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Theorem signsplypnf 28690
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 22680 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
2 ffn 5639 . . . . 5  |-  ( F : CC --> CC  ->  F  Fn  CC )
31, 2syl 16 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  F  Fn  CC )
4 ovex 6224 . . . . . 6  |-  ( x ^ D )  e. 
_V
54rgenw 2743 . . . . 5  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
6 signsplypnf.g . . . . . 6  |-  G  =  ( x  e.  RR+  |->  ( x ^ D
) )
76fnmpt 5615 . . . . 5  |-  ( A. x  e.  RR+  ( x ^ D )  e. 
_V  ->  G  Fn  RR+ )
85, 7mp1i 12 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  G  Fn  RR+ )
9 cnex 9484 . . . . 5  |-  CC  e.  _V
109a1i 11 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  CC  e.  _V )
11 reex 9494 . . . . . 6  |-  RR  e.  _V
12 rpssre 11149 . . . . . 6  |-  RR+  C_  RR
1311, 12ssexi 4510 . . . . 5  |-  RR+  e.  _V
1413a1i 11 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  RR+  e.  _V )
15 ax-resscn 9460 . . . . . 6  |-  RR  C_  CC
1612, 15sstri 3426 . . . . 5  |-  RR+  C_  CC
17 dfss1 3617 . . . . 5  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
1816, 17mpbi 208 . . . 4  |-  ( CC 
i^i  RR+ )  =  RR+
19 signsply0.c . . . . 5  |-  C  =  (coeff `  F )
20 signsply0.d . . . . 5  |-  D  =  (deg `  F )
2119, 20coeid2 22721 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  CC )  ->  ( F `  x )  =  sum_ k  e.  ( 0 ... D ) ( ( C `  k )  x.  (
x ^ k ) ) )
226fvmpt2 5865 . . . . . 6  |-  ( ( x  e.  RR+  /\  (
x ^ D )  e.  _V )  -> 
( G `  x
)  =  ( x ^ D ) )
234, 22mpan2 669 . . . . 5  |-  ( x  e.  RR+  ->  ( G `
 x )  =  ( x ^ D
) )
2423adantl 464 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( G `  x )  =  ( x ^ D ) )
253, 8, 10, 14, 18, 21, 24offval 6446 . . 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 11986 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
0 ... D )  e. 
Fin )
2716a1i 11 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  RR+  C_  CC )
2827sselda 3417 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  x  e.  CC )
29 dgrcl 22715 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
3020, 29syl5eqel 2474 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  D  e.  NN0 )
3130adantr 463 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  D  e.  NN0 )
3228, 31expcld 12212 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  e.  CC )
3319coef3 22714 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> CC )
3433ad2antrr 723 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  C : NN0 --> CC )
35 elfznn0 11693 . . . . . . . . 9  |-  ( k  e.  ( 0 ... D )  ->  k  e.  NN0 )
3635adantl 464 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  k  e.  NN0 )
3734, 36ffvelrnd 5934 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  ( C `  k )  e.  CC )
3828adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  x  e.  CC )
3938, 36expcld 12212 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ k )  e.  CC )
4037, 39mulcld 9527 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
( C `  k
)  x.  ( x ^ k ) )  e.  CC )
41 rpne0 11154 . . . . . . . 8  |-  ( x  e.  RR+  ->  x  =/=  0 )
4241adantl 464 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  x  =/=  0 )
4330nn0zd 10882 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  D  e.  ZZ )
4443adantr 463 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  D  e.  ZZ )
4528, 42, 44expne0d 12218 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  =/=  0 )
4626, 32, 40, 45fsumdivc 13603 . . . . 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 11754 . . . . . . . 8  |-  ( ( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  (/)
48 fzosn 11785 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  ( D..^ ( D  +  1 ) )  =  { D } )
4948ineq2d 3614 . . . . . . . 8  |-  ( D  e.  ZZ  ->  (
( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  ( ( 0..^ D )  i^i  { D }
) )
5047, 49syl5reqr 2438 . . . . . . 7  |-  ( D  e.  ZZ  ->  (
( 0..^ D )  i^i  { D }
)  =  (/) )
5144, 50syl 16 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( 0..^ D )  i^i  { D }
)  =  (/) )
52 fzval3 11784 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  (
0 ... D )  =  ( 0..^ ( D  +  1 ) ) )
5343, 52syl 16 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( 0..^ ( D  + 
1 ) ) )
54 nn0uz 11035 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
5530, 54syl6eleq 2480 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  D  e.  (
ZZ>= `  0 ) )
56 fzosplitsn 11817 . . . . . . . . 9  |-  ( D  e.  ( ZZ>= `  0
)  ->  ( 0..^ ( D  +  1 ) )  =  ( ( 0..^ D )  u.  { D }
) )
5755, 56syl 16 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  ( 0..^ ( D  +  1 ) )  =  ( ( 0..^ D )  u. 
{ D } ) )
5853, 57eqtrd 2423 . . . . . . 7  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( ( 0..^ D )  u.  { D }
) )
5958adantr 463 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
0 ... D )  =  ( ( 0..^ D )  u.  { D } ) )
6032adantr 463 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  e.  CC )
6142adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  x  =/=  0 )
6244adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  D  e.  ZZ )
6338, 61, 62expne0d 12218 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  =/=  0 )
6440, 60, 63divcld 10237 . . . . . 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 13564 . . . . 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 2423 . . . 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 4453 . . 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 2423 . 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 13512 . . . . 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 13512 . . . . 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 11987 . . . . . . 7  |-  ( 0..^ D )  e.  Fin
7574a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( 0..^ D )  e.  Fin )
76 ovex 6224 . . . . . . 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 723 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  C : NN0
--> CC )
79 elfzonn0 11762 . . . . . . . . . . 11  |-  ( k  e.  ( 0..^ D )  ->  k  e.  NN0 )
8079ad2antlr 724 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e. 
NN0 )
8178, 80ffvelrnd 5934 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( C `
 k )  e.  CC )
8228adantlr 712 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  e.  CC )
8382, 80expcld 12212 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ k )  e.  CC )
8432adantlr 712 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  e.  CC )
8541adantl 464 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  =/=  0 )
8644adantlr 712 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  ZZ )
8782, 85, 86expne0d 12218 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  =/=  0 )
8881, 83, 84, 87divassd 10272 . . . . . . . 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 4453 . . . . . . 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 5784 . . . . . . . . . 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 6224 . . . . . . . . . 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 463 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  C : NN0
--> CC )
9579adantl 464 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  NN0 )
9694, 95ffvelrnd 5934 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( C `  k )  e.  CC )
97 rlimconst 13369 . . . . . . . . . 10  |-  ( (
RR+  C_  RR  /\  ( C `  k )  e.  CC )  ->  (
x  e.  RR+  |->  ( C `
 k ) )  ~~> r  ( C `  k ) )
9812, 96, 97sylancr 661 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( C `  k ) )  ~~> r  ( C `  k ) )
9980nn0zd 10882 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  ZZ )
10086, 99zsubcld 10889 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( D  -  k )  e.  ZZ )
10182, 85, 100cxpexpzd 23179 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x  ^c  ( D  -  k ) )  =  ( x ^
( D  -  k
) ) )
102101oveq2d 6212 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
10382, 85, 100expnegd 12219 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
10486zcnd 10885 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  CC )
10580nn0cnd 10771 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  CC )
106104, 105negsubdi2d 9860 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  -u ( D  -  k )  =  ( k  -  D ) )
107106oveq2d 6212 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( x ^
( k  -  D
) ) )
108102, 103, 1073eqtr2d 2429 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( x ^
( k  -  D
) ) )
10982, 85, 86, 99expsubd 12223 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ ( k  -  D ) )  =  ( ( x ^
k )  /  (
x ^ D ) ) )
110108, 109eqtrd 2423 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( ( x ^ k )  / 
( x ^ D
) ) )
111110mpteq2dva 4453 . . . . . . . . . 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 10770 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  RR )
11330adantr 463 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  NN0 )
114113nn0red 10770 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  RR )
115 elfzolt2 11731 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0..^ D )  ->  k  <  D )
116115adantl 464 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  <  D )
117 difrp 11173 . . . . . . . . . . . . 13  |-  ( ( k  e.  RR  /\  D  e.  RR )  ->  ( k  <  D  <->  ( D  -  k )  e.  RR+ ) )
118117biimpa 482 . . . . . . . . . . . 12  |-  ( ( ( k  e.  RR  /\  D  e.  RR )  /\  k  <  D
)  ->  ( D  -  k )  e.  RR+ )
119112, 114, 116, 118syl21anc 1225 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( D  -  k )  e.  RR+ )
120 cxplim 23418 . . . . . . . . . . 11  |-  ( ( D  -  k )  e.  RR+  ->  ( x  e.  RR+  |->  ( 1  /  ( x  ^c  ( D  -  k ) ) ) )  ~~> r  0 )
121119, 120syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( 1  / 
( x  ^c 
( D  -  k
) ) ) )  ~~> r  0 )
122111, 121eqbrtrrd 4389 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( x ^ k )  / 
( x ^ D
) ) )  ~~> r  0 )
12391, 93, 98, 122rlimmul 13469 . . . . . . . 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 9690 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( ( C `  k )  x.  0 )  =  0 )
125123, 124breqtrd 4391 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( C `
 k )  x.  ( ( x ^
k )  /  (
x ^ D ) ) ) )  ~~> r  0 )
12689, 125eqbrtrd 4387 . . . . . 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 13627 . . . . 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 391 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( ( 0..^ D )  C_  ( ZZ>=
`  0 )  \/  ( 0..^ D )  e.  Fin ) )
129 sumz 13546 . . . . . 6  |-  ( ( ( 0..^ D ) 
C_  ( ZZ>= `  0
)  \/  ( 0..^ D )  e.  Fin )  ->  sum_ k  e.  ( 0..^ D ) 0  =  0 )
130128, 129syl 16 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  sum_ k  e.  ( 0..^ D ) 0  =  0 )
131127, 130breqtrd 4391 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  ( 0..^ D ) ( ( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  0 )
13233, 30ffvelrnd 5934 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  RR )  ->  ( C `  D )  e.  CC )
133132adantr 463 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( C `  D )  e.  CC )
134133, 32mulcld 9527 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( C `  D
)  x.  ( x ^ D ) )  e.  CC )
135134, 32, 45divcld 10237 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  e.  CC )
136 fveq2 5774 . . . . . . . . . . 11  |-  ( k  =  D  ->  ( C `  k )  =  ( C `  D ) )
137 oveq2 6204 . . . . . . . . . . 11  |-  ( k  =  D  ->  (
x ^ k )  =  ( x ^ D ) )
138136, 137oveq12d 6214 . . . . . . . . . 10  |-  ( k  =  D  ->  (
( C `  k
)  x.  ( x ^ k ) )  =  ( ( C `
 D )  x.  ( x ^ D
) ) )
139138oveq1d 6211 . . . . . . . . 9  |-  ( k  =  D  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( ( ( C `  D )  x.  ( x ^ D ) )  / 
( x ^ D
) ) )
140139sumsn 13565 . . . . . . . 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 659 . . . . . . 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 10243 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  =  ( C `  D ) )
143141, 142eqtrd 2423 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( C `  D ) )
144143mpteq2dva 4453 . . . . 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 13369 . . . . . 6  |-  ( (
RR+  C_  RR  /\  ( C `  D )  e.  CC )  ->  (
x  e.  RR+  |->  ( C `
 D ) )  ~~> r  ( C `  D ) )
14612, 132, 145sylancr 661 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  ( C `  D ) )  ~~> r  ( C `  D ) )
147144, 146eqbrtrd 4387 . . . 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 13467 . . 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 9692 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  ( C `  D ) )
150 signsply0.b . . . 4  |-  B  =  ( C `  D
)
151149, 150syl6eqr 2441 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  B )
152148, 151breqtrd 4391 . 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 4387 1  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  G )  ~~> r  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   _Vcvv 3034    u. cun 3387    i^i cin 3388    C_ wss 3389   (/)c0 3711   {csn 3944   class class class wbr 4367    |-> cmpt 4425    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196    oFcof 6437   Fincfn 7435   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408    < clt 9539    - cmin 9718   -ucneg 9719    / cdiv 10123   NN0cn0 10712   ZZcz 10781   ZZ>=cuz 11001   RR+crp 11139   ...cfz 11593  ..^cfzo 11717   ^cexp 12069    ~~> r crli 13310   sum_csu 13510  Polycply 22666  coeffccoe 22668  degcdgr 22669    ^c ccxp 23028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ioc 11455  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-fac 12256  df-bc 12283  df-hash 12308  df-shft 12902  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-limsup 13296  df-clim 13313  df-rlim 13314  df-sum 13511  df-ef 13805  df-sin 13807  df-cos 13808  df-pi 13810  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-fbas 18529  df-fg 18530  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-lp 19723  df-perf 19724  df-cn 19814  df-cnp 19815  df-haus 19902  df-tx 20148  df-hmeo 20341  df-fil 20432  df-fm 20524  df-flim 20525  df-flf 20526  df-xms 20908  df-ms 20909  df-tms 20910  df-cncf 21467  df-0p 22162  df-limc 22355  df-dv 22356  df-ply 22670  df-coe 22672  df-dgr 22673  df-log 23029  df-cxp 23030
This theorem is referenced by:  signsply0  28691
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