Theorem signsply0 30628

Description: Lemma for the rule of signs, based on Bolzano's intermediate value theorem for polynomials : If the lowest and highest coefficient A and B are of opposite signs, the polynomial admits a positive root. (Contributed by Thierry Arnoux, 19-Sep-2018.)

Hypotheses

Ref	Expression
signsply0.d	|- D = (deg ` F )
signsply0.c	|- C = (coeff ` F )
signsply0.b	|- B = ( C ` D )
signsply0.a	|- A = ( C ` 0 )
signsply0.1	|- ( ph -> F e. (Poly ` RR ) )
signsply0.2	|- ( ph -> F =/= 0p )
signsply0.3	|- ( ph -> ( A x. B ) < 0 )

Assertion

Ref	Expression
signsply0	|- ( ph -> E. z e. RR+ ( F ` z ) = 0 )

Distinct variable groups:   z, B   z, F   ph, z

Allowed substitution hints:   A( z)   C( z)   D( z)

Proof of Theorem signsply0

Dummy variables e d f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 792	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> d e. RR+ )
2		simpr 477	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) )
3		rpxr 11840	. . . . . . . 8 |- ( d e. RR+ -> d e. RR* )
4		xrleid 11983	. . . . . . . 8 |- ( d e. RR* -> d <_ d )
5	3, 4	syl 17	. . . . . . 7 |- ( d e. RR+ -> d <_ d )
6	5	ad2antlr 763	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> d <_ d )
7		id 22	. . . . . . 7 |- ( d e. RR+ -> d e. RR+ )
8		simpr 477	. . . . . . . . 9 |- ( ( d e. RR+ /\ f = d ) -> f = d )
9	8	breq2d 4665	. . . . . . . 8 |- ( ( d e. RR+ /\ f = d ) -> ( d <_ f <-> d <_ d ) )
10	8	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( d e. RR+ /\ f = d ) -> ( F ` f ) = ( F ` d ) )
11	8	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( d e. RR+ /\ f = d ) -> ( f ^ D ) = ( d ^ D ) )
12	10, 11	oveq12d 6668	. . . . . . . . . . 11 |- ( ( d e. RR+ /\ f = d ) -> ( ( F ` f ) / ( f ^ D ) ) = ( ( F ` d ) / ( d ^ D ) ) )
13	12	oveq1d 6665	. . . . . . . . . 10 |- ( ( d e. RR+ /\ f = d ) -> ( ( ( F ` f ) / ( f ^ D ) ) - B ) = ( ( ( F ` d ) / ( d ^ D ) ) - B ) )
14	13	fveq2d 6195	. . . . . . . . 9 |- ( ( d e. RR+ /\ f = d ) -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) = ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) )
15	14	breq1d 4663	. . . . . . . 8 |- ( ( d e. RR+ /\ f = d ) -> ( ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B <-> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) )
16	9, 15	imbi12d 334	. . . . . . 7 |- ( ( d e. RR+ /\ f = d ) -> ( ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) <-> ( d <_ d -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) ) )
17	7, 16	rspcdv 3312	. . . . . 6 |- ( d e. RR+ -> ( A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) -> ( d <_ d -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) ) )
18	1, 2, 6, 17	syl3c 66	. . . . 5 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B )
19		signsply0.1	. . . . . . . . . . . 12 |- ( ph -> F e. (Poly ` RR ) )
20	19	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> F e. (Poly ` RR ) )
21		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> d e. RR+ )
22	21	rpred 11872	. . . . . . . . . . 11 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> d e. RR )
23	20, 22	plyrecld 30626	. . . . . . . . . 10 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( F ` d ) e. RR )
24		signsply0.d	. . . . . . . . . . . . 13 |- D = (deg ` F )
25		dgrcl 23989	. . . . . . . . . . . . . 14 |- ( F e. (Poly ` RR ) -> (deg ` F ) e. NN0 )
26	19, 25	syl 17	. . . . . . . . . . . . 13 |- ( ph -> (deg ` F ) e. NN0 )
27	24, 26	syl5eqel 2705	. . . . . . . . . . . 12 |- ( ph -> D e. NN0 )
28	27	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> D e. NN0 )
29	22, 28	reexpcld 13025	. . . . . . . . . 10 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) e. RR )
30	21	rpcnd 11874	. . . . . . . . . . 11 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> d e. CC )
31	21	rpne0d 11877	. . . . . . . . . . 11 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> d =/= 0 )
32	27	nn0zd 11480	. . . . . . . . . . . 12 |- ( ph -> D e. ZZ )
33	32	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> D e. ZZ )
34	30, 31, 33	expne0d 13014	. . . . . . . . . 10 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) =/= 0 )
35	23, 29, 34	redivcld 10853	. . . . . . . . 9 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( F ` d ) / ( d ^ D ) ) e. RR )
36		signsply0.b	. . . . . . . . . . . 12 |- B = ( C ` D )
37		0re 10040	. . . . . . . . . . . . . 14 |- 0 e. RR
38		signsply0.c	. . . . . . . . . . . . . . 15 |- C = (coeff ` F )
39	38	coef2 23987	. . . . . . . . . . . . . 14 |- ( ( F e. (Poly ` RR ) /\ 0 e. RR ) -> C : NN0 --> RR )
40	37, 39	mpan2 707	. . . . . . . . . . . . 13 |- ( F e. (Poly ` RR ) -> C : NN0 --> RR )
41	40	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` RR ) /\ D e. NN0 ) -> ( C ` D ) e. RR )
42	36, 41	syl5eqel 2705	. . . . . . . . . . 11 |- ( ( F e. (Poly ` RR ) /\ D e. NN0 ) -> B e. RR )
43	19, 27, 42	syl2anc 693	. . . . . . . . . 10 |- ( ph -> B e. RR )
44	43	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> B e. RR )
45	44	renegcld 10457	. . . . . . . . 9 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> -u B e. RR )
46	35, 44, 45	absdifltd 14172	. . . . . . . 8 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B <-> ( ( B - -u B ) < ( ( F ` d ) / ( d ^ D ) ) /\ ( ( F ` d ) / ( d ^ D ) ) < ( B + -u B ) ) ) )
47	46	simplbda 654	. . . . . . 7 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( ( F ` d ) / ( d ^ D ) ) < ( B + -u B ) )
48	43	recnd 10068	. . . . . . . . . 10 |- ( ph -> B e. CC )
49	48	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> B e. CC )
50	49	negidd 10382	. . . . . . . 8 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( B + -u B ) = 0 )
51	50	adantr 481	. . . . . . 7 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( B + -u B ) = 0 )
52	47, 51	breqtrd 4679	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( ( F ` d ) / ( d ^ D ) ) < 0 )
53	21, 33	rpexpcld 13032	. . . . . . . . . 10 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) e. RR+ )
54	23, 53	ge0divd 11910	. . . . . . . . 9 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( 0 <_ ( F ` d ) <-> 0 <_ ( ( F ` d ) / ( d ^ D ) ) ) )
55	54	notbid 308	. . . . . . . 8 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( -. 0 <_ ( F ` d ) <-> -. 0 <_ ( ( F ` d ) / ( d ^ D ) ) ) )
56		0red 10041	. . . . . . . . 9 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> 0 e. RR )
57	23, 56	ltnled 10184	. . . . . . . 8 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( F ` d ) < 0 <-> -. 0 <_ ( F ` d ) ) )
58	35, 56	ltnled 10184	. . . . . . . 8 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( ( F ` d ) / ( d ^ D ) ) < 0 <-> -. 0 <_ ( ( F ` d ) / ( d ^ D ) ) ) )
59	55, 57, 58	3bitr4d 300	. . . . . . 7 |- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( F ` d ) < 0 <-> ( ( F ` d ) / ( d ^ D ) ) < 0 ) )
60	59	adantr 481	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( ( F ` d ) < 0 <-> ( ( F ` d ) / ( d ^ D ) ) < 0 ) )
61	52, 60	mpbird 247	. . . . 5 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( F ` d ) < 0 )
62	18, 61	syldan 487	. . . 4 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> ( F ` d ) < 0 )
63		0red 10041	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> 0 e. RR )
64		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> d e. RR+ )
65	64	rpred 11872	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> d e. RR )
66	64	rpgt0d 11875	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> 0 < d )
67		iccssre 12255	. . . . . . . 8 |- ( ( 0 e. RR /\ d e. RR ) -> ( 0 [,] d ) C_ RR )
68	37, 65, 67	sylancr 695	. . . . . . 7 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( 0 [,] d ) C_ RR )
69		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
70	68, 69	syl6ss 3615	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( 0 [,] d ) C_ CC )
71		plycn 24017	. . . . . . . 8 |- ( F e. (Poly ` RR ) -> F e. ( CC -cn-> CC ) )
72	19, 71	syl 17	. . . . . . 7 |- ( ph -> F e. ( CC -cn-> CC ) )
73	72	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> F e. ( CC -cn-> CC ) )
74	19	ad4antr 768	. . . . . . 7 |- ( ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) /\ x e. ( 0 [,] d ) ) -> F e. (Poly ` RR ) )
75	68	sselda 3603	. . . . . . 7 |- ( ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) /\ x e. ( 0 [,] d ) ) -> x e. RR )
76	74, 75	plyrecld 30626	. . . . . 6 |- ( ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) /\ x e. ( 0 [,] d ) ) -> ( F ` x ) e. RR )
77		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( F ` d ) < 0 )
78		simplll 798	. . . . . . . . 9 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ph )
79	78, 43	syl 17	. . . . . . . . . 10 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> B e. RR )
80		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ -u B e. RR+ ) -> -u B e. RR+ )
81	80	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> -u B e. RR+ )
82		negelrp 11864	. . . . . . . . . . 11 |- ( B e. RR -> ( -u B e. RR+ <-> B < 0 ) )
83	82	biimpa 501	. . . . . . . . . 10 |- ( ( B e. RR /\ -u B e. RR+ ) -> B < 0 )
84	79, 81, 83	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> B < 0 )
85		signsply0.a	. . . . . . . . . . . 12 |- A = ( C ` 0 )
86	19, 37, 39	sylancl 694	. . . . . . . . . . . . 13 |- ( ph -> C : NN0 --> RR )
87		0nn0 11307	. . . . . . . . . . . . . 14 |- 0 e. NN0
88	87	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 0 e. NN0 )
89	86, 88	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ph -> ( C ` 0 ) e. RR )
90	85, 89	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> A e. RR )
91		signsply0.3	. . . . . . . . . . 11 |- ( ph -> ( A x. B ) < 0 )
92	90, 43, 91	mul2lt0rlt0 11932	. . . . . . . . . 10 |- ( ( ph /\ B < 0 ) -> 0 < A )
93	92, 85	syl6breq 4694	. . . . . . . . 9 |- ( ( ph /\ B < 0 ) -> 0 < ( C ` 0 ) )
94	78, 84, 93	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> 0 < ( C ` 0 ) )
95	38	coefv0 24004	. . . . . . . . . 10 |- ( F e. (Poly ` RR ) -> ( F ` 0 ) = ( C ` 0 ) )
96	19, 95	syl 17	. . . . . . . . 9 |- ( ph -> ( F ` 0 ) = ( C ` 0 ) )
97	96	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( F ` 0 ) = ( C ` 0 ) )
98	94, 97	breqtrrd 4681	. . . . . . 7 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> 0 < ( F ` 0 ) )
99	77, 98	jca 554	. . . . . 6 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( ( F ` d ) < 0 /\ 0 < ( F ` 0 ) ) )
100	63, 65, 63, 66, 70, 73, 76, 99	ivth2 23224	. . . . 5 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> E. z e. ( 0 (,) d ) ( F ` z ) = 0 )
101		0le0 11110	. . . . . . . 8 |- 0 <_ 0
102		pnfge 11964	. . . . . . . . 9 |- ( d e. RR* -> d <_ +oo )
103	3, 102	syl 17	. . . . . . . 8 |- ( d e. RR+ -> d <_ +oo )
104		0xr 10086	. . . . . . . . 9 |- 0 e. RR*
105		pnfxr 10092	. . . . . . . . 9 |- +oo e. RR*
106		ioossioo 12265	. . . . . . . . 9 |- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 <_ 0 /\ d <_ +oo ) ) -> ( 0 (,) d ) C_ ( 0 (,) +oo ) )
107	104, 105, 106	mpanl12 718	. . . . . . . 8 |- ( ( 0 <_ 0 /\ d <_ +oo ) -> ( 0 (,) d ) C_ ( 0 (,) +oo ) )
108	101, 103, 107	sylancr 695	. . . . . . 7 |- ( d e. RR+ -> ( 0 (,) d ) C_ ( 0 (,) +oo ) )
109		ioorp 12251	. . . . . . 7 |- ( 0 (,) +oo ) = RR+
110	108, 109	syl6sseq 3651	. . . . . 6 |- ( d e. RR+ -> ( 0 (,) d ) C_ RR+ )
111		ssrexv 3667	. . . . . 6 |- ( ( 0 (,) d ) C_ RR+ -> ( E. z e. ( 0 (,) d ) ( F ` z ) = 0 -> E. z e. RR+ ( F ` z ) = 0 ) )
112	64, 110, 111	3syl 18	. . . . 5 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( E. z e. ( 0 (,) d ) ( F ` z ) = 0 -> E. z e. RR+ ( F ` z ) = 0 ) )
113	100, 112	mpd 15	. . . 4 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> E. z e. RR+ ( F ` z ) = 0 )
114	62, 113	syldan 487	. . 3 |- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> E. z e. RR+ ( F ` z ) = 0 )
115		plyf 23954	. . . . . . . . . . 11 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
116	19, 115	syl 17	. . . . . . . . . 10 |- ( ph -> F : CC --> CC )
117		ffn 6045	. . . . . . . . . 10 |- ( F : CC --> CC -> F Fn CC )
118	116, 117	syl 17	. . . . . . . . 9 |- ( ph -> F Fn CC )
119		ovex 6678	. . . . . . . . . . 11 |- ( x ^ D ) e. _V
120	119	rgenw 2924	. . . . . . . . . 10 |- A. x e. RR+ ( x ^ D ) e. _V
121		eqid 2622	. . . . . . . . . . 11 |- ( x e. RR+ |-> ( x ^ D ) ) = ( x e. RR+ |-> ( x ^ D ) )
122	121	fnmpt 6020	. . . . . . . . . 10 |- ( A. x e. RR+ ( x ^ D ) e. _V -> ( x e. RR+ |-> ( x ^ D ) ) Fn RR+ )
123	120, 122	mp1i 13	. . . . . . . . 9 |- ( ph -> ( x e. RR+ |-> ( x ^ D ) ) Fn RR+ )
124		cnex 10017	. . . . . . . . . 10 |- CC e. _V
125	124	a1i 11	. . . . . . . . 9 |- ( ph -> CC e. _V )
126		rpssre 11843	. . . . . . . . . . . 12 |- RR+ C_ RR
127	126, 69	sstri 3612	. . . . . . . . . . 11 |- RR+ C_ CC
128	124, 127	ssexi 4803	. . . . . . . . . 10 |- RR+ e. _V
129	128	a1i 11	. . . . . . . . 9 |- ( ph -> RR+ e. _V )
130		sseqin2 3817	. . . . . . . . . 10 |- ( RR+ C_ CC <-> ( CC i^i RR+ ) = RR+ )
131	127, 130	mpbi 220	. . . . . . . . 9 |- ( CC i^i RR+ ) = RR+
132		eqidd 2623	. . . . . . . . 9 |- ( ( ph /\ f e. CC ) -> ( F ` f ) = ( F ` f ) )
133		eqidd 2623	. . . . . . . . . 10 |- ( ( ph /\ f e. RR+ ) -> ( x e. RR+ |-> ( x ^ D ) ) = ( x e. RR+ |-> ( x ^ D ) ) )
134		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. RR+ ) /\ x = f ) -> x = f )
135	134	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ph /\ f e. RR+ ) /\ x = f ) -> ( x ^ D ) = ( f ^ D ) )
136		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ f e. RR+ ) -> f e. RR+ )
137		ovexd 6680	. . . . . . . . . 10 |- ( ( ph /\ f e. RR+ ) -> ( f ^ D ) e. _V )
138	133, 135, 136, 137	fvmptd 6288	. . . . . . . . 9 |- ( ( ph /\ f e. RR+ ) -> ( ( x e. RR+ |-> ( x ^ D ) ) ` f ) = ( f ^ D ) )
139	118, 123, 125, 129, 131, 132, 138	offval 6904	. . . . . . . 8 |- ( ph -> ( F oF / ( x e. RR+ |-> ( x ^ D ) ) ) = ( f e. RR+ |-> ( ( F ` f ) / ( f ^ D ) ) ) )
140		oveq1 6657	. . . . . . . . . . 11 |- ( x = f -> ( x ^ D ) = ( f ^ D ) )
141	140	cbvmptv 4750	. . . . . . . . . 10 |- ( x e. RR+ |-> ( x ^ D ) ) = ( f e. RR+ |-> ( f ^ D ) )
142	24, 38, 36, 141	signsplypnf 30627	. . . . . . . . 9 |- ( F e. (Poly ` RR ) -> ( F oF / ( x e. RR+ |-> ( x ^ D ) ) ) ~~> r B )
143	19, 142	syl 17	. . . . . . . 8 |- ( ph -> ( F oF / ( x e. RR+ |-> ( x ^ D ) ) ) ~~> r B )
144	139, 143	eqbrtrrd 4677	. . . . . . 7 |- ( ph -> ( f e. RR+ |-> ( ( F ` f ) / ( f ^ D ) ) ) ~~> r B )
145	116	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ f e. RR+ ) -> F : CC --> CC )
146	136	rpcnd 11874	. . . . . . . . . . 11 |- ( ( ph /\ f e. RR+ ) -> f e. CC )
147	145, 146	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ f e. RR+ ) -> ( F ` f ) e. CC )
148	27	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ f e. RR+ ) -> D e. NN0 )
149	146, 148	expcld 13008	. . . . . . . . . 10 |- ( ( ph /\ f e. RR+ ) -> ( f ^ D ) e. CC )
150	136	rpne0d 11877	. . . . . . . . . . 11 |- ( ( ph /\ f e. RR+ ) -> f =/= 0 )
151	32	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ f e. RR+ ) -> D e. ZZ )
