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Theorem signsply0 28705
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 simplr 755 . . . . . 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 461 . . . . . 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 11252 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  e. 
RR* )
4 xrleid 11381 . . . . . . . 8  |-  ( d  e.  RR*  ->  d  <_ 
d )
53, 4syl 16 . . . . . . 7  |-  ( d  e.  RR+  ->  d  <_ 
d )
65ad2antlr 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 )
7 id 22 . . . . . . 7  |-  ( d  e.  RR+  ->  d  e.  RR+ )
8 simpr 461 . . . . . . . . 9  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
f  =  d )
98breq2d 4468 . . . . . . . 8  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( d  <_  f  <->  d  <_  d ) )
108fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( F `  f
)  =  ( F `
 d ) )
118oveq1d 6311 . . . . . . . . . . . 12  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( f ^ D
)  =  ( d ^ D ) )
1210, 11oveq12d 6314 . . . . . . . . . . 11  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( F `  f )  /  (
f ^ D ) )  =  ( ( F `  d )  /  ( d ^ D ) ) )
1312oveq1d 6311 . . . . . . . . . 10  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( ( F `
 f )  / 
( f ^ D
) )  -  B
)  =  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )
1413fveq2d 5876 . . . . . . . . 9  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  =  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) ) )
1514breq1d 4466 . . . . . . . 8  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  -u B  <->  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  -u B ) )
169, 15imbi12d 320 . . . . . . 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
) ) )
177, 16rspcdv 3213 . . . . . 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 ) ) )
181, 2, 6, 17syl3c 61 . . . . 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 ) )
2019ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  F  e.  (Poly `  RR )
)
21 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR+ )
2221rpred 11281 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR )
2320, 22plyrecld 28703 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( F `  d )  e.  RR )
24 signsply0.d . . . . . . . . . . . . 13  |-  D  =  (deg `  F )
25 dgrcl 22756 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
2619, 25syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  F )  e.  NN0 )
2724, 26syl5eqel 2549 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  NN0 )
2827ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  NN0 )
2922, 28reexpcld 12330 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR )
3021rpcnd 11283 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  CC )
3121rpne0d 11286 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  =/=  0 )
3227nn0zd 10988 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ZZ )
3332ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  ZZ )
3430, 31, 33expne0d 12319 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  =/=  0 )
3523, 29, 34redivcld 10393 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  /  ( d ^ D ) )  e.  RR )
36 signsply0.b . . . . . . . . . . . 12  |-  B  =  ( C `  D
)
37 0re 9613 . . . . . . . . . . . . . 14  |-  0  e.  RR
38 signsply0.c . . . . . . . . . . . . . . 15  |-  C  =  (coeff `  F )
3938coef2 22754 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  C : NN0 --> RR )
4037, 39mpan2 671 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> RR )
4140ffvelrnda 6032 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  D  e. 
NN0 )  ->  ( C `  D )  e.  RR )
4236, 41syl5eqel 2549 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  D  e. 
NN0 )  ->  B  e.  RR )
4319, 27, 42syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
4443ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  RR )
4544renegcld 10007 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  -u B  e.  RR )
4635, 44, 45absdifltd 13277 . . . . . . . 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 ) ) ) )
4746simplbda 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 ) )
4843recnd 9639 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
4948ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  CC )
5049negidd 9940 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( B  +  -u B )  =  0 )
5150adantr 465 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( B  +  -u B )  =  0 )
5247, 51breqtrd 4480 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( ( F `  d )  /  ( d ^ D ) )  <  0 )
5321, 33rpexpcld 12336 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR+ )
5423, 53ge0divd 11315 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
0  <_  ( F `  d )  <->  0  <_  ( ( F `  d
)  /  ( d ^ D ) ) ) )
5554notbid 294 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( -.  0  <_  ( F `
 d )  <->  -.  0  <_  ( ( F `  d )  /  (
d ^ D ) ) ) )
56 0red 9614 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  0  e.  RR )
5723, 56ltnled 9749 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  <  0  <->  -.  0  <_  ( F `  d
) ) )
5835, 56ltnled 9749 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( ( F `  d )  /  (
d ^ D ) )  <  0  <->  -.  0  <_  ( ( F `
