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Theorem signsply0 26950
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.
StepHypRef Expression
1 rpxr 10996 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  e. 
RR* )
2 xrleid 11125 . . . . . . . 8  |-  ( d  e.  RR*  ->  d  <_ 
d )
31, 2syl 16 . . . . . . 7  |-  ( d  e.  RR+  ->  d  <_ 
d )
43ad2antlr 726 . . . . . 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 )
5 id 22 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  e.  RR+ )
65ad2antlr 726 . . . . . . 7  |-  ( ( ( ( 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+ )
7 simpr 461 . . . . . . 7  |-  ( ( ( ( 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
) )
8 simpr 461 . . . . . . . . . . 11  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
f  =  d )
98breq2d 4302 . . . . . . . . . 10  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( d  <_  f  <->  d  <_  d ) )
108fveq2d 5693 . . . . . . . . . . . . . 14  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( F `  f
)  =  ( F `
 d ) )
118oveq1d 6104 . . . . . . . . . . . . . 14  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( f ^ D
)  =  ( d ^ D ) )
1210, 11oveq12d 6107 . . . . . . . . . . . . 13  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( F `  f )  /  (
f ^ D ) )  =  ( ( F `  d )  /  ( d ^ D ) ) )
1312oveq1d 6104 . . . . . . . . . . . 12  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( ( F `
 f )  / 
( f ^ D
) )  -  B
)  =  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )
1413fveq2d 5693 . . . . . . . . . . 11  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  =  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) ) )
1514breq1d 4300 . . . . . . . . . 10  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  -u B  <->  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  -u B ) )
169, 15imbi12d 320 . . . . . . . . 9  |-  ( ( 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
) ) )
175, 16rspcdv 3074 . . . . . . . 8  |-  ( 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 ) ) )
1817imp 429 . . . . . . 7  |-  ( ( 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
) )
196, 7, 18syl2anc 661 . . . . . 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  ->  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  -u B ) )
204, 19mpd 15 . . . . 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 )
21 signsply0.1 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  (Poly `  RR ) )
2221ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  F  e.  (Poly `  RR )
)
23 rpssre 10999 . . . . . . . . . . . 12  |-  RR+  C_  RR
24 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR+ )
2523, 24sseldi 3352 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR )
2622, 25plyrecld 26948 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( F `  d )  e.  RR )
27 signsply0.d . . . . . . . . . . . . 13  |-  D  =  (deg `  F )
28 dgrcl 21699 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
2921, 28syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  F )  e.  NN0 )
3027, 29syl5eqel 2525 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  NN0 )
3130ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  NN0 )
3225, 31reexpcld 12023 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR )
33 ax-resscn 9337 . . . . . . . . . . . 12  |-  RR  C_  CC
3433, 25sseldi 3352 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  CC )
3524rpne0d 11030 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  =/=  0 )
3630nn0zd 10743 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ZZ )
3736ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  ZZ )
3834, 35, 37expne0d 12012 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  =/=  0 )
3926, 32, 38redivcld 10157 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  /  ( d ^ D ) )  e.  RR )
40 signsply0.b . . . . . . . . . . . 12  |-  B  =  ( C `  D
)
41 0re 9384 . . . . . . . . . . . . . 14  |-  0  e.  RR
42 signsply0.c . . . . . . . . . . . . . . 15  |-  C  =  (coeff `  F )
4342coef2 21697 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  C : NN0 --> RR )
4441, 43mpan2 671 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> RR )
4544ffvelrnda 5841 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  D  e. 
NN0 )  ->  ( C `  D )  e.  RR )
4640, 45syl5eqel 2525 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  D  e. 