152	146, 150, 151	expne0d 13014	. . . . . . . . . 10 |- ( ( ph /\ f e. RR+ ) -> ( f ^ D ) =/= 0 )
153	147, 149, 152	divcld 10801	. . . . . . . . 9 |- ( ( ph /\ f e. RR+ ) -> ( ( F ` f ) / ( f ^ D ) ) e. CC )
154	153	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. f e. RR+ ( ( F ` f ) / ( f ^ D ) ) e. CC )
155	126	a1i 11	. . . . . . . 8 |- ( ph -> RR+ C_ RR )
156		1red 10055	. . . . . . . 8 |- ( ph -> 1 e. RR )
157	154, 155, 48, 156	rlim3 14229	. . . . . . 7 |- ( ph -> ( ( f e. RR+ |-> ( ( F ` f ) / ( f ^ D ) ) ) ~~> r B <-> A. e e. RR+ E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) )
158	144, 157	mpbid 222	. . . . . 6 |- ( ph -> A. e e. RR+ E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) )
159		0lt1 10550	. . . . . . . . . 10 |- 0 < 1
160		pnfge 11964	. . . . . . . . . . 11 |- ( +oo e. RR* -> +oo <_ +oo )
161	105, 160	ax-mp 5	. . . . . . . . . 10 |- +oo <_ +oo
162		icossioo 12264	. . . . . . . . . 10 |- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 < 1 /\ +oo <_ +oo ) ) -> ( 1 [,) +oo ) C_ ( 0 (,) +oo ) )
163	104, 105, 159, 161, 162	mp4an 709	. . . . . . . . 9 |- ( 1 [,) +oo ) C_ ( 0 (,) +oo )
164	163, 109	sseqtri 3637	. . . . . . . 8 |- ( 1 [,) +oo ) C_ RR+
165		ssrexv 3667	. . . . . . . 8 |- ( ( 1 [,) +oo ) C_ RR+ -> ( E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) )
166	164, 165	ax-mp 5	. . . . . . 7 |- ( E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) )
167	166	ralimi 2952	. . . . . 6 |- ( A. e e. RR+ E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) )
168	158, 167	syl 17	. . . . 5 |- ( ph -> A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) )
169	168	adantr 481	. . . 4 |- ( ( ph /\ -u B e. RR+ ) -> A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) )
170		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ -u B e. RR+ ) /\ e = -u B ) -> e = -u B )
171	170	breq2d 4665	. . . . . . 7 |- ( ( ( ph /\ -u B e. RR+ ) /\ e = -u B ) -> ( ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e <-> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) )
172	171	imbi2d 330	. . . . . 6 |- ( ( ( ph /\ -u B e. RR+ ) /\ e = -u B ) -> ( ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) <-> ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) )
173	172	rexralbidv 3058	. . . . 5 |- ( ( ( ph /\ -u B e. RR+ ) /\ e = -u B ) -> ( E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) <-> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) )
174	80, 173	rspcdv 3312	. . . 4 |- ( ( ph /\ -u B e. RR+ ) -> ( A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) )
175	169, 174	mpd 15	. . 3 |- ( ( ph /\ -u B e. RR+ ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) )
176	114, 175	r19.29a 3078	. 2 |- ( ( ph /\ -u B e. RR+ ) -> E. z e. RR+ ( F ` z ) = 0 )
177		simplr 792	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> d e. RR+ )
178		simpr 477	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) )
179	5	ad2antlr 763	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> d <_ d )
180	14	breq1d 4663	. . . . . . . 8 |- ( ( d e. RR+ /\ f = d ) -> ( ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B <-> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) )
181	9, 180	imbi12d 334	. . . . . . 7 |- ( ( d e. RR+ /\ f = d ) -> ( ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) <-> ( d <_ d -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) ) )
182	7, 181	rspcdv 3312	. . . . . 6 |- ( d e. RR+ -> ( A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) -> ( d <_ d -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) ) )
183	177, 178, 179, 182	syl3c 66	. . . . 5 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B )
184	48	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> B e. CC )
185	184	subidd 10380	. . . . . . . 8 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( B - B ) = 0 )
186	185	adantr 481	. . . . . . 7 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> ( B - B ) = 0 )
187	19	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> F e. (Poly ` RR ) )
188	126	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ B e. RR+ ) -> RR+ C_ RR )
189	188	sselda 3603	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> d e. RR )
190	187, 189	plyrecld 30626	. . . . . . . . . 10 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( F ` d ) e. RR )
191	27	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> D e. NN0 )
192	189, 191	reexpcld 13025	. . . . . . . . . 10 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) e. RR )
193	189	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> d e. CC )
194		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> d e. RR+ )
195	194	rpne0d 11877	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> d =/= 0 )
196	32	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> D e. ZZ )
197	193, 195, 196	expne0d 13014	. . . . . . . . . 10 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) =/= 0 )
198	190, 192, 197	redivcld 10853	. . . . . . . . 9 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( ( F ` d ) / ( d ^ D ) ) e. RR )
199	43	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> B e. RR )
200	198, 199, 199	absdifltd 14172	. . . . . . . 8 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B <-> ( ( B - B ) < ( ( F ` d ) / ( d ^ D ) ) /\ ( ( F ` d ) / ( d ^ D ) ) < ( B + B ) ) ) )
201	200	simprbda 653	. . . . . . 7 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> ( B - B ) < ( ( F ` d ) / ( d ^ D ) ) )
202	186, 201	eqbrtrrd 4677	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> 0 < ( ( F ` d ) / ( d ^ D ) ) )
203	194, 196	rpexpcld 13032	. . . . . . . 8 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) e. RR+ )
204	190, 203	gt0divd 11909	. . . . . . 7 |- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( 0 < ( F ` d ) <-> 0 < ( ( F ` d ) / ( d ^ D ) ) ) )
205	204	adantr 481	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> ( 0 < ( F ` d ) <-> 0 < ( ( F ` d ) / ( d ^ D ) ) ) )
206	202, 205	mpbird 247	. . . . 5 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> 0 < ( F ` d ) )
207	183, 206	syldan 487	. . . 4 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> 0 < ( F ` d ) )
208		0red 10041	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> 0 e. RR )
209		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> d e. RR+ )
210	209	rpred 11872	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> d e. RR )
211	209	rpgt0d 11875	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> 0 < d )
212	37, 210, 67	sylancr 695	. . . . . . 7 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( 0 [,] d ) C_ RR )
213	212, 69	syl6ss 3615	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( 0 [,] d ) C_ CC )
214	72	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> F e. ( CC -cn-> CC ) )
215	19	ad4antr 768	. . . . . . 7 |- ( ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) /\ x e. ( 0 [,] d ) ) -> F e. (Poly ` RR ) )
216	212	sselda 3603	. . . . . . 7 |- ( ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) /\ x e. ( 0 [,] d ) ) -> x e. RR )
217	215, 216	plyrecld 30626	. . . . . 6 |- ( ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) /\ x e. ( 0 [,] d ) ) -> ( F ` x ) e. RR )
218	96	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( F ` 0 ) = ( C ` 0 ) )
219		simplll 798	. . . . . . . . . 10 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ph )
220		simpr1 1067	. . . . . . . . . . . 12 |- ( ( ph /\ ( B e. RR+ /\ d e. RR+ /\ 0 < ( F ` d ) ) ) -> B e. RR+ )