 d )  / 
( d ^ D
) ) ) )
5955, 57, 583bitr4d 285 . . . . . . 7  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  <  0  <->  ( ( F `  d )  /  ( d ^ D ) )  <  0 ) )
6059adantr 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
) )
6152, 60mpbird 232 . . . . 5  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( F `  d )  <  0
)
6218, 61syldan 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 )
63 0red 9614 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  e.  RR )
64 simplr 755 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  d  e.  RR+ )
6564rpred 11281 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  d  e.  RR )
6664rpgt0d 11284 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  d
)
67 iccssre 11631 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  d  e.  RR )  ->  ( 0 [,] d
)  C_  RR )
6837, 65, 67sylancr 663 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( 0 [,] d )  C_  RR )
69 ax-resscn 9566 . . . . . . 7  |-  RR  C_  CC
7068, 69syl6ss 3511 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( 0 [,] d )  C_  CC )
71 plycn 22784 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  F  e.  ( CC -cn-> CC ) )
7219, 71syl 16 . . . . . . 7  |-  ( ph  ->  F  e.  ( CC
-cn-> CC ) )
7372ad3antrrr 729 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  F  e.  ( CC -cn-> CC ) )
7419ad4antr 731 . . . . . . 7  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  F  e.  (Poly `  RR ) )
7568sselda 3499 . . . . . . 7  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  x  e.  RR )
7674, 75plyrecld 28703 . . . . . 6  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  ( F `  x )  e.  RR )
77 simpr 461 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( F `  d )  <  0
)
78 simplll 759 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ph )
7978, 43syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  B  e.  RR )
80 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  -u B  e.  RR+ )  ->  -u B  e.  RR+ )
8180ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  -u B  e.  RR+ )
82 negelrp 27721 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  ( -u B  e.  RR+  <->  B  <  0 ) )
8382biimpa 484 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  -u B  e.  RR+ )  ->  B  <  0 )
8479, 81, 83syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  B  <  0
)
85 signsply0.a . . . . . . . . . . . 12  |-  A  =  ( C `  0
)
8619, 37, 39sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  C : NN0 --> RR )
87 0nn0 10831 . . . . . . . . . . . . . 14  |-  0  e.  NN0
8887a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  NN0 )
8986, 88ffvelrnd 6033 . . . . . . . . . . . 12  |-  ( ph  ->  ( C `  0
)  e.  RR )
9085, 89syl5eqel 2549 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
91 signsply0.3 . . . . . . . . . . 11  |-  ( ph  ->  ( A  x.  B
)  <  0 )
9290, 43, 91mul2lt0rlt0 27722 . . . . . . . . . 10  |-  ( (
ph  /\  B  <  0 )  ->  0  <  A )
9392, 85syl6breq 4495 . . . . . . . . 9  |-  ( (
ph  /\  B  <  0 )  ->  0  <  ( C `  0
) )
9478, 84, 93syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  ( C `  0 )
)
9538coefv0 22771 . . . . . . . . . 10  |-  ( F  e.  (Poly `  RR )  ->  ( F ` 
0 )  =  ( C `  0 ) )
9619, 95syl 16 . . . . . . . . 9  |-  ( ph  ->  ( F `  0
)  =  ( C `
 0 ) )
9796ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( F ` 
0 )  =  ( C `  0 ) )
9894, 97breqtrrd 4482 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  ( F `  0 )
)
9977, 98jca 532 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( ( F `
 d )  <  0  /\  0  < 
( F `  0
) ) )
10063, 65, 63, 66, 70, 73, 76, 99ivth2 21993 . . . . 5  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  E. z  e.  ( 0 (,) d ) ( F `  z
)  =  0 )
101 0le0 10646 . . . . . . . 8  |-  0  <_  0
102 pnfge 11364 . . . . . . . . 9  |-  ( d  e.  RR*  ->  d  <_ +oo )
1033, 102syl 16 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  <_ +oo )
104 0xr 9657 . . . . . . . . 9  |-  0  e.  RR*
105 pnfxr 11346 . . . . . . . . 9  |- +oo  e.  RR*
106 ioossioo 11641 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <_  0  /\  d  <_ +oo ) )  -> 
( 0 (,) d
)  C_  ( 0 (,) +oo ) )
107104, 105, 106mpanl12 682 . . . . . . . 8  |-  ( ( 0  <_  0  /\  d  <_ +oo )  ->  (
0 (,) d ) 
C_  ( 0 (,) +oo ) )
108101, 103, 107sylancr 663 . . . . . . 7  |-  ( d  e.  RR+  ->  ( 0 (,) d )  C_  ( 0 (,) +oo ) )
109 ioorp 11627 . . . . . . 7  |-  ( 0 (,) +oo )  = 
RR+
110108, 109syl6sseq 3545 . . . . . 6  |-  ( d  e.  RR+  ->  ( 0 (,) d )  C_  RR+ )
111 ssrexv 3561 . . . . . 6  |-  ( ( 0 (,) d ) 
C_  RR+  ->  ( E. z  e.  ( 0 (,) d ) ( F `  z )  =  0  ->  E. z  e.  RR+  ( F `  z )  =  0 ) )
11264, 110, 1113syl 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 ) )
113100, 112mpd 15 . . . 4  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
11462, 113syldan 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 )
115 plyf 22721 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
11619, 115syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : CC --> CC )
117 ffn 5737 . . . . . . . . . 10  |-  ( F : CC --> CC  ->  F  Fn  CC )
118116, 117syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  CC )
119 ovex 6324 . . . . . . . . . . 11  |-  ( x ^ D )  e. 
_V
120119rgenw 2818 . . . . . . . . . 10  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
121 eqid 2457 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( x  e.  RR+  |->  ( x ^ D ) )
122121fnmpt 5713 . . . . . . . . . 10  |-  ( A. x  e.  RR+  ( x ^ D )  e. 
_V  ->  ( x  e.  RR+  |->  ( x ^ D ) )  Fn  RR+ )
123120, 122mp1i 12 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  RR+  |->  ( x ^ D
) )  Fn  RR+ )
124 cnex 9590 . . . . . . . . . 10  |-  CC  e.  _V
125124a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
126 rpssre 11255 . . . . . . . . . . . 12  |-  RR+  C_  RR
127126, 69sstri 3508 . . . . . . . . . . 11  |-  RR+  C_  CC
128124, 127ssexi 4601 . . . . . . . . . 10  |-  RR+  e.  _V
129128a1i 11 . . . . . . . . 9  |-  ( ph  -> 
RR+  e.  _V )
130 dfss1 3699 . . . . . . . . . 10  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
131127, 130mpbi 208 . . . . . . . . 9  |-  ( CC 
i^i  RR+ )  =  RR+
132 eqidd 2458 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  CC )  ->  ( F `
 f )  =  ( F `  f
) )
133 eqidd 2458 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( x  e.  RR+  |->  ( x ^ D
) ) )
134 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  RR+ )  /\  x  =  f )  ->  x  =  f )
135134oveq1d 6311 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  RR+ )  /\  x  =  f )  -> 
( x ^ D
)  =  ( f ^ D ) )
136 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  e.  RR+ )
137 ovex 6324 . . . . . . . . . . 11  |-  ( f ^ D )  e. 
_V
138137a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  e.  _V )
139133, 135, 136, 138fvmptd 5961 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( (
x  e.  RR+  |->  ( x ^ D ) ) `
 f )  =  ( f ^ D
) )
140118, 123, 125, 129, 131, 132, 139offval 6546 . . . . . . . 8  |-  ( ph  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  =  ( f  e.  RR+  |->  ( ( F `
 f )  / 
( f ^ D
) ) ) )
141 oveq1 6303 . . . . . . . . . . 11  |-  ( x  =  f  ->  (
x ^ D )  =  ( f ^ D ) )
142141cbvmptv 4548 . . . . . . . . . 10  |-  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( f  e.  RR+  |->  ( f ^ D ) )
14324, 38, 36, 142signsplypnf 28704 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  ~~> r  B )
14419, 143syl 16 . . . . . . . 8  |-  ( ph  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  ~~> r  B )
145140, 144eqbrtrrd 4478 . . . . . . 7  |-  ( ph  ->  ( f  e.  RR+  |->  ( ( F `  f )  /  (
f ^ D ) ) )  ~~> r  B
)
146116adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  F : CC
--> CC )
147136rpcnd 11283 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  e.  CC )
148146, 147ffvelrnd 6033 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( F `  f )  e.  CC )
14927adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  D  e.  NN0 )
150147, 149expcld 12313 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  e.  CC )
151136rpne0d 11286 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  =/=  0 )
15232adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  D  e.  ZZ )
153147, 151, 152expne0d 12319 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  =/=  0
)
154148, 150, 153divcld 10341 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( ( F `  f )  /  ( f ^ D ) )  e.  CC )
155154ralrimiva 2871 . . . . . . . 8  |-  ( ph  ->  A. f  e.  RR+  ( ( F `  f )  /  (
f ^ D ) )  e.  CC )
156126a1i 11 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
157 1red 9628 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
158155, 156, 48, 157rlim3 13333 . . . . . . 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 ) ) )
159145, 158mpbid 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 ) )
160 0lt1 10096 . . . . . . . . . 10  |-  0  <  1
161 pnfge 11364 . . . . . . . . . . 11  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
162105, 161ax-mp 5 . . . . . . . . . 10  |- +oo  <_ +oo
163 icossioo 11640 . . . . . . . . . 10  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <  1  /\ +oo 
<_ +oo ) )  -> 
( 1 [,) +oo )  C_  ( 0 (,) +oo ) )
164104, 105, 160, 162, 163mp4an 673 . . . . . . . . 9  |-  ( 1 [,) +oo )  C_  ( 0 (,) +oo )
165164, 109sseqtri 3531 . . . . . . . 8  |-  ( 1 [,) +oo )  C_  RR+
166 ssrexv 3561 . . . . . . . 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 ) ) )
167165, 166ax-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 ) )
168167ralimi 2850 . . . . . 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
) )
169159, 168syl 16 . . . . 5  |-  ( ph  ->  A. e  e.  RR+  E. d  e.  RR+  A. f  e.  RR+  ( d  <_ 
f  ->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  e