NN0 )  ->  B  e.  RR )
4721, 30, 46syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
4847ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  RR )
4948renegcld 9773 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  -u B  e.  RR )
5039, 48, 49absdifltd 12918 . . . . . . . 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 ) ) ) )
5150simplbda 624 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( ( F `  d )  /  ( d ^ D ) )  < 
( B  +  -u B ) )
5233, 47sseldi 3352 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
5352ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  CC )
5453negidd 9707 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( B  +  -u B )  =  0 )
5554adantr 465 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( B  +  -u B )  =  0 )
5651, 55breqtrd 4314 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( ( F `  d )  /  ( d ^ D ) )  <  0 )
5724, 37rpexpcld 12029 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR+ )
5826, 57ge0divd 11059 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
0  <_  ( F `  d )  <->  0  <_  ( ( F `  d
)  /  ( d ^ D ) ) ) )
5958notbid 294 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( -.  0  <_  ( F `
 d )  <->  -.  0  <_  ( ( F `  d )  /  (
d ^ D ) ) ) )
6041a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  0  e.  RR )
6126, 60ltnled 9519 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  <  0  <->  -.  0  <_  ( F `  d
) ) )
6239, 60ltnled 9519 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( ( F `  d )  /  (
d ^ D ) )  <  0  <->  -.  0  <_  ( ( F `
 d )  / 
( d ^ D
) ) ) )
6359, 61, 623bitr4d 285 . . . . . . 7  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  <  0  <->  ( ( F `  d )  /  ( d ^ D ) )  <  0 ) )
6463adantr 465 . . . . . 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
) )
6556, 64mpbird 232 . . . . 5  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( F `  d )  <  0
)
6620, 65syldan 470 . . . 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 )
6741a1i 11 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  e.  RR )
68 simplr 754 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  d  e.  RR+ )
6923, 68sseldi 3352 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  d  e.  RR )
7068rpgt0d 11028 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  d
)
71 iccssre 11375 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  d  e.  RR )  ->  ( 0 [,] d
)  C_  RR )
7267, 69, 71syl2anc 661 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( 0 [,] d )  C_  RR )
7372, 33syl6ss 3366 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( 0 [,] d )  C_  CC )
74 plycn 21726 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  F  e.  ( CC -cn-> CC ) )
7521, 74syl 16 . . . . . . 7  |-  ( ph  ->  F  e.  ( CC
-cn-> CC ) )
7675ad3antrrr 729 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  F  e.  ( CC -cn-> CC ) )
7721ad4antr 731 . . . . . . 7  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  F  e.  (Poly `  RR ) )
7872sselda 3354 . . . . . . 7  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  x  e.  RR )
7977, 78plyrecld 26948 . . . . . 6  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  ( F `  x )  e.  RR )
80 simpr 461 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( F `  d )  <  0
)
81 simplll 757 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ph )
8281, 47syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  B  e.  RR )
83 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  -u B  e.  RR+ )  ->  -u B  e.  RR+ )
8483ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  -u B  e.  RR+ )
85 negelrp 26035 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  ( -u B  e.  RR+  <->  B  <  0 ) )
8685biimpa 484 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  -u B  e.  RR+ )  ->  B  <  0 )
8782, 84, 86syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  B  <  0
)
88 signsply0.a . . . . . . . . . . . 12  |-  A  =  ( C `  0
)
8921, 41, 43sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  C : NN0 --> RR )
90 0nn0 10592 . . . . . . . . . . . . . 14  |-  0  e.  NN0
9190a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  NN0 )
9289, 91ffvelrnd 5842 . . . . . . . . . . . 12  |-  ( ph  ->  ( C `  0
)  e.  RR )
9388, 92syl5eqel 2525 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
94 signsply0.3 . . . . . . . . . . 11  |-  ( ph  ->  ( A  x.  B
)  <  0 )
9593, 47, 94mul2lt0rlt0 26036 . . . . . . . . . 10  |-  ( (
ph  /\  B  <  0 )  ->  0  <  A )
9695, 88syl6breq 4329 . . . . . . . . 9  |-  ( (
ph  /\  B  <  0 )  ->  0  <  ( C `  0
) )
9781, 87, 96syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  ( C `  0 )
)
9842coefv0 21713 . . . . . . . . . 10  |-  ( F  e.  (Poly `  RR )  ->  ( F ` 
0 )  =  ( C `  0 ) )
9921, 98syl 16 . . . . . . . . 9  |-  ( ph  ->  ( F `  0
)  =  ( C `
 0 ) )
10099ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( F ` 
0 )  =  ( C `  0 ) )
10197, 100breqtrrd 4316 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  ( F `  0 )
)
10280, 101jca 532 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( ( F `
 d )  <  0  /\  0  < 
( F `  0
) ) )
10367, 69, 67, 70, 73, 76, 79, 102ivth2 20937 . . . . 5  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  E. z  e.  ( 0 (,) d ) ( F `  z
)  =  0 )
1045ad2antlr 726 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  d  e.  RR+ )
10541leidi 9872 . . . . . . . 8  |-  0  <_  0
106 pnfge 11108 . . . . . . . . 9  |-  ( d  e.  RR*  ->  d  <_ +oo )
1071, 106syl 16 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  <_ +oo )
108 ressxr 9425 . . . . . . . . . 10  |-  RR  C_  RR*
109108, 41sselii 3351 . . . . . . . . 9  |-  0  e.  RR*
110 pnfxr 11090 . . . . . . . . 9  |- +oo  e.  RR*
111 ioossioo 26061 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <_  0  /\  d  <_ +oo ) )  -> 
( 0 (,) d
)  C_  ( 0 (,) +oo ) )
112109, 110, 111mpanl12 682 . . . . . . . 8  |-  ( ( 0  <_  0  /\  d  <_ +oo )  ->  (
0 (,) d ) 
C_  ( 0 (,) +oo ) )
113105, 107, 112sylancr 663 . . . . . . 7  |-  ( d  e.  RR+  ->  ( 0 (,) d )  C_  ( 0 (,) +oo ) )
114 ioorp 11371 . . . . . . 7  |-  ( 0 (,) +oo )  = 
RR+
115113, 114syl6sseq 3400 . . . . . 6  |-  ( d  e.  RR+  ->  ( 0 (,) d )  C_  RR+ )
116 ssrexv 3415 . . . . . 6  |-  ( ( 0 (,) d ) 
C_  RR+  ->  ( E. z  e.  ( 0 (,) d ) ( F `  z )  =  0  ->  E. z  e.  RR+  ( F `  z )  =  0 ) )
117104, 115, 1163syl 20 . . . . 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 ) )
118103, 117mpd 15 . . . 4  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
11966, 118syldan 470 . . 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 )
120 oveq1 6096 . . . . . . . . . . 11  |-  ( x  =  f  ->  (
x ^ D )  =  ( f ^ D ) )
121120cbvmptv 4381 . . . . . . . . . 10  |-  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( f  e.  RR+  |->  ( f ^ D ) )
12227, 42, 40, 121signsplypnf 26949 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  ~~> r  B )
12321, 122syl 16 . . . . . . . 8  |-  ( ph  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  ~~> r  B )
124 plyf 21664 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
12521, 124syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F : CC --> CC )
126 ffn 5557 . . . . . . . . . . 11  |-  ( F : CC --> CC  ->  F  Fn  CC )
127125, 126syl 16 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  CC )
128 ovex 6114 . . . . . . . . . . . 12  |-  ( x ^ D )  e. 
_V
129128rgenw 2781 . . . . . . . . . . 11  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
130 eqid 2441 . . . . . . . . . . . 12  |-  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( x  e.  RR+  |->  ( x ^ D ) )
131130fnmpt 5535 . . . . . . . . . . 11  |-  ( A. x  e.  RR+  ( x ^ D )  e. 
_V  ->  ( x  e.  RR+  |->  ( x ^ D ) )  Fn  RR+ )
132129, 131mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  RR+  |->  ( x ^ D
) )  Fn  RR+ )
133 cnex 9361 . . . . . . . . . . 11  |-  CC  e.  _V
134133a1i 11 . . . . . . . . . 10  |-  ( ph  ->  CC  e.  _V )
13523, 33sstri 3363 . . . . . . . . . . . 12  |-  RR+  C_  CC
136133, 135ssexi 4435 . . . . . . . . . . 11  |-  RR+  e.  _V
137136a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
RR+  e.  _V )
138 dfss1 3553 . . . . . . . . . . 11  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
139135, 138mpbi 208 . . . . . . . . . 10  |-  ( CC 
i^i  RR+ )  =  RR+
140 eqidd 2442 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  CC )  ->  ( F `
 f )  =  ( F `  f
) )
141130a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( x  e.  RR+  |->  ( x ^ D
) ) )
142 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  RR+ )  /\  x  =  f )  ->  x  =  f )
143142oveq1d 6104 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  RR+ )  /\  x  =  f )  -> 
( x ^ D
)  =  ( f ^ D ) )
144 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  e.  RR+ )
145 ovex 6114 . . . . . . . . . . . 12  |-  ( f ^ D )  e. 
_V
146145a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  e.  _V )
147141, 143, 144, 146fvmptd 5777 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( (
x  e.  RR+  |->  ( x ^ D ) ) `
 f )  =  ( f ^ D
) )
148127, 132, 134, 137, 139, 140, 147offval 6325 . . . . . . . . 9  |-  ( ph  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  =  ( f  e.  RR+  |->  ( ( F `
 f )  / 
( f ^ D
) ) ) )
149148breq1d 4300 . . . . . . . 8  |-  ( ph  ->  ( ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  ~~> r  B  <->  ( f  e.  RR+  |->  ( ( F `
 f )  / 
( f ^ D
) ) )  ~~> r  B
) )
150123, 149mpbid 210 . . . . . . 7  |-  ( ph  ->  ( f  e.  RR+  |->  ( ( F `  f )  /  (
f ^ D ) ) )  ~~> r  B
)
151125adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  F : CC
--> CC )
152135, 144sseldi 3352 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  e.  CC )
153151, 152ffvelrnd 5842 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( F `  f )  e.  CC )
15430adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  D  e.  NN0 )
155152, 154expcld 12006 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  e.  CC )
156144rpne0d 11030 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  =/=  0 )
15736adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  D  e.  ZZ )
158152, 156, 157expne0d 12012 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  =/=  0
)
159153, 155, 158divcld 10105 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( ( F `  f )  /  ( f ^ D ) )  e.  CC )
160159ralrimiva 2797 . . . . . . . 8  |-  ( ph  ->  A. f  e.  RR+  ( ( F `  f )  /  (
f ^ D ) )  e.  CC )
16123a1i 11 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
162 1re 9383 . . . . . . . . 9  |-  1  e.  RR
163162a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
164160, 161, 52, 163rlim3 12974 . . . . . . 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 ) ) )
165150, 164mpbid 210 . . . . . 6  |-  ( ph  ->  A. e  e.  RR+  E. d  e.  ( 1 [,) +oo ) A. f  e.  RR+  ( d  <_  f  ->  ( abs `  ( ( ( F `  f )  /  ( f ^ D ) )  -  B ) )  < 
e ) )
166 0lt1 9860 . . . . . . . . . 10  |-  0  <  1
167 pnfge 11108 . . . . . . . . . . 11  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
168110, 167ax-mp 5 . . . . . . . . . 10  |- +oo  <_ +oo
169 icossioo 26060 . . . . . . . . . 10  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <  1  /\ +oo 
<_ +oo ) )  -> 
( 1 [,) +oo )  C_  ( 0 (,) +oo ) )
170109, 110, 166, 168, 169mp4an 673 . . . . . . . . 9  |-  ( 1 [,) +oo )  C_  ( 0 (,) +oo )
171170, 114sseqtri 3386 . . . . . . . 8  |-  ( 1 [,) +oo )  C_  RR+
172 ssrexv 3415 . . . . . . . 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 ) ) )
173171, 172ax-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 ) )
174173ralimi 2789 . . . . . 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
) )
175165, 174syl 16 . . . . 5  |-  ( ph  ->  A. e  e.  RR+  E. d  e.  RR+  A. f  e.  RR+  ( d  <_ 
f  ->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  e
) )
176175adantr 465 . . . 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 ) )
177 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  e  =  -u B )  -> 
e  =  -u B
)
178177breq2d 4302 . . . . . . 7  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  e  =  -u B )  -> 
( ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  e  <->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  -u B
) )
179178imbi2d 316 . . . . . 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
) ) )
180179rexralbidv 2757 . . . . 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
) ) )
18183, 180rspcdv 3074 . . . 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 ) ) )
182176, 181mpd 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
) )
183119, 182r19.29a 2860 . 2  |-  ( (
ph  /\  -u B  e.  RR+ )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
1845ad2antlr 726 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  A. f  e.  RR+  (
d  <_  f  ->  ( abs `  ( ( ( F `  f
)  /  ( f ^ D ) )  -  B ) )  <  B ) )  ->  d  e.  RR+ )
185184, 3syl 16 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  A. f  e.  RR+  (
d  <_  f  ->  ( abs `  ( ( ( F `  f
)  /  ( f ^ D ) )  -  B ) )  <  B ) )  ->  d  <_  d
)
186 simpr 461 . . . . . . 7  |-  ( ( ( ( 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 ) )
18714breq1d 4300 . . . . . . . . . 10  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  B  <->  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) )  <  B
) )
1889, 187imbi12d 320 . . . . . . . . 9  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( d  <_ 
f  ->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  B
)  <->  ( d  <_ 
d  ->  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) )  <  B
) ) )
1895, 188rspcdv 3074 . . . . . . . 8  |-  ( 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 ) ) )
190189imp 429 . . . . . . 7  |-  ( ( 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 ) )
191184, 186, 190syl2anc 661 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  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
) )
192185, 191mpd 15 . . . . 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 )
19352ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  CC )
194193subidd 9705 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( B  -  B )  =  0 )
195194adantr 465 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
( B  -  B
)  =  0 )
19621ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  F  e.  (Poly `  RR )
)
19723a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  RR+ )  ->  RR+  C_  RR )
198197sselda 3354 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR )
199196, 198plyrecld 26948 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( F `  d )  e.  RR )
20030ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  NN0 )
201198, 200reexpcld 12023 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR )
20233, 198sseldi 3352 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  CC )
203 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR+ )
204203rpne0d 11030 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  =/=  0 )
20536ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  ZZ )
206202, 204, 205expne0d 12012 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  =/=  0 )
207199, 201, 206redivcld 10157 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  /  ( d ^ D ) )  e.  RR )
20847ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  RR )
209207, 208, 208absdifltd 12918 . . . . . . . 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
) ) ) )
210209simprbda 623 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
( B  -  B
)  <  ( ( F `  d )  /  ( d ^ D ) ) )
211195, 210eqbrtrrd 4312 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
0  <  ( ( F `  d )  /  ( d ^ D ) ) )
212203rpgt0d 11028 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  0  <  d )
213 expgt0 11895 . . . . . . . . . 10  |-  ( ( d  e.  RR  /\  D  e.  ZZ  /\  0  <  d )  ->  0  <  ( d ^ D
) )
214198, 205, 212, 213syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  0  <  ( d ^ D
) )
215201, 214elrpd 11023 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR+ )
216199, 215gt0divd 11058 . . . . . . 7  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
0  <  ( F `  d )  <->  0  <  ( ( F `  d
)  /  ( d ^ D ) ) ) )
217216adantr 465 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
( 0  <  ( F `  d )  <->  0  <  ( ( F `
 d )  / 
( d ^ D
) ) ) )
218211, 217mpbird 232 . . . . 5  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
0  <  ( F `  d ) )
219192, 218syldan 470 . . . 4  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  A. f  e.  RR+  (
d  <_  f  ->  ( abs `  ( ( ( F `  f
)  /  ( f ^ D ) )  -  B ) )  <  B ) )  ->  0  <  ( F `  d )
)
22041a1i 11 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  e.  RR )
221 simplr 754 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
d  e.  RR+ )
22223, 221sseldi 3352 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
d  e.  RR )
223221rpgt0d 11028 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  d )
224220, 222, 71syl2anc 661 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( 0 [,] d
)  C_  RR )
225224, 33syl6ss 3366 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( 0 [,] d
)  C_  CC )
22675ad3antrrr 729 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  F  e.  ( CC -cn-> CC ) )
22721ad4antr 731 . . . . . . 7  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  F  e.  (Poly `  RR ) )
228224sselda 3354 . . . . . . 7  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  x  e.  RR )
229227, 228plyrecld 26948 . . . . . 6  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  ( F `  x )  e.  RR )
23099ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( F `  0
)  =  ( C `
 0 ) )
231 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  ph )
232 simpr1 994 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( B  e.  RR+  /\  d  e.  RR+  /\  0  <  ( F `  d )
) )  ->  B  e.  RR+ )
233232rpgt0d 11028 . . . . . . . . . . 11  |-  ( (
ph  /\  ( B  e.  RR+  /\  d  e.  RR+  /\  0  <  ( F `  d )
) )  ->  0  <  B )
2342333anassrs 1209 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  B )
23593, 47, 94mul2lt0rgt0 26037 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  B )  ->  A  <  0 )
236231, 234, 235syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  A  <  0 )
23788, 236syl5eqbrr 4324 . . . . . . . 8  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( C `  0
)  <  0 )
238230, 237eqbrtrd 4310 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( F `  0
)  <  0 )
239 simpr 461 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  ( F `  d ) )
240238, 239jca 532 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( ( F ` 
0 )  <  0  /\  0  <  ( F `
 d ) ) )
241220, 222, 220, 223, 225, 226, 229, 240ivth 20936 . . . . 5  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  E. z  e.  (
0 (,) d ) ( F `  z
)  =  0 )
242221, 115, 1163syl 20 . . . . 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 ) )
243241, 242mpd 15 . . . 4  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
244219, 243syldan 470 . . 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 )
245175adantr 465 . . . 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 ) )
246 simpr 461 . . . . 5  |-  ( (
ph  /\  B  e.  RR+ )  ->  B  e.  RR+ )
247 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  e  =  B )  ->  e  =  B )
248247breq2d 4302 . . . . . . 7  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  e  =  B )  ->  (
( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  e  <->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  B
) )
249248imbi2d 316 . . . . . 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 ) ) )
250249rexralbidv 2757 . . . . 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 ) ) )
251246, 250rspcdv 3074 . . . 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 ) ) )
252245, 251mpd 15 . . 3  |-  ( (
ph  /\  B  e.  RR+ )  ->  E. d  e.  RR+  A. f  e.  RR+  ( d  <_  f  ->  ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  B ) )
253244, 252r19.29a 2860 . 2  |-  ( (
ph  /\  B  e.  RR+ )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
254 signsply0.2 . . . . 5  |-  ( ph  ->  F  =/=  0p )
25527, 42dgreq0 21730 . . . . . . 7  |-  ( F  e.  (Poly `  RR )  ->  ( F  =  0p  <->  ( C `  D )  =  0 ) )
25621, 255syl 16 . . . . . 6  |-  ( ph  ->  ( F  =  0p  <->  ( C `  D )  =  0 ) )
257256necon3bid 2641 . . . . 5  |-  ( ph  ->  ( F  =/=  0p 
<->  ( C `  D
)  =/=  0 ) )
258254, 257mpbid 210 . . . 4  |-  ( ph  ->  ( C `  D
)  =/=  0 )
25940neeq1i 2616 . . . 4  |-  ( B  =/=  0  <->  ( C `  D )  =/=  0
)
260258, 259sylibr 212 . . 3  |-  ( ph  ->  B  =/=  0 )
261 rpneg 11018 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( B  e.  RR+  <->  -.  -u B  e.  RR+ )
)
262261biimprd 223 . . . 4  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( -.  -u B  e.  RR+  ->  B  e.  RR+ ) )
263262orrd 378 . . 3  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( -u B  e.  RR+  \/  B  e.  RR+ )
)
26447, 260, 263syl2anc 661 . 2  |-  ( ph  ->  ( -u B  e.  RR+  \/  B  e.  RR+ ) )
265183, 253, 264mpjaodan 784 1  |-  ( ph  ->  E. z  e.  RR+  ( F `  z )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970    i^i cin 3325    C_ wss 3326   class class class wbr 4290    e. cmpt 4348    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089    oFcof 6316   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281    + caddc 9283    x. cmul 9285   +oocpnf 9413   RR*cxr 9415    < clt 9416    <_ cle 9417    - cmin 9593   -ucneg 9594    / cdiv 9991   NN0cn0 10577   ZZcz 10644   RR+crp 10989   (,)cioo 11298   [,)cico 11300   [,]cicc 11301   ^cexp 11863   abscabs 12721    ~~> r crli 12961   -cn->ccncf 20450   0pc0p 21145  Polycply 21650  coeffccoe 21652  degcdgr 21653
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359  ax-mulf 9360
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 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-fi 7659  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ioo 11302  df-ioc 11303  df-ico 11304  df-icc 11305  df-fz 11436  df-fzo 11547  df-fl 11640  df-mod 11707  df-seq 11805  df-exp 11864  df-fac 12050  df-bc 12077  df-hash 12102  df-shft 12554  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-limsup 12947  df-clim 12964  df-rlim 12965  df-sum 13162  df-ef 13351  df-sin 13353  df-cos 13354  df-pi 13356  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-rest 14359  df-topn 14360  df-0g 14378  df-gsum 14379  df-topgen 14380  df-pt 14381  df-prds 14384  df-xrs 14438  df-qtop 14443  df-imas 14444  df-xps 14446  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-submnd 15463  df-mulg 15546  df-cntz 15833  df-cmn 16277  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810  df-mopn 17811  df-fbas 17812  df-fg 17813  df-cnfld 17817  df-top 18501  df-bases 18503  df-topon 18504  df-topsp 18505  df-cld 18621  df-ntr 18622  df-cls 18623  df-nei 18700  df-lp 18738  df-perf 18739  df-cn 18829  df-cnp 18830  df-haus 18917  df-tx 19133  df-hmeo 19326  df-fil 19417  df-fm 19509  df-flim 19510  df-flf 19511  df-xms 19893  df-ms 19894  df-tms 19895  df-cncf 20452  df-0p 21146  df-limc 21339  df-dv 21340  df-ply 21654  df-coe 21656  df-dgr 21657  df-log 22006  df-cxp 22007
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator