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Theorem signsplypnf 29511
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 23231 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
2 ffn 5739 . . . . 5  |-  ( F : CC --> CC  ->  F  Fn  CC )
31, 2syl 17 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  F  Fn  CC )
4 ovex 6336 . . . . . 6  |-  ( x ^ D )  e. 
_V
54rgenw 2768 . . . . 5  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
6 signsplypnf.g . . . . . 6  |-  G  =  ( x  e.  RR+  |->  ( x ^ D
) )
76fnmpt 5714 . . . . 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 9638 . . . . 5  |-  CC  e.  _V
109a1i 11 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  CC  e.  _V )
11 reex 9648 . . . . . 6  |-  RR  e.  _V
12 rpssre 11335 . . . . . 6  |-  RR+  C_  RR
1311, 12ssexi 4541 . . . . 5  |-  RR+  e.  _V
1413a1i 11 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  RR+  e.  _V )
15 ax-resscn 9614 . . . . . 6  |-  RR  C_  CC
1612, 15sstri 3427 . . . . 5  |-  RR+  C_  CC
17 dfss1 3628 . . . . 5  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
1816, 17mpbi 213 . . . 4  |-  ( CC 
i^i  RR+ )  =  RR+
19 signsply0.c . . . . 5  |-  C  =  (coeff `  F )
20 signsply0.d . . . . 5  |-  D  =  (deg `  F )
2119, 20coeid2 23272 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  CC )  ->  ( F `  x )  =  sum_ k  e.  ( 0 ... D ) ( ( C `  k )  x.  (
x ^ k ) ) )
226fvmpt2 5972 . . . . . 6  |-  ( ( x  e.  RR+  /\  (
x ^ D )  e.  _V )  -> 
( G `  x
)  =  ( x ^ D ) )
234, 22mpan2 685 . . . . 5  |-  ( x  e.  RR+  ->  ( G `
 x )  =  ( x ^ D
) )
2423adantl 473 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( G `  x )  =  ( x ^ D ) )
253, 8, 10, 14, 18, 21, 24offval 6557 . . 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 12224 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
0 ... D )  e. 
Fin )
2716a1i 11 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  RR+  C_  CC )
2827sselda 3418 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  x  e.  CC )
29 dgrcl 23266 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
3020, 29syl5eqel 2553 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  D  e.  NN0 )
3130adantr 472 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  D  e.  NN0 )
3228, 31expcld 12454 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  e.  CC )
3319coef3 23265 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> CC )
3433ad2antrr 740 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  C : NN0 --> CC )
35 elfznn0 11913 . . . . . . . . 9  |-  ( k  e.  ( 0 ... D )  ->  k  e.  NN0 )
3635adantl 473 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  k  e.  NN0 )
3734, 36ffvelrnd 6038 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  ( C `  k )  e.  CC )
3828adantr 472 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  x  e.  CC )
3938, 36expcld 12454 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ k )  e.  CC )
4037, 39mulcld 9681 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
( C `  k
)  x.  ( x ^ k ) )  e.  CC )
41 rpne0 11340 . . . . . . . 8  |-  ( x  e.  RR+  ->  x  =/=  0 )
4241adantl 473 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  x  =/=  0 )
4330nn0zd 11061 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  D  e.  ZZ )
4443adantr 472 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  D  e.  ZZ )
4528, 42, 44expne0d 12460 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  =/=  0 )
4626, 32, 40, 45fsumdivc 13924 . . . . 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 11979 . . . . . . . 8  |-  ( ( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  (/)
48 fzosn 12013 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  ( D..^ ( D  +  1 ) )  =  { D } )
4948ineq2d 3625 . . . . . . . 8  |-  ( D  e.  ZZ  ->  (
( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  ( ( 0..^ D )  i^i  { D }
) )
5047, 49syl5reqr 2520 . . . . . . 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 12012 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  (
0 ... D )  =  ( 0..^ ( D  +  1 ) ) )
5343, 52syl 17 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( 0..^ ( D  + 
1 ) ) )
54 nn0uz 11217 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
5530, 54syl6eleq 2559 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  D  e.  (
ZZ>= `  0 ) )
56 fzosplitsn 12048 . . . . . . . . 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 2505 . . . . . . 7  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( ( 0..^ D )  u.  { D }
) )
5958adantr 472 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
0 ... D )  =  ( ( 0..^ D )  u.  { D } ) )
6032adantr 472 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  e.  CC )
6142adantr 472 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  x  =/=  0 )
6244adantr 472 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  D  e.  ZZ )
6338, 61, 62expne0d 12460 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  =/=  0 )
6440, 60, 63divcld 10405 . . . . . 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 13883 . . . . 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 2505 . . . 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 4482 . . 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 2505 . 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 13831 . . . . 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 13831 . . . . 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 12225 . . . . . . 7  |-  ( 0..^ D )  e.  Fin
7574a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( 0..^ D )  e.  Fin )
76 ovex 6336 . . . . . . 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 740 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  C : NN0
--> CC )
79 elfzonn0 11988 . . . . . . . . . . 11  |-  ( k  e.  ( 0..^ D )  ->  k  e.  NN0 )
8079ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e. 
NN0 )
8178, 80ffvelrnd 6038 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( C `
 k )  e.  CC )
8228adantlr 729 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  e.  CC )
8382, 80expcld 12454 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ k )  e.  CC )
8432adantlr 729 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  e.  CC )
8541adantl 473 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  =/=  0 )
8644adantlr 729 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  ZZ )
8782, 85, 86expne0d 12460 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  =/=  0 )
8881, 83, 84, 87divassd 10440 . . . . . . . 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 4482 . . . . . . 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 6336 . . . . . . . . . 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 472 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  C : NN0
--> CC )
9579adantl 473 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  NN0 )
9694, 95ffvelrnd 6038 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( C `  k )  e.  CC )
97 rlimconst 13685 . . . . . . . . . 10  |-  ( (
RR+  C_  RR  /\  ( C `  k )  e.  CC )  ->  (
x  e.  RR+  |->  ( C `
 k ) )  ~~> r  ( C `  k ) )
9812, 96, 97sylancr 676 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( C `  k ) )  ~~> r  ( C `  k ) )
9980nn0zd 11061 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  ZZ )
10086, 99zsubcld 11068 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( D  -  k )  e.  ZZ )
10182, 85, 100cxpexpzd 23735 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x  ^c  ( D  -  k ) )  =  ( x ^
( D  -  k
) ) )
102101oveq2d 6324 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
10382, 85, 100expnegd 12461 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
10486zcnd 11064 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  CC )
10580nn0cnd 10951 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  CC )
106104, 105negsubdi2d 10021 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  -u ( D  -  k )  =  ( k  -  D ) )
107106oveq2d 6324 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( x ^
( k  -  D
) ) )
108102, 103, 1073eqtr2d 2511 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( x ^
( k  -  D
) ) )
10982, 85, 86, 99expsubd 12465 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ ( k  -  D ) )  =  ( ( x ^
k )  /  (
x ^ D ) ) )
110108, 109eqtrd 2505 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( ( x ^ k )  / 
( x ^ D
) ) )
111110mpteq2dva 4482 . . . . . . . . . 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 10950 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  RR )
11330adantr 472 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  NN0 )
114113nn0red 10950 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  RR )
115 elfzolt2 11956 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0..^ D )  ->  k  <  D )
116115adantl 473 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  <  D )
117 difrp 11360 . . . . . . . . . . . . 13  |-  ( ( k  e.  RR  /\  D  e.  RR )  ->  ( k  <  D  <->  ( D  -  k )  e.  RR+ ) )
118117biimpa 492 . . . . . . . . . . . 12  |-  ( ( ( k  e.  RR  /\  D  e.  RR )  /\  k  <  D
)  ->  ( D  -  k )  e.  RR+ )
119112, 114, 116, 118syl21anc 1291 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( D  -  k )  e.  RR+ )
120 cxplim 23976 . . . . . . . . . . 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 4418 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( x ^ k )  / 
( x ^ D
) ) )  ~~> r  0 )
12391, 93, 98, 122rlimmul 13785 . . . . . . . 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 9850 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( ( C `  k )  x.  0 )  =  0 )
125123, 124breqtrd 4420 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( C `
 k )  x.  ( ( x ^
k )  /  (
x ^ D ) ) ) )  ~~> r  0 )
12689, 125eqbrtrd 4416 . . . . . 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 13948 . . . . 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 400 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( ( 0..^ D )  C_  ( ZZ>=
`  0 )  \/  ( 0..^ D )  e.  Fin ) )
129 sumz 13865 . . . . . 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 4420 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  ( 0..^ D ) ( ( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  0 )
13233, 30ffvelrnd 6038 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  RR )  ->  ( C `  D )  e.  CC )
133132adantr 472 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( C `  D )  e.  CC )
134133, 32mulcld 9681 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( C `  D
)  x.  ( x ^ D ) )  e.  CC )
135134, 32, 45divcld 10405 . . . . . . . 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 6316 . . . . . . . . . . 11  |-  ( k  =  D  ->  (
x ^ k )  =  ( x ^ D ) )
138136, 137oveq12d 6326 . . . . . . . . . 10  |-  ( k  =  D  ->  (
( C `  k
)  x.  ( x ^ k ) )  =  ( ( C `
 D )  x.  ( x ^ D
) ) )
139138oveq1d 6323 . . . . . . . . 9  |-  ( k  =  D  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( ( ( C `  D )  x.  ( x ^ D ) )  / 
( x ^ D
) ) )
140139sumsn 13884 . . . . . . . 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 673 . . . . . . 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 10411 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  =  ( C `  D ) )
143141, 142eqtrd 2505 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( C `  D ) )
144143mpteq2dva 4482 . . . . 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 13685 . . . . . 6  |-  ( (
RR+  C_  RR  /\  ( C `  D )  e.  CC )  ->  (
x  e.  RR+  |->  ( C `
 D ) )  ~~> r  ( C `  D ) )
14612, 132, 145sylancr 676 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  ( C `  D ) )  ~~> r  ( C `  D ) )
147144, 146eqbrtrd 4416 . . . 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 13783 . . 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 9852 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  ( C `  D ) )
150 signsply0.b . . . 4  |-  B  =  ( C `  D
)
151149, 150syl6eqr 2523 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  B )
152148, 151breqtrd 4420 . 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 4416 1  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  G )  ~~> r  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959   class class class wbr 4395    |-> cmpt 4454    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    - cmin 9880   -ucneg 9881    / cdiv 10291   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   ...cfz 11810  ..^cfzo 11942   ^cexp 12310    ~~> r crli 13626   sum_csu 13829  Polycply 23217  coeffccoe 23219  degcdgr 23220    ^c ccxp 23584
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  ax-mulf 9637
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-iin 4272  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-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  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-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-0p 22707  df-limc 22900  df-dv 22901  df-ply 23221  df-coe 23223  df-dgr 23224  df-log 23585  df-cxp 23586
This theorem is referenced by:  signsply0  29512
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