  ( x ^
k ) )  / 
( x ^ D
) ) )  =  ( x  e.  RR+  |->  ( ( C `  k )  x.  (
( x ^ k
)  /  ( x ^ D ) ) ) ) )
90 fvex 5889 . . . . . . . . . 10  |-  ( C `
 k )  e. 
_V
9190a1i 11 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( C `
 k )  e. 
_V )
92 ovex 6331 . . . . . . . . . 10  |-  ( ( x ^ k )  /  ( x ^ D ) )  e. 
_V
9392a1i 11 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( ( x ^ k )  /  ( x ^ D ) )  e. 
_V )
9433adantr 467 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  C : NN0
--> CC )
9579adantl 468 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  NN0 )
9694, 95ffvelrnd 6036 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( C `  k )  e.  CC )
97 rlimconst 13601 . . . . . . . . . 10  |-  ( (
RR+  C_  RR  /\  ( C `  k )  e.  CC )  ->  (
x  e.  RR+  |->  ( C `
 k ) )  ~~> r  ( C `  k ) )
9812, 96, 97sylancr 668 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( C `  k ) )  ~~> r  ( C `  k ) )
9980nn0zd 11040 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  ZZ )
10086, 99zsubcld 11047 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( D  -  k )  e.  ZZ )
10182, 85, 100cxpexpzd 23648 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x  ^c  ( D  -  k ) )  =  ( x ^
( D  -  k
) ) )
102101oveq2d 6319 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
10382, 85, 100expnegd 12424 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
10486zcnd 11043 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  CC )
10580nn0cnd 10929 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  CC )
106104, 105negsubdi2d 10004 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  -u ( D  -  k )  =  ( k  -  D ) )
107106oveq2d 6319 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( x ^
( k  -  D
) ) )
108102, 103, 1073eqtr2d 2470 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( x ^
( k  -  D
) ) )
10982, 85, 86, 99expsubd 12428 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ ( k  -  D ) )  =  ( ( x ^
k )  /  (
x ^ D ) ) )
110108, 109eqtrd 2464 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( ( x ^ k )  / 
( x ^ D
) ) )
111110mpteq2dva 4508 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( 1  / 
( x  ^c 
( D  -  k
) ) ) )  =  ( x  e.  RR+  |->  ( ( x ^ k )  / 
( x ^ D
) ) ) )
11295nn0red 10928 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  RR )
11330adantr 467 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  NN0 )
114113nn0red 10928 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  RR )
115 elfzolt2 11931 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0..^ D )  ->  k  <  D )
116115adantl 468 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  <  D )
117 difrp 11339 . . . . . . . . . . . . 13  |-  ( ( k  e.  RR  /\  D  e.  RR )  ->  ( k  <  D  <->  ( D  -  k )  e.  RR+ ) )
118117biimpa 487 . . . . . . . . . . . 12  |-  ( ( ( k  e.  RR  /\  D  e.  RR )  /\  k  <  D
)  ->  ( D  -  k )  e.  RR+ )
119112, 114, 116, 118syl21anc 1264 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( D  -  k )  e.  RR+ )
120 cxplim 23889 . . . . . . . . . . 11  |-  ( ( D  -  k )  e.  RR+  ->  ( x  e.  RR+  |->  ( 1  /  ( x  ^c  ( D  -  k ) ) ) )  ~~> r  0 )
121119, 120syl 17 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( 1  / 
( x  ^c 
( D  -  k
) ) ) )  ~~> r  0 )
122111, 121eqbrtrrd 4444 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( x ^ k )  / 
( x ^ D
) ) )  ~~> r  0 )
12391, 93, 98, 122rlimmul 13701 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( C `
 k )  x.  ( ( x ^
k )  /  (
x ^ D ) ) ) )  ~~> r  ( ( C `  k
)  x.  0 ) )
12496mul01d 9834 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( ( C `  k )  x.  0 )  =  0 )
125123, 124breqtrd 4446 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( C `
 k )  x.  ( ( x ^
k )  /  (
x ^ D ) ) ) )  ~~> r  0 )
12689, 125eqbrtrd 4442 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( ( C `  k )  x.  ( x ^
k ) )  / 
( x ^ D
) ) )  ~~> r  0 )
12773, 75, 77, 126fsumrlim 13864 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  ( 0..^ D ) ( ( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  sum_ k  e.  ( 0..^ D ) 0 )
12875olcd 395 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( ( 0..^ D )  C_  ( ZZ>=
`  0 )  \/  ( 0..^ D )  e.  Fin ) )