221	220	rpgt0d 11875	. . . . . . . . . . 11 |- ( ( ph /\ ( B e. RR+ /\ d e. RR+ /\ 0 < ( F ` d ) ) ) -> 0 < B )
222	221	3anassrs 1290	. . . . . . . . . 10 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> 0 < B )
223	90, 43, 91	mul2lt0rgt0 11933	. . . . . . . . . 10 |- ( ( ph /\ 0 < B ) -> A < 0 )
224	219, 222, 223	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> A < 0 )
225	85, 224	syl5eqbrr 4689	. . . . . . . 8 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( C ` 0 ) < 0 )
226	218, 225	eqbrtrd 4675	. . . . . . 7 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( F ` 0 ) < 0 )
227		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> 0 < ( F ` d ) )
228	226, 227	jca 554	. . . . . 6 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( ( F ` 0 ) < 0 /\ 0 < ( F ` d ) ) )
229	208, 210, 208, 211, 213, 214, 217, 228	ivth 23223	. . . . 5 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> E. z e. ( 0 (,) d ) ( F ` z ) = 0 )
230	209, 110, 111	3syl 18	. . . . 5 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( E. z e. ( 0 (,) d ) ( F ` z ) = 0 -> E. z e. RR+ ( F ` z ) = 0 ) )
231	229, 230	mpd 15	. . . 4 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> E. z e. RR+ ( F ` z ) = 0 )
232	207, 231	syldan 487	. . 3 |- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> E. z e. RR+ ( F ` z ) = 0 )
233	168	adantr 481	. . . 4 |- ( ( ph /\ B e. RR+ ) -> A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) )
234		simpr 477	. . . . 5 |- ( ( ph /\ B e. RR+ ) -> B e. RR+ )
235		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ B e. RR+ ) /\ e = B ) -> e = B )
236	235	breq2d 4665	. . . . . . 7 |- ( ( ( ph /\ B e. RR+ ) /\ e = B ) -> ( ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e <-> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) )
237	236	imbi2d 330	. . . . . 6 |- ( ( ( ph /\ B e. RR+ ) /\ e = B ) -> ( ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) <-> ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) )
238	237	rexralbidv 3058	. . . . 5 |- ( ( ( ph /\ B e. RR+ ) /\ e = B ) -> ( E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) <-> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) )
239	234, 238	rspcdv 3312	. . . 4 |- ( ( ph /\ B e. RR+ ) -> ( A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) )
240	233, 239	mpd 15	. . 3 |- ( ( ph /\ B e. RR+ ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) )
241	232, 240	r19.29a 3078	. 2 |- ( ( ph /\ B e. RR+ ) -> E. z e. RR+ ( F ` z ) = 0 )
242		signsply0.2	. . . . 5 |- ( ph -> F =/= 0p )
243	24, 38	dgreq0 24021	. . . . . . 7 |- ( F e. (Poly ` RR ) -> ( F = 0p <-> ( C ` D ) = 0 ) )
244	19, 243	syl 17	. . . . . 6 |- ( ph -> ( F = 0p <-> ( C ` D ) = 0 ) )
245	244	necon3bid 2838	. . . . 5 |- ( ph -> ( F =/= 0p <-> ( C ` D ) =/= 0 ) )
246	242, 245	mpbid 222	. . . 4 |- ( ph -> ( C ` D ) =/= 0 )
247	36	neeq1i 2858	. . . 4 |- ( B =/= 0 <-> ( C ` D ) =/= 0 )
248	246, 247	sylibr 224	. . 3 |- ( ph -> B =/= 0 )
249		rpneg 11863	. . . . 5 |- ( ( B e. RR /\ B =/= 0 ) -> ( B e. RR+ <-> -. -u B e. RR+ ) )
250	249	biimprd 238	. . . 4 |- ( ( B e. RR /\ B =/= 0 ) -> ( -. -u B e. RR+ -> B e. RR+ ) )
251	250	orrd 393	. . 3 |- ( ( B e. RR /\ B =/= 0 ) -> ( -u B e. RR+ \/ B e. RR+ ) )
252	43, 248, 251	syl2anc 693	. 2 |- ( ph -> ( -u B e. RR+ \/ B e. RR+ ) )
253	176, 241, 252	mpjaodan 827	1 |- ( ph -> E. z e. RR+ ( F ` z ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NN0cn0 11292   ZZcz 11377   RR+crp 11832   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   ^cexp 12860   abscabs 13974   ~~> r crli 14216   -cn->ccncf 22679   0pc0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943

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: (None)