) )
170169adantr 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 ) )
171 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  e  =  -u B )  -> 
e  =  -u B
)
172171breq2d 4468 . . . . . . 7  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  e  =  -u B )  -> 
( ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  e  <->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  -u B
) )
173172imbi2d 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
) ) )
174173rexralbidv 2976 . . . . 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
) ) )
17580, 174rspcdv 3213 . . . 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 ) ) )
176170, 175mpd 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
) )
177114, 176r19.29a 2999 . 2  |-  ( (
ph  /\  -u B  e.  RR+ )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
178 simplr 755 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  A. f  e.  RR+  (
d  <_  f  ->  ( abs `  ( ( ( F `  f
)  /  ( f ^ D ) )  -  B ) )  <  B ) )  ->  d  e.  RR+ )
179 simpr 461 . . . . . 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 ) )
1805ad2antlr 726 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  A. f  e.  RR+  (
d  <_  f  ->  ( abs `  ( ( ( F `  f
)  /  ( f ^ D ) )  -  B ) )  <  B ) )  ->  d  <_  d
)
18114breq1d 4466 . . . . . . . 8  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  B  <->  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) )  <  B
) )
1829, 181imbi12d 320 . . . . . . 7  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( d  <_ 
f  ->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  B
)  <->  ( d  <_ 
d  ->  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) )  <  B
) ) )
1837, 182rspcdv 3213 . . . . . 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 ) ) )
184178, 179, 180, 183syl3c 61 . . . . 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 )
18548ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  CC )
186185subidd 9938 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( B  -  B )  =  0 )
187186adantr 465 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
( B  -  B
)  =  0 )
18819ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  F  e.  (Poly `  RR )
)
189126a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  RR+ )  ->  RR+  C_  RR )
190189sselda 3499 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR )
191188, 190plyrecld 28703 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( F `  d )  e.  RR )
19227ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  NN0 )
193190, 192reexpcld 12330 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR )
194190recnd 9639 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  CC )
195 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR+ )
196195rpne0d 11286 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  =/=  0 )
19732ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  ZZ )
198194, 196, 197expne0d 12319 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  =/=  0 )
199191, 193, 198redivcld 10393 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  /  ( d ^ D ) )  e.  RR )
20043ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  RR )
201199, 200, 200absdifltd 13277 . . . . . . . 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
) ) ) )
202201simprbda 623 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
( B  -  B
)  <  ( ( F `  d )  /  ( d ^ D ) ) )
203187, 202eqbrtrrd 4478 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
0  <  ( ( F `  d )  /  ( d ^ D ) ) )
204195, 197rpexpcld 12336 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR+ )
205191, 204gt0divd 11314 . . . . . . 7  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
0  <  ( F `  d )  <->  0  <  ( ( F `  d
)  /  ( d ^ D ) ) ) )
206205adantr 465 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
( 0  <  ( F `  d )  <->  0  <  ( ( F `
 d )  / 
( d ^ D
) ) ) )
207203, 206mpbird 232 . . . . 5  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
0  <  ( F `  d ) )
208184, 207syldan 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 )
)
209 0red 9614 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  e.  RR )
210 simplr 755 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
d  e.  RR+ )
211210rpred 11281 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
d  e.  RR )
212210rpgt0d 11284 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  d )
21337, 211, 67sylancr 663 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( 0 [,] d
)  C_  RR )