129 sumz 13781 . . . . . 6  |-  ( ( ( 0..^ D ) 
C_  ( ZZ>= `  0
)  \/  ( 0..^ D )  e.  Fin )  ->  sum_ k  e.  ( 0..^ D ) 0  =  0 )
130128, 129syl 17 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  sum_ k  e.  ( 0..^ D ) 0  =  0 )
131127, 130breqtrd 4446 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  ( 0..^ D ) ( ( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  0 )
13233, 30ffvelrnd 6036 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  RR )  ->  ( C `  D )  e.  CC )
133132adantr 467 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( C `  D )  e.  CC )
134133, 32mulcld 9665 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( C `  D
)  x.  ( x ^ D ) )  e.  CC )
135134, 32, 45divcld 10385 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  e.  CC )
136 fveq2 5879 . . . . . . . . . . 11  |-  ( k  =  D  ->  ( C `  k )  =  ( C `  D ) )
137 oveq2 6311 . . . . . . . . . . 11  |-  ( k  =  D  ->  (
x ^ k )  =  ( x ^ D ) )
138136, 137oveq12d 6321 . . . . . . . . . 10  |-  ( k  =  D  ->  (
( C `  k
)  x.  ( x ^ k ) )  =  ( ( C `
 D )  x.  ( x ^ D
) ) )
139138oveq1d 6318 . . . . . . . . 9  |-  ( k  =  D  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( ( ( C `  D )  x.  ( x ^ D ) )  / 
( x ^ D
) ) )
140139sumsn 13800 . . . . . . . 8  |-  ( ( D  e.  NN0  /\  ( ( ( C `
 D )  x.  ( x ^ D
) )  /  (
x ^ D ) )  e.  CC )  ->  sum_ k  e.  { D }  ( (
( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) )  =  ( ( ( C `
 D )  x.  ( x ^ D
) )  /  (
x ^ D ) ) )
14131, 135, 140syl2anc 666 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( ( ( C `  D )  x.  ( x ^ D ) )  / 
( x ^ D
) ) )
142133, 32, 45divcan4d 10391 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  =  ( C `  D ) )
143141, 142eqtrd 2464 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( C `  D ) )
144143mpteq2dva 4508 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  { D }  ( (
( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) ) )  =  ( x  e.  RR+  |->  ( C `  D ) ) )
145 rlimconst 13601 . . . . . 6  |-  ( (
RR+  C_  RR  /\  ( C `  D )  e.  CC )  ->  (
x  e.  RR+  |->  ( C `
 D ) )  ~~> r  ( C `  D ) )
14612, 132, 145sylancr 668 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  ( C `  D ) )  ~~> r  ( C `  D ) )
147144, 146eqbrtrd 4442 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  { D }  ( (
( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  ( C `  D ) )
14870, 72, 131, 147rlimadd 13699 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  ( sum_ k  e.  ( 0..^ D ) ( ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) )  +  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) ) )  ~~> r  ( 0  +  ( C `
 D ) ) )
149132addid2d 9836 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  ( C `  D ) )
150 signsply0.b . . . 4  |-  B  =  ( C `  D
)
151149, 150syl6eqr 2482 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  B )
152148, 151breqtrd 4446 . 2  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  ( sum_ k  e.  ( 0..^ D ) ( ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) )  +  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) ) )  ~~> r  B
)
15368, 152eqbrtrd 4442 1  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  G )  ~~> r  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   _Vcvv 3082    u. cun 3435    i^i cin 3436    C_ wss 3437   (/)c0 3762   {csn 3997   class class class wbr 4421    |-> cmpt 4480    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303    oFcof 6541   Fincfn 7575   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542    + caddc 9544    x. cmul 9546    < clt 9677    - cmin 9862   -ucneg 9863    / cdiv 10271   NN0cn0 10871   ZZcz 10939   ZZ>=cuz 11161   RR+crp 11304   ...cfz 11786  ..^cfzo 11917   ^cexp 12273    ~~> r crli 13542   sum_csu 13745  Polycply 23130  coeffccoe 23132  degcdgr 23133    ^c ccxp 23497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-haus 20323  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-xms 21327  df-ms 21328  df-tms 21329  df-cncf 21902  df-0p 22620  df-limc 22813  df-dv 22814  df-ply 23134  df-coe 23136  df-dgr 23137  df-log 23498  df-cxp 23499
This theorem is referenced by:  signsply0  29442
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