Theorem signsplypnf 30627

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.

Step	Hyp	Ref	Expression
1		plyf 23954	. . . . 5 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
2		ffn 6045	. . . . 5 |- ( F : CC --> CC -> F Fn CC )
3	1, 2	syl 17	. . . 4 |- ( F e. (Poly ` RR ) -> F Fn CC )
4		ovex 6678	. . . . . 6 |- ( x ^ D ) e. _V
5	4	rgenw 2924	. . . . 5 |- A. x e. RR+ ( x ^ D ) e. _V
6		signsplypnf.g	. . . . . 6 |- G = ( x e. RR+ |-> ( x ^ D ) )
7	6	fnmpt 6020	. . . . 5 |- ( A. x e. RR+ ( x ^ D ) e. _V -> G Fn RR+ )
8	5, 7	mp1i 13	. . . 4 |- ( F e. (Poly ` RR ) -> G Fn RR+ )
9		cnex 10017	. . . . 5 |- CC e. _V
10	9	a1i 11	. . . 4 |- ( F e. (Poly ` RR ) -> CC e. _V )
11		reex 10027	. . . . . 6 |- RR e. _V
12		rpssre 11843	. . . . . 6 |- RR+ C_ RR
13	11, 12	ssexi 4803	. . . . 5 |- RR+ e. _V
14	13	a1i 11	. . . 4 |- ( F e. (Poly ` RR ) -> RR+ e. _V )
15		ax-resscn 9993	. . . . . 6 |- RR C_ CC
16	12, 15	sstri 3612	. . . . 5 |- RR+ C_ CC
17		sseqin2 3817	. . . . 5 |- ( RR+ C_ CC <-> ( CC i^i RR+ ) = RR+ )
18	16, 17	mpbi 220	. . . 4 |- ( CC i^i RR+ ) = RR+
19		signsply0.c	. . . . 5 |- C = (coeff ` F )
20		signsply0.d	. . . . 5 |- D = (deg ` F )
21	19, 20	coeid2 23995	. . . 4 |- ( ( F e. (Poly ` RR ) /\ x e. CC ) -> ( F ` x ) = sum_ k e. ( 0 ... D ) ( ( C ` k ) x. ( x ^ k ) ) )
22	6	fvmpt2 6291	. . . . . 6 |- ( ( x e. RR+ /\ ( x ^ D ) e. _V ) -> ( G ` x ) = ( x ^ D ) )
23	4, 22	mpan2 707	. . . . 5 |- ( x e. RR+ -> ( G ` x ) = ( x ^ D ) )
24	23	adantl 482	. . . 4 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( G ` x ) = ( x ^ D ) )
25	3, 8, 10, 14, 18, 21, 24	offval 6904	. . 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 12772	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( 0 ... D ) e. Fin )
27	16	a1i 11	. . . . . . . 8 |- ( F e. (Poly ` RR ) -> RR+ C_ CC )
28	27	sselda 3603	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> x e. CC )
29		dgrcl 23989	. . . . . . . . 9 |- ( F e. (Poly ` RR ) -> (deg ` F ) e. NN0 )
30	20, 29	syl5eqel 2705	. . . . . . . 8 |- ( F e. (Poly ` RR ) -> D e. NN0 )
31	30	adantr 481	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> D e. NN0 )
32	28, 31	expcld 13008	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( x ^ D ) e. CC )
33	19	coef3 23988	. . . . . . . . 9 |- ( F e. (Poly ` RR ) -> C : NN0 --> CC )
34	33	ad2antrr 762	. . . . . . . 8 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> C : NN0 --> CC )
35		elfznn0 12433	. . . . . . . . 9 |- ( k e. ( 0 ... D ) -> k e. NN0 )
36	35	adantl 482	. . . . . . . 8 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> k e. NN0 )
37	34, 36	ffvelrnd 6360	. . . . . . 7 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> ( C ` k ) e. CC )
38	28	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> x e. CC )
39	38, 36	expcld 13008	. . . . . . 7 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> ( x ^ k ) e. CC )
40	37, 39	mulcld 10060	. . . . . 6 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> ( ( C ` k ) x. ( x ^ k ) ) e. CC )
41		rpne0 11848	. . . . . . . 8 |- ( x e. RR+ -> x =/= 0 )
42	41	adantl 482	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> x =/= 0 )
43	30	nn0zd 11480	. . . . . . . 8 |- ( F e. (Poly ` RR ) -> D e. ZZ )
44	43	adantr 481	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> D e. ZZ )
45	28, 42, 44	expne0d 13014	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( x ^ D ) =/= 0 )
46	26, 32, 40, 45	fsumdivc 14518	. . . . 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 12502	. . . . . . . 8 |- ( ( 0..^ D ) i^i ( D..^ ( D + 1 ) ) ) = (/)
48		fzosn 12538	. . . . . . . . 9 |- ( D e. ZZ -> ( D..^ ( D + 1 ) ) = { D } )
49	48	ineq2d 3814	. . . . . . . 8 |- ( D e. ZZ -> ( ( 0..^ D ) i^i ( D..^ ( D + 1 ) ) ) = ( ( 0..^ D ) i^i { D } ) )
50	47, 49	syl5reqr 2671	. . . . . . 7 |- ( D e. ZZ -> ( ( 0..^ D ) i^i { D } ) = (/) )
51	44, 50	syl 17	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( ( 0..^ D ) i^i { D } ) = (/) )
52		fzval3 12536	. . . . . . . . 9 |- ( D e. ZZ -> ( 0 ... D ) = ( 0..^ ( D + 1 ) ) )
53	43, 52	syl 17	. . . . . . . 8 |- ( F e. (Poly ` RR ) -> ( 0 ... D ) = ( 0..^ ( D + 1 ) ) )
54		nn0uz 11722	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
55	30, 54	syl6eleq 2711	. . . . . . . . 9 |- ( F e. (Poly ` RR ) -> D e. ( ZZ>= ` 0 ) )
56		fzosplitsn 12576	. . . . . . . . 9 |- ( D e. ( ZZ>= ` 0 ) -> ( 0..^ ( D + 1 ) ) = ( ( 0..^ D ) u. { D } ) )
57	55, 56	syl 17	. . . . . . . 8 |- ( F e. (Poly ` RR ) -> ( 0..^ ( D + 1 ) ) = ( ( 0..^ D ) u. { D } ) )
58	53, 57	eqtrd 2656	. . . . . . 7 |- ( F e. (Poly ` RR ) -> ( 0 ... D ) = ( ( 0..^ D ) u. { D } ) )
59	58	adantr 481	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( 0 ... D ) = ( ( 0..^ D ) u. { D } ) )
60	32	adantr 481	. . . . . . 7 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> ( x ^ D ) e. CC )
61	42	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> x =/= 0 )
62	44	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> D e. ZZ )
63	38, 61, 62	expne0d 13014	. . . . . . 7 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> ( x ^ D ) =/= 0 )
64	40, 60, 63	divcld 10801	. . . . . 6 |- ( ( ( F e. (Poly ` RR ) /\ x e. RR+ ) /\ k e. ( 0 ... D ) ) -> ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) e. CC )
65	51, 59, 26, 64	fsumsplit 14471	. . . . 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 ) ) ) )
66	46, 65	eqtrd 2656	. . . 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 ) ) ) )
67	66	mpteq2dva 4744	. . 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 ) ) ) ) )
68	25, 67	eqtrd 2656	. 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 14418	. . . . 5 |- sum_ k e. ( 0..^ D ) ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) e. _V
70	69	a1i 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 14418	. . . . 5 |- sum_ k e. { D } ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) e. _V
72	71	a1i 11	. . . 4 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> sum_ k e. { D } ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) e. _V )
73	12	a1i 11	. . . . . 6 |- ( F e. (Poly ` RR ) -> RR+ C_ RR )
74		fzofi 12773	. . . . . . 7 |- ( 0..^ D ) e. Fin
75	74	a1i 11	. . . . . 6 |- ( F e. (Poly ` RR ) -> ( 0..^ D ) e. Fin )
76		ovexd 6680	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ ( x e. RR+ /\ k e. ( 0..^ D ) ) ) -> ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) e. _V )
77	33	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> C : NN0 --> CC )
78		elfzonn0 12512	. . . . . . . . . . 11 |- ( k e. ( 0..^ D ) -> k e. NN0 )
79	78	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> k e. NN0 )
80	77, 79	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( C ` k ) e. CC )
81	28	adantlr 751	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> x e. CC )
82	81, 79	expcld 13008	. . . . . . . . 9 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( x ^ k ) e. CC )
83	32	adantlr 751	. . . . . . . . 9 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( x ^ D ) e. CC )
84	41	adantl 482	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> x =/= 0 )
85	44	adantlr 751	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> D e. ZZ )
86	81, 84, 85	expne0d 13014	. . . . . . . . 9 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( x ^ D ) =/= 0 )
87	80, 82, 83, 86	divassd 10836	. . . . . . . 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 ) ) ) )
88	87	mpteq2dva 4744	. . . . . . 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 ) ) ) ) )
89		fvexd 6203	. . . . . . . . 9 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( C ` k ) e. _V )
90		ovexd 6680	. . . . . . . . 9 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( ( x ^ k ) / ( x ^ D ) ) e. _V )
91	33	adantr 481	. . . . . . . . . . 11 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> C : NN0 --> CC )
92	78	adantl 482	. . . . . . . . . . 11 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> k e. NN0 )
93	91, 92	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( C ` k ) e. CC )
94		rlimconst 14275	. . . . . . . . . 10 |- ( ( RR+ C_ RR /\ ( C ` k ) e. CC ) -> ( x e. RR+ |-> ( C ` k ) ) ~~> r ( C ` k ) )
95	12, 93, 94	sylancr 695	. . . . . . . . 9 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( x e. RR+ |-> ( C ` k ) ) ~~> r ( C ` k ) )
96	79	nn0zd 11480	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> k e. ZZ )
97	85, 96	zsubcld 11487	. . . . . . . . . . . . . . 15 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( D - k ) e. ZZ )
98	81, 84, 97	cxpexpzd 24457	. . . . . . . . . . . . . 14 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( x ^c ( D - k ) ) = ( x ^ ( D - k ) ) )
99	98	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( 1 / ( x ^c ( D - k ) ) ) = ( 1 / ( x ^ ( D - k ) ) ) )
100	81, 84, 97	expnegd 13015	. . . . . . . . . . . . 13 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( x ^ -u ( D - k ) ) = ( 1 / ( x ^ ( D - k ) ) ) )
101	85	zcnd 11483	. . . . . . . . . . . . . . 15 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> D e. CC )
102	79	nn0cnd 11353	. . . . . . . . . . . . . . 15 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> k e. CC )
103	101, 102	negsubdi2d 10408	. . . . . . . . . . . . . 14 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> -u ( D - k ) = ( k - D ) )
104	103	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( x ^ -u ( D - k ) ) = ( x ^ ( k - D ) ) )
105	99, 100, 104	3eqtr2d 2662	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( 1 / ( x ^c ( D - k ) ) ) = ( x ^ ( k - D ) ) )
106	81, 84, 85, 96	expsubd 13019	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( x ^ ( k - D ) ) = ( ( x ^ k ) / ( x ^ D ) ) )
107	105, 106	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) /\ x e. RR+ ) -> ( 1 / ( x ^c ( D - k ) ) ) = ( ( x ^ k ) / ( x ^ D ) ) )
108	107	mpteq2dva 4744	. . . . . . . . . 10 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( x e. RR+ |-> ( 1 / ( x ^c ( D - k ) ) ) ) = ( x e. RR+ |-> ( ( x ^ k ) / ( x ^ D ) ) ) )
109	92	nn0red 11352	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> k e. RR )
110	30	adantr 481	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> D e. NN0 )
111	110	nn0red 11352	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> D e. RR )
112		elfzolt2 12479	. . . . . . . . . . . . 13 |- ( k e. ( 0..^ D ) -> k < D )
113	112	adantl 482	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> k < D )
114		difrp 11868	. . . . . . . . . . . . 13 |- ( ( k e. RR /\ D e. RR ) -> ( k < D <-> ( D - k ) e. RR+ ) )
115	114	biimpa 501	. . . . . . . . . . . 12 |- ( ( ( k e. RR /\ D e. RR ) /\ k < D ) -> ( D - k ) e. RR+ )
116	109, 111, 113, 115	syl21anc 1325	. . . . . . . . . . 11 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( D - k ) e. RR+ )
117		cxplim 24698	. . . . . . . . . . 11 |- ( ( D - k ) e. RR+ -> ( x e. RR+ |-> ( 1 / ( x ^c ( D - k ) ) ) ) ~~> r 0 )
118	116, 117	syl 17	. . . . . . . . . 10 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( x e. RR+ |-> ( 1 / ( x ^c ( D - k ) ) ) ) ~~> r 0 )
119	108, 118	eqbrtrrd 4677	. . . . . . . . 9 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( x e. RR+ |-> ( ( x ^ k ) / ( x ^ D ) ) ) ~~> r 0 )
120	89, 90, 95, 119	rlimmul 14375	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( x e. RR+ |-> ( ( C ` k ) x. ( ( x ^ k ) / ( x ^ D ) ) ) ) ~~> r ( ( C ` k ) x. 0 ) )
121	93	mul01d 10235	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( ( C ` k ) x. 0 ) = 0 )
122	120, 121	breqtrd 4679	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( x e. RR+ |-> ( ( C ` k ) x. ( ( x ^ k ) / ( x ^ D ) ) ) ) ~~> r 0 )
123	88, 122	eqbrtrd 4675	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ k e. ( 0..^ D ) ) -> ( x e. RR+ |-> ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) ) ~~> r 0 )
124	73, 75, 76, 123	fsumrlim 14543	. . . . 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 )
125	75	olcd 408	. . . . . 6 |- ( F e. (Poly ` RR ) -> ( ( 0..^ D ) C_ ( ZZ>= ` 0 ) \/ ( 0..^ D ) e. Fin ) )
126		sumz 14453	. . . . . 6 |- ( ( ( 0..^ D ) C_ ( ZZ>= ` 0 ) \/ ( 0..^ D ) e. Fin ) -> sum_ k e. ( 0..^ D ) 0 = 0 )
127	125, 126	syl 17	. . . . 5 |- ( F e. (Poly ` RR ) -> sum_ k e. ( 0..^ D ) 0 = 0 )
128	124, 127	breqtrd 4679	. . . 4 |- ( F e. (Poly ` RR ) -> ( x e. RR+ |-> sum_ k e. ( 0..^ D ) ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) ) ~~> r 0 )
129	33, 30	ffvelrnd 6360	. . . . . . . . . . 11 |- ( F e. (Poly ` RR ) -> ( C ` D ) e. CC )
130	129	adantr 481	. . . . . . . . . 10 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( C ` D ) e. CC )
131	130, 32	mulcld 10060	. . . . . . . . 9 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( ( C ` D ) x. ( x ^ D ) ) e. CC )
132	131, 32, 45	divcld 10801	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( ( ( C ` D ) x. ( x ^ D ) ) / ( x ^ D ) ) e. CC )
133		fveq2 6191	. . . . . . . . . . 11 |- ( k = D -> ( C ` k ) = ( C ` D ) )
134		oveq2 6658	. . . . . . . . . . 11 |- ( k = D -> ( x ^ k ) = ( x ^ D ) )
135	133, 134	oveq12d 6668	. . . . . . . . . 10 |- ( k = D -> ( ( C ` k ) x. ( x ^ k ) ) = ( ( C ` D ) x. ( x ^ D ) ) )
136	135	oveq1d 6665	. . . . . . . . 9 |- ( k = D -> ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) = ( ( ( C ` D ) x. ( x ^ D ) ) / ( x ^ D ) ) )
137	136	sumsn 14475	. . . . . . . 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 ) ) )
138	31, 132, 137	syl2anc 693	. . . . . . 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 ) ) )
139	130, 32, 45	divcan4d 10807	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> ( ( ( C ` D ) x. ( x ^ D ) ) / ( x ^ D ) ) = ( C ` D ) )
140	138, 139	eqtrd 2656	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ x e. RR+ ) -> sum_ k e. { D } ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) = ( C ` D ) )
141	140	mpteq2dva 4744	. . . . 5 |- ( F e. (Poly ` RR ) -> ( x e. RR+ |-> sum_ k e. { D } ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) ) = ( x e. RR+ |-> ( C ` D ) ) )
142		rlimconst 14275	. . . . . 6 |- ( ( RR+ C_ RR /\ ( C ` D ) e. CC ) -> ( x e. RR+ |-> ( C ` D ) ) ~~> r ( C ` D ) )
143	12, 129, 142	sylancr 695	. . . . 5 |- ( F e. (Poly ` RR ) -> ( x e. RR+ |-> ( C ` D ) ) ~~> r ( C ` D ) )
144	141, 143	eqbrtrd 4675	. . . 4 |- ( F e. (Poly ` RR ) -> ( x e. RR+ |-> sum_ k e. { D } ( ( ( C ` k ) x. ( x ^ k ) ) / ( x ^ D ) ) ) ~~> r ( C ` D ) )
145	70, 72, 128, 144	rlimadd 14373	. . 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 ) ) )
146	129	addid2d 10237	. . . 4 |- ( F e. (Poly ` RR ) -> ( 0 + ( C ` D ) ) = ( C ` D ) )
147		signsply0.b	. . . 4 |- B = ( C ` D )
148	146, 147	syl6eqr 2674	. . 3 |- ( F e. (Poly ` RR ) -> ( 0 + ( C ` D ) ) = B )
149	145, 148	breqtrd 4679	. 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 )
150	68, 149	eqbrtrd 4675	1 |- ( F e. (Poly ` RR ) -> ( F oF / G ) ~~> r B )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326  ..^cfzo 12465   ^cexp 12860   ~~> r crli 14216   sum_csu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-0p 23437  df-limc 23630  df-dv 23631  df-ply 23944  df-coe 23946  df-dgr 23947  df-log 24303  df-cxp 24304

This theorem is referenced by:  signsply0  30628