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Theorem signsplypnf 28147
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 13469 . . . . 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 13469 . . . . 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 11226 . . . . . . 7  |-  RR+  C_  RR
65a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  RR+  C_  RR )
7 fzofi 12048 . . . . . . 7  |-  ( 0..^ D )  e.  Fin
87a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( 0..^ D )  e.  Fin )
9 ovex 6307 . . . . . . 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 5874 . . . . . . . . . 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 6307 . . . . . . . . . 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 22364 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> CC )
1716adantr 465 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  C : NN0
--> CC )
18 elfzouz 11797 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0..^ D )  ->  k  e.  ( ZZ>= `  0 )
)
19 nn0uz 11112 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
2018, 19syl6eleqr 2566 . . . . . . . . . . . 12  |-  ( k  e.  ( 0..^ D )  ->  k  e.  NN0 )
2120adantl 466 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  NN0 )
2217, 21ffvelrnd 6020 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( C `  k )  e.  CC )
23 rlimconst 13326 . . . . . . . . . 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 10849 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  e.  RR )
26 signsply0.d . . . . . . . . . . . . . . 15  |-  D  =  (deg `  F )
27 dgrcl 22365 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
2826, 27syl5eqel 2559 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  RR )  ->  D  e.  NN0 )
2928adantr 465 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  NN0 )
3029nn0red 10849 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  D  e.  RR )
31 elfzolt2 11801 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0..^ D )  ->  k  <  D )
3231adantl 466 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  k  <  D )
33 difrp 11249 . . . . . . . . . . . . 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 1227 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( D  -  k )  e.  RR+ )
36 cxplim 23029 . . . . . . . . . . 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 9545 . . . . . . . . . . . . . . . . . . . 20  |-  RR  C_  CC
395, 38sstri 3513 . . . . . . . . . . . . . . . . . . 19  |-  RR+  C_  CC
4039a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  (Poly `  RR )  ->  RR+  C_  CC )
4140sselda 3504 . . . . . . . . . . . . . . . . 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 11227 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR+  ->  0  < 
x )
4443gt0ne0d 10113 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR+  ->  x  =/=  0 )
4544adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  x  =/=  0 )
4628nn0zd 10960 . . . . . . . . . . . . . . . . . . 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 10960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  ZZ )
5148, 50zsubcld 10967 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( D  -  k )  e.  ZZ )
5242, 45, 51cxpexpzd 22820 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x  ^c  ( D  -  k ) )  =  ( x ^
( D  -  k
) ) )
5352oveq2d 6298 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
5442, 45, 51expnegd 12281 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( 1  / 
( x ^ ( D  -  k )
) ) )
5548zcnd 10963 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  D  e.  CC )
5649nn0cnd 10850 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  k  e.  CC )
5755, 56negsubdi2d 9942 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  -u ( D  -  k )  =  ( k  -  D ) )
5857oveq2d 6298 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ -u ( D  -  k ) )  =  ( x ^
( k  -  D
) ) )
5953, 54, 583eqtr2d 2514 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( x ^
( k  -  D
) ) )
6042, 45, 48, 50expsubd 12285 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ ( k  -  D ) )  =  ( ( x ^
k )  /  (
x ^ D ) ) )
6159, 60eqtrd 2508 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( 1  /  ( x  ^c  ( D  -  k ) ) )  =  ( ( x ^ k )  / 
( x ^ D
) ) )
6261mpteq2dva 4533 . . . . . . . . . . 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 4457 . . . . . . . . . 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 13426 . . . . . . . 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 9774 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  ->  ( ( C `  k )  x.  0 )  =  0 )
6765, 66breqtrd 4471 . . . . . . 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 6020 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( C `
 k )  e.  CC )
7042, 49expcld 12274 . . . . . . . . . 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 12274 . . . . . . . . . . 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 12280 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  RR )  /\  k  e.  ( 0..^ D ) )  /\  x  e.  RR+ )  ->  ( x ^ D )  =/=  0 )
7569, 70, 73, 74divassd 10351 . . . . . . . . 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 4533 . . . . . . . 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 4457 . . . . . . 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 13584 . . . . 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 13503 . . . . . 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 4471 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( x  e.  RR+  |->  sum_ k  e.  ( 0..^ D ) ( ( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) ) )  ~~> r  0 )