214213, 69syl6ss 3511 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( 0 [,] d
)  C_  CC )
21572ad3antrrr 729 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  F  e.  ( CC -cn-> CC ) )
21619ad4antr 731 . . . . . . 7  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  F  e.  (Poly `  RR ) )
217213sselda 3499 . . . . . . 7  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  x  e.  RR )
218216, 217plyrecld 28703 . . . . . 6  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  ( F `  x )  e.  RR )
21996ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( F `  0
)  =  ( C `
 0 ) )
220 simplll 759 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  ph )
221 simpr1 1002 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( B  e.  RR+  /\  d  e.  RR+  /\  0  <  ( F `  d )
) )  ->  B  e.  RR+ )
222221rpgt0d 11284 . . . . . . . . . . 11  |-  ( (
ph  /\  ( B  e.  RR+  /\  d  e.  RR+  /\  0  <  ( F `  d )
) )  ->  0  <  B )
2232223anassrs 1218 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  B )
22490, 43, 91mul2lt0rgt0 27723 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  B )  ->  A  <  0 )
225220, 223, 224syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  A  <  0 )
22685, 225syl5eqbrr 4490 . . . . . . . 8  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( C `  0
)  <  0 )
227219, 226eqbrtrd 4476 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( F `  0
)  <  0 )
228 simpr 461 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  ( F `  d ) )
229227, 228jca 532 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( ( F ` 
0 )  <  0  /\  0  <  ( F `
 d ) ) )
230209, 211, 209, 212, 214, 215, 218, 229ivth 21992 . . . . 5  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  E. z  e.  (
0 (,) d ) ( F `  z
)  =  0 )
231210, 110, 1113syl 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 ) )
232230, 231mpd 15 . . . 4  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
233208, 232syldan 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 )
234169adantr 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 ) )
235 simpr 461 . . . . 5  |-  ( (
ph  /\  B  e.  RR+ )  ->  B  e.  RR+ )
236 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  e  =  B )  ->  e  =  B )
237236breq2d 4468 . . . . . . 7  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  e  =  B )  ->  (
( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  e  <->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  B
) )
238237imbi2d 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 ) ) )
239238rexralbidv 2976 . . . . 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 ) ) )
240235, 239rspcdv 3213 . . . 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 ) ) )
241234, 240mpd 15 . . 3  |-  ( (
ph  /\  B  e.  RR+ )  ->  E. d  e.  RR+  A. f  e.  RR+  ( d  <_  f  ->  ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  B ) )
242233, 241r19.29a 2999 . 2  |-  ( (
ph  /\  B  e.  RR+ )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
243 signsply0.2 . . . . 5  |-  ( ph  ->  F  =/=  0p )
24424, 38dgreq0 22788 . . . . . . 7  |-  ( F  e.  (Poly `  RR )  ->  ( F  =  0p  <->  ( C `  D )  =  0 ) )
24519, 244syl 16 . . . . . 6  |-  ( ph  ->  ( F  =  0p  <->  ( C `  D )  =  0 ) )
246245necon3bid 2715 . . . . 5  |-  ( ph  ->  ( F  =/=  0p 
<->  ( C `  D
)  =/=  0 ) )
247243, 246mpbid 210 . . . 4  |-  ( ph  ->  ( C `  D
)  =/=  0 )
24836neeq1i 2742 . . . 4  |-  ( B  =/=  0  <->  ( C `  D )  =/=  0
)
249247, 248sylibr 212 . . 3  |-  ( ph  ->  B  =/=  0 )
250 rpneg 11274 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( B  e.  RR+  <->  -.  -u B  e.  RR+ )
)
251250biimprd 223 . . . 4  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( -.  -u B  e.  RR+  ->  B  e.  RR+ ) )
252251orrd 378 . . 3  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( -u B  e.  RR+  \/  B  e.  RR+ )
)
25343, 249, 252syl2anc 661 . 2  |-  ( ph  ->  ( -u B  e.  RR+  \/  B  e.  RR+ ) )
254177, 242, 253mpjaodan 786 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824   -ucneg 9825    / cdiv 10227   NN0cn0 10816   ZZcz 10885   RR+crp 11245   (,)cioo 11554   [,)cico 11556   [,]cicc 11557   ^cexp 12169   abscabs 13079    ~~> r crli 13320   -cn->ccncf 21506   0pc0p 22202  Polycply 22707  coeffccoe 22709  degcdgr 22710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-pi 13820  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-0p 22203  df-limc 22396  df-dv 22397  df-ply 22711  df-coe 22713  df-dgr 22714  df-log 23070  df-cxp 23071
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator