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Theorem signsply0 29228
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 760 . . . . . 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 462 . . . . . 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 11309 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  e. 
RR* )
4 xrleid 11449 . . . . . . . 8  |-  ( d  e.  RR*  ->  d  <_ 
d )
53, 4syl 17 . . . . . . 7  |-  ( d  e.  RR+  ->  d  <_ 
d )
65ad2antlr 731 . . . . . 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 23 . . . . . . 7  |-  ( d  e.  RR+  ->  d  e.  RR+ )
8 simpr 462 . . . . . . . . 9  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
f  =  d )
98breq2d 4438 . . . . . . . 8  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( d  <_  f  <->  d  <_  d ) )
108fveq2d 5885 . . . . . . . . . . . 12  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( F `  f
)  =  ( F `
 d ) )
118oveq1d 6320 . . . . . . . . . . . 12  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( f ^ D
)  =  ( d ^ D ) )
1210, 11oveq12d 6323 . . . . . . . . . . 11  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( F `  f )  /  (
f ^ D ) )  =  ( ( F `  d )  /  ( d ^ D ) ) )
1312oveq1d 6320 . . . . . . . . . 10  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( ( F `
 f )  / 
( f ^ D
) )  -  B
)  =  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )
1413fveq2d 5885 . . . . . . . . 9  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  =  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) ) )
1514breq1d 4436 . . . . . . . 8  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  -u B  <->  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  -u B ) )
169, 15imbi12d 321 . . . . . . 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 3191 . . . . . 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 63 . . . . 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 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  F  e.  (Poly `  RR )
)
21 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR+ )
2221rpred 11341 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR )
2320, 22plyrecld 29226 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( F `  d )  e.  RR )
24 signsply0.d . . . . . . . . . . . . 13  |-  D  =  (deg `  F )
25 dgrcl 23055 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
2619, 25syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  F )  e.  NN0 )
2724, 26syl5eqel 2521 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  NN0 )
2827ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  NN0 )
2922, 28reexpcld 12430 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR )
3021rpcnd 11343 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  CC )
3121rpne0d 11346 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  =/=  0 )
3227nn0zd 11038 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ZZ )
3332ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  ZZ )
3430, 31, 33expne0d 12419 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  =/=  0 )
3523, 29, 34redivcld 10434 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  /  ( d ^ D ) )  e.  RR )
36 signsply0.b . . . . . . . . . . . 12  |-  B  =  ( C `  D
)
37 0re 9642 . . . . . . . . . . . . . 14  |-  0  e.  RR
38 signsply0.c . . . . . . . . . . . . . . 15  |-  C  =  (coeff `  F )
3938coef2 23053 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  C : NN0 --> RR )
4037, 39mpan2 675 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> RR )
4140ffvelrnda 6037 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  D  e. 
NN0 )  ->  ( C `  D )  e.  RR )
4236, 41syl5eqel 2521 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  RR )  /\  D  e. 
NN0 )  ->  B  e.  RR )
4319, 27, 42syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
4443ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  RR )
4544renegcld 10045 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  -u B  e.  RR )
4635, 44, 45absdifltd 13474 . . . . . . . 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 628 . . . . . . 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 9668 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
4948ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  CC )
5049negidd 9975 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( B  +  -u B )  =  0 )
5150adantr 466 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( B  +  -u B )  =  0 )
5247, 51breqtrd 4450 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( ( F `  d )  /  ( d ^ D ) )  <  0 )
5321, 33rpexpcld 12436 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR+ )
5423, 53ge0divd 11376 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
0  <_  ( F `  d )  <->  0  <_  ( ( F `  d
)  /  ( d ^ D ) ) ) )
5554notbid 295 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( -.  0  <_  ( F `
 d )  <->  -.  0  <_  ( ( F `  d )  /  (
d ^ D ) ) ) )
56 0red 9643 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  0  e.  RR )
5723, 56ltnled 9781 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  <  0  <->  -.  0  <_  ( F `  d
) ) )
5835, 56ltnled 9781 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( ( F `  d )  /  (
d ^ D ) )  <  0  <->  -.  0  <_  ( ( F `
 d )  / 
( d ^ D
) ) ) )
5955, 57, 583bitr4d 288 . . . . . . 7  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  <  0  <->  ( ( F `  d )  /  ( d ^ D ) )  <  0 ) )
6059adantr 466 . . . . . 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 235 . . . . 5  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( F `  d )  <  0
)
6218, 61syldan 472 . . . 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 9643 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  e.  RR )
64 simplr 760 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  d  e.  RR+ )
6564rpred 11341 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  d  e.  RR )
6664rpgt0d 11344 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  d
)
67 iccssre 11716 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  d  e.  RR )  ->  ( 0 [,] d
)  C_  RR )
6837, 65, 67sylancr 667 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( 0 [,] d )  C_  RR )
69 ax-resscn 9595 . . . . . . 7  |-  RR  C_  CC
7068, 69syl6ss 3482 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( 0 [,] d )  C_  CC )
71 plycn 23083 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  F  e.  ( CC -cn-> CC ) )
7219, 71syl 17 . . . . . . 7  |-  ( ph  ->  F  e.  ( CC
-cn-> CC ) )
7372ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  F  e.  ( CC -cn-> CC ) )
7419ad4antr 736 . . . . . . 7  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  F  e.  (Poly `  RR ) )
7568sselda 3470 . . . . . . 7  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  x  e.  RR )
7674, 75plyrecld 29226 . . . . . 6  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  ( F `  x )  e.  RR )
77 simpr 462 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( F `  d )  <  0
)
78 simplll 766 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ph )
7978, 43syl 17 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  B  e.  RR )
80 simpr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  -u B  e.  RR+ )  ->  -u B  e.  RR+ )
8180ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  -u B  e.  RR+ )
82 negelrp 11333 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  ( -u B  e.  RR+  <->  B  <  0 ) )
8382biimpa 486 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  -u B  e.  RR+ )  ->  B  <  0 )
8479, 81, 83syl2anc 665 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  B  <  0
)
85 signsply0.a . . . . . . . . . . . 12  |-  A  =  ( C `  0
)
8619, 37, 39sylancl 666 . . . . . . . . . . . . 13  |-  ( ph  ->  C : NN0 --> RR )
87 0nn0 10884 . . . . . . . . . . . . . 