8416, 28ffvelrnd 6020 . . . . . . . . . . 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 9612 . . . . . . . . 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 12280 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
x ^ D )  =/=  0 )
8986, 72, 88divcld 10316 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  e.  CC )
90 fveq2 5864 . . . . . . . . . . 11  |-  ( k  =  D  ->  ( C `  k )  =  ( C `  D ) )
91 oveq2 6290 . . . . . . . . . . 11  |-  ( k  =  D  ->  (
x ^ k )  =  ( x ^ D ) )
9290, 91oveq12d 6300 . . . . . . . . . 10  |-  ( k  =  D  ->  (
( C `  k
)  x.  ( x ^ k ) )  =  ( ( C `
 D )  x.  ( x ^ D
) ) )
9392oveq1d 6297 . . . . . . . . 9  |-  ( k  =  D  ->  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( ( ( C `  D )  x.  ( x ^ D ) )  / 
( x ^ D
) ) )
9493sumsn 13522 . . . . . . . 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 10228 . . . . . . . 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 1228 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  (
( ( C `  D )  x.  (
x ^ D ) )  /  ( x ^ D ) )  =  ( C `  D ) )
9895, 97eqtrd 2508 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  ->  sum_ k  e.  { D }  (
( ( C `  k )  x.  (
x ^ k ) )  /  ( x ^ D ) )  =  ( C `  D ) )
9998mpteq2dva 4533 . . . . 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 13326 . . . . . 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 4467 . . . 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 13424 . . 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 9776 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  ( C `  D ) )
105 signsply0.b . . . 4  |-  B  =  ( C `  D
)
106104, 105syl6eqr 2526 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( 0  +  ( C `  D
) )  =  B )
107103, 106breqtrd 4471 . 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 22330 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
109 ffn 5729 . . . . . 6  |-  ( F : CC --> CC  ->  F  Fn  CC )
110108, 109syl 16 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  F  Fn  CC )
111 ovex 6307 . . . . . . 7  |-  ( x ^ D )  e. 
_V
112111rgenw 2825 . . . . . 6  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
113 signsplypnf.g . . . . . . 7  |-  G  =  ( x  e.  RR+  |->  ( x ^ D
) )
114113fnmpt 5705 . . . . . 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 9569 . . . . . 6  |-  CC  e.  _V
117116a1i 11 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  CC  e.  _V )
118 reex 9579 . . . . . . 7  |-  RR  e.  _V
119118, 5ssexi 4592 . . . . . 6  |-  RR+  e.  _V
120119a1i 11 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  RR+  e.  _V )
121 dfss1 3703 . . . . . 6  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
12239, 121mpbi 208 . . . . 5  |-  ( CC 
i^i  RR+ )  =  RR+
12315, 26coeid2 22371 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  x  e.  CC )  ->  ( F `  x )  =  sum_ k  e.  ( 0 ... D ) ( ( C `  k )  x.  (
x ^ k ) ) )
124113fvmpt2 5955 . . . . . . 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 6529 . . . 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 12047 . . . . . . 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 11766 . . . . . . . . . 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 6020 . . . . . . . 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 12274 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ k )  e.  CC )
135132, 134mulcld 9612 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
( C `  k
)  x.  ( x ^ k ) )  e.  CC )
136128, 72, 135, 88fsumdivc 13560 . . . . . 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 11823 . . . . . . . . 9  |-  ( ( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  (/)
138 fzosn 11850 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  ( D..^ ( D  +  1 ) )  =  { D } )
139138ineq2d 3700 . . . . . . . . 9  |-  ( D  e.  ZZ  ->  (
( 0..^ D )  i^i  ( D..^ ( D  +  1 ) ) )  =  ( ( 0..^ D )  i^i  { D }
) )
140137, 139syl5reqr 2523 . . . . . . . 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 11849 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  (
0 ... D )  =  ( 0..^ ( D  +  1 ) ) )
14346, 142syl 16 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  ( 0 ... D )  =  ( 0..^ ( D  + 
1 ) ) )
14428, 19syl6eleq 2565 . . . . . . . . . 10  |-  ( F  e.  (Poly `  RR )  ->  D  e.  (
ZZ>= `  0 ) )
145 fzosplitsn 11882 . . . . . . . . . 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 2508 . . . . . . . 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 12280 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  RR )  /\  x  e.  RR+ )  /\  k  e.  ( 0 ... D
) )  ->  (
x ^ D )  =/=  0 )
153135, 149, 152divcld 10316 . . . . . . 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 13521 . . . . . 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 2508 . . . . 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 4533 . . . 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 2508 . . 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 4457 . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447    |-> cmpt 4505    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520   Fincfn 7513   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    - cmin 9801   -ucneg 9802    / cdiv 10202   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   RR+crp 11216   ...cfz 11668  ..^cfzo 11788   ^cexp 12130    ~~> r crli 13267   sum_csu 13467  Polycply 22316  coeffccoe 22318  degcdgr 22319    ^c ccxp 22671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-pi 13666  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-0p 21812  df-limc 22005  df-dv 22006  df-ply 22320  df-coe 22322  df-dgr 22323  df-log 22672  df-cxp 22673
This theorem is referenced by:  signsply0  28148
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