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Theorem signsplypnf 26903
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 sumex 13157 . . . . 5  |-  sum_ k  e.  ( 0..^ D ) ( ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) )  e.  _V
21a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  ( 0..^ D ) ( ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) )  e.  _V )
3 sumex 13157 . . . . 5  |-  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  e.  _V
43a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  e.  _V )
5 rpssre 10993 . . . . . . 7  |-  RR+  C_  RR
65a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  RR+  C_  RR )
7 fzofi 11788 . . . . . . 7  |-  ( 0..^ D )  e.  Fin
87a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( 0..^ D )  e.  Fin )
9 ovex 6111 . . . . . . 7  |-  ( ( ( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) )  e. 
_V
109a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  (
x  e.  RR+  /\  k  e.  ( 0..^ D ) ) )  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  e.  _V )
11 fvex 5696 . . . . . . . . . 10  |-  ( C `
 k )  e. 
_V
1211a1i 11 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( C `
 k )  e. 
_V )
13 ovex 6111 . . . . . . . . . 10  |-  ( ( x ^ k )  /  ( x ^ D ) )  e. 
_V
1413a1i 11 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( ( x ^ k )  /  ( x ^ D ) )  e. 
_V )
15 signsply0.c . . . . . . . . . . . . 13  |-  C  =  (coeff `  F )
1615coef3 21675 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> CC )
1716adantr 465 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  C : NN0
--> CC )
18 elfzouz 11549 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0..^ D )  ->  k  e.  ( ZZ>= `  0 )
)
19 nn0uz 10887 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
2018, 19syl6eleqr 2529 . . . . . . . . . . . 12  |-  ( k  e.  ( 0..^ D )  ->  k  e.  NN0 )
2120adantl 466 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  NN0 )
2217, 21ffvelrnd 5839 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( C `  k )  e.  CC )
23 rlimconst 13014 . . . . . . . . . 10  |-  ( (
RR+  C_  RR  /\  ( C `  k )  e.  CC )  ->  (
x  e.  RR+  |->  ( C `
 k ) )  ~~> r  ( C `  k ) )
245, 22, 23sylancr 663 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( C `  k ) )  ~~> r  ( C `  k ) )
2521nn0red 10629 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  RR )
26 signsply0.d . . . . . . . . . . . . . . 15  |-  D  =  (deg `  F )
27 dgrcl 21676 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
2826, 27syl5eqel 2522 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  RR )  ->  D  e.  NN0 )
2928adantr 465 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  NN0 )
3029nn0red 10629 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  RR )
31 elfzolt2 11553 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0..^ D )  ->  k  <  D )
3231adantl 466 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  <  D )
33 difrp 11016 . . . . . . . . . . . . 13  |-  ( ( k  e.  RR  /\  D  e.  RR )  ->  ( k  <  D  <->  ( D  -  k )  e.  RR+ ) )
3433biimpa 484 . . . . . . . . . . . 12  |-  ( ( ( k  e.  RR  /\  D  e.  RR )  /\  k  <  D
)  ->  ( D  -  k )  e.  RR+ )
3525, 30, 32, 34syl21anc 1217 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( D  -  k )  e.  RR+ )
36 cxplim 22340 . . . . . . . . . . 11  |-  ( ( D  -  k )  e.  RR+  ->  ( x  e.  RR+  |->  ( 1  /  ( x  ^c  ( D  -  k ) ) ) )  ~~> r  0 )
3735, 36syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( 1  / 
( x  ^c 
( D  -  k
) ) ) )  ~~> r  0 )
38 ax-resscn 9331 . . . . . . . . . . . . . . . . . . . 20  |-  RR  C_  CC
395, 38sstri 3360 . . . . . . . . . . . . . . . . . . 19  |-  RR+  C_  CC
4039a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  (Poly `  RR )  ->  RR+  C_  CC )
4140sselda 3351 . . . . . . . . . . . . . . . . 17  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  x  e.  CC )
4241adantlr 714 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  e.  CC )
43 rpgt0 10994 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR+  ->  0  < 
x )
4443gt0ne0d 9896 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR+  ->  x  =/=  0 )
4544adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  =/=  0 )
4628nn0zd 10737 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  (Poly `  RR )  ->  D  e.  ZZ )
4746adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  D  e.  ZZ )
4847adantlr 714 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  ZZ )
4920ad2antlr 726 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e. 
NN0 )
5049nn0zd 10737 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  ZZ )
5148, 50zsubcld 10744 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( D  -  k )  e.  ZZ )
5242, 45, 51cxpexpzd 22131 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x  ^c  ( D  -  k ) )  =  ( x ^
( D  -  k
) ) )
5352oveq2d 6102 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
5442, 45, 51expnegd 12007 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
5548zcnd 10740 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  CC )
5649nn0cnd 10630 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  CC )
5755, 56negsubdi2d 9727 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  -u ( D  -  k )  =  ( k  -  D ) )
5857oveq2d 6102 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( x ^
( k  -  D
) ) )
5953, 54, 583eqtr2d 2476 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( x ^
( k  -  D
) ) )
6042, 45, 48, 50expsubd 12011 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ ( k  -  D ) )  =  ( ( x ^
k )  /  (
x ^ D ) ) )
6159, 60eqtrd 2470 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( ( x ^ k )  / 
( x ^ D
) ) )
6261mpteq2dva 4373 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( 1  / 
( x  ^c 
( D  -  k
) ) ) )  =  ( x  e.  RR+  |->  ( ( x ^ k )  / 
( x ^ D
) ) ) )
6362breq1d 4297 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( (
x  e.  RR+  |->  ( 1  /  ( x  ^c  ( D  -  k ) ) ) )  ~~> r  0  <->  (
x  e.  RR+  |->  ( ( x ^ k )  /  ( x ^ D ) ) )  ~~> r  0 ) )
6437, 63mpbid 210 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( x ^ k )  / 
( x ^ D
) ) )  ~~> r  0 )
6512, 14, 24, 64rlimmul 13114 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( C `
 k )  x.  ( ( x ^
k )  /  (
x ^ D ) ) ) )  ~~> r  ( ( C `  k
)  x.  0 ) )
6622mul01d 9560 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( ( C `  k )  x.  0 )  =  0 )
6765, 66breqtrd 4311 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( C `
 k )  x.  ( ( x ^
k )  /  (
x ^ D ) ) ) )  ~~> r  0 )
6816ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  C : NN0
--> CC )
6968, 49ffvelrnd 5839 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( C `
 k )  e.  CC )
7042, 49expcld 12000 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ k )  e.  CC )
7128adantr 465 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  D  e.  NN0 )
7241, 71expcld 12000 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  e.  CC )
7372adantlr 714 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  e.  CC )
7442, 45, 48expne0d 12006 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  =/=  0 )
7569, 70, 73, 74divassd 10134 . . . . . . . . 9  |-  ( ( ( 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 ) ) ) )
7675mpteq2dva 4373 . . . . . . . 8  |-  ( ( 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 ) ) ) ) )
7776breq1d 4297 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( (
x  e.  RR+  |->  ( ( ( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  0  <->  ( x  e.  RR+  |->  ( ( C `
 k )  x.  ( ( x ^
k )  /  (
x ^ D ) ) ) )  ~~> r  0 ) )
7867, 77mpbird 232 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( x  e.  RR+  |->  ( ( ( C `  k )  x.  ( x ^
k ) )  / 
( x ^ D
) ) )  ~~> r  0 )
796, 8, 10, 78fsumrlim 13266 . . . . 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 )
808olcd 393 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( ( 0..^ D )  C_  ( ZZ>=
`  0 )  \/  ( 0..^ D )  e.  Fin ) )
81 sumz 13191 . . . . . 6  |-  ( ( ( 0..^ D ) 
C_  ( ZZ>= `  0
)  \/  ( 0..^ D )  e.  Fin )  ->  sum_ k  e.  ( 0..^ D ) 0  =  0 )
8280, 81syl 16 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  sum_ k  e.  ( 0..^ D ) 0  =  0 )
8379, 82breqtrd 4311 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  ( 0..^ D ) ( ( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  0 )
8416, 28ffvelrnd 5839 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  RR )  ->  ( C `  D )  e.  CC )
8584adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( C `  D )  e.  CC )
8685, 72mulcld 9398 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( C `  D
)  x.  ( x ^ D ) )  e.  CC )
8744adantl 466 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  x  =/=  0 )
8841, 87, 47expne0d 12006 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  =/=  0 )
8986, 72, 88divcld 10099 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  e.  CC )
90 fveq2 5686 . . . . . . . . . . 11  |-  ( k  =  D  ->  ( C `  k )  =  ( C `  D ) )
91 oveq2 6094 . . . . . . . . . . 11  |-  ( k  =  D  ->  (
x ^ k )  =  ( x ^ D ) )
9290, 91oveq12d 6104 . . . . . . . . . 10  |-  ( k  =  D  ->  (
( C `  k
)  x.  ( x ^ k ) )  =  ( ( C `
 D )  x.  ( x ^ D
) ) )
9392oveq1d 6101 . . . . . . . . 9  |-  ( k  =  D  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( ( ( C `  D )  x.  ( x ^ D ) )  / 
( x ^ D
) ) )
9493sumsn 13209 . . . . . . . 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 ) ) )
9571, 89, 94syl2anc 661 . . . . . . 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
) ) )
96 divcan4 10011 . . . . . . . 8  |-  ( ( ( C `  D
)  e.  CC  /\  ( x ^ D
)  e.  CC  /\  ( x ^ D
)  =/=  0 )  ->  ( ( ( C `  D )  x.  ( x ^ D ) )  / 
( x ^ D
) )  =  ( C `  D ) )
9785, 72, 88, 96syl3anc 1218 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  =  ( C `  D ) )
9895, 97eqtrd 2470 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( C `  D ) )
9998mpteq2dva 4373 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  { D }  ( (
( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) ) )  =  ( x  e.  RR+  |->  ( C `  D ) ) )
100 rlimconst 13014 . . . . . 6  |-  ( (
RR+  C_  RR  /\  ( C `  D )  e.  CC )  ->  (
x  e.  RR+  |->  ( C `
 D ) )  ~~> r  ( C `  D ) )
1016, 84, 100syl2anc 661 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  ( C `  D ) )  ~~> r  ( C `  D ) )
10299, 101eqbrtrd 4307 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  { D }  ( (
( C `  k
)  x.  ( x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  ( C `  D ) )
1032, 4, 83, 102rlimadd 13112 . . 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 ) ) )
10484addid2d 9562 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  ( C `  D ) )
105 signsply0.b . . . 4  |-  B  =  ( C `  D
)
106104, 105syl6eqr 2488 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  B )
107103, 106breqtrd 4311 . 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
)
108 plyf 21641 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
109 ffn 5554 . . . . . 6  |-  ( F : CC --> CC  ->  F  Fn  CC )
110108, 109syl 16 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  F  Fn  CC )
111 ovex 6111 . . . . . . 7  |-  ( x ^ D )  e. 
_V
112111rgenw 2778 . . . . . 6  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
113 signsplypnf.g . . . . . . 7  |-  G  =  ( x  e.  RR+  |->  ( x ^ D
) )
114113fnmpt 5532 . . . . . 6  |-  ( A. x  e.  RR+  ( x ^ D )  e. 
_V  ->  G  Fn  RR+ )
115112, 114mp1i 12 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  G  Fn  RR+ )
116 cnex 9355 . . . . . 6  |-  CC  e.  _V
117116a1i 11 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  CC  e.  _V )
118 reex 9365 . . . . . . 7  |-  RR  e.  _V
119118, 5ssexi 4432 . . . . . 6  |-  RR+  e.  _V
120119a1i 11 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  RR+  e.  _V )
121 dfss1 3550 . . . . . 6  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
12239, 121mpbi 208 . . . . 5  |-  ( CC 
i^i  RR+ )  =  RR+
12315, 26coeid2 21682 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  CC )  ->  ( F `  x )  =  sum_ k  e.  ( 0 ... D ) ( ( C `  k )  x.  (
x ^ k ) ) )
124113fvmpt2 5776 . . . . . . 7  |-  ( ( x  e.  RR+  /\  (
x ^ D )  e.  _V )  -> 
( G `  x
)  =  ( x ^ D ) )
125111, 124mpan2 671 . . . . . 6  |-  ( x  e.  RR+  ->  ( G `
 x )  =  ( x ^ D
) )
126125adantl 466 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  ( G `  x )  =  ( x ^ D ) )
127110, 115, 117, 120, 122, 123, 126offval 6322 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  G )  =  ( x  e.  RR+  |->  ( sum_ k  e.  ( 0 ... D
) ( ( C `
 k )  x.  ( x ^ k
) )  /  (
x ^ D ) ) ) )
128 fzfid 11787 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
0 ... D )  e. 
Fin )
12916ad2antrr 725 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  C : NN0 --> CC )
130 elfznn0 11473 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... D )  ->  k  e.  NN0 )
131130adantl 466 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  k  e.  NN0 )
132129, 131ffvelrnd 5839 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  ( C `  k )  e.  CC )
13341adantr 465 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  x  e.  CC )
134133, 131expcld 12000 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ k )  e.  CC )
135132, 134mulcld 9398 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
( C `  k
)  x.  ( x ^ k ) )  e.  CC )
136128, 72, 135, 88fsumdivc 13245 . . . . . 6  |-  ( ( 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 ) ) )
137 fzodisj 11575 . . . . . . . . 9  |-  ( ( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  (/)
138 fzosn 11598 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  ( D..^ ( D  +  1 ) )  =  { D } )
139138ineq2d 3547 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  (
( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  ( ( 0..^ D )  i^i  { D }
) )
140137, 139syl5reqr 2485 . . . . . . . 8  |-  ( D  e.  ZZ  ->  (
( 0..^ D )  i^i  { D }
)  =  (/) )
14147, 140syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( 0..^ D )  i^i  { D }
)  =  (/) )
142 fzval3 11597 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  (
0 ... D )  =  ( 0..^ ( D  +  1 ) ) )
14346, 142syl 16 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( 0..^ ( D  + 
1 ) ) )
14428, 19syl6eleq 2528 . . . . . . . . . 10  |-  ( F  e.  (Poly `  RR )  ->  D  e.  (
ZZ>= `  0 ) )
145 fzosplitsn 11625 . . . . . . . . . 10  |-  ( D  e.  ( ZZ>= `  0
)  ->  ( 0..^ ( D  +  1 ) )  =  ( ( 0..^ D )  u.  { D }
) )
146144, 145syl 16 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  ( 0..^ ( D  +  1 ) )  =  ( ( 0..^ D )  u. 
{ D } ) )
147143, 146eqtrd 2470 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( ( 0..^ D )  u.  { D }
) )
148147adantr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
0 ... D )  =  ( ( 0..^ D )  u.  { D } ) )
14972adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  e.  CC )
15087adantr 465 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  x  =/=  0 )
15147adantr 465 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  D  e.  ZZ )
152133, 150, 151expne0d 12006 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  =/=  0 )
153135, 149, 152divcld 10099 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  e.  CC )
154141, 148, 128, 153fsumsplit 13208 . . . . . 6  |-  ( ( 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 ) ) ) )
155136, 154eqtrd 2470 . . . . 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 ) ) ) )
156155mpteq2dva 4373 . . . 4  |-  ( 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 ) ) ) ) )
157127, 156eqtrd 2470 . . 3  |-  ( 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 ) ) ) ) )
158157breq1d 4297 . 2  |-  ( F  e.  (Poly `  RR )  ->  ( ( F  oF  /  G
)  ~~> r  B  <->  ( 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
) )
159107, 158mpbird 232 1  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  G )  ~~> r  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   _Vcvv 2967    u. cun 3321    i^i cin 3322    C_ wss 3323   (/)c0 3632   {csn 3872   class class class wbr 4287    e. cmpt 4345    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    oFcof 6313   Fincfn 7302   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    - cmin 9587   -ucneg 9588    / cdiv 9985   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   RR+crp 10983   ...cfz 11429  ..^cfzo 11540   ^cexp 11857    ~~> r crli 12955   sum_csu 13155  Polycply 21627  coeffccoe 21629  degcdgr 21630    ^c ccxp 21982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-0p 21123  df-limc 21316  df-dv 21317  df-ply 21631  df-coe 21633  df-dgr 21634  df-log 21983  df-cxp 21984
This theorem is referenced by:  signsply0  26904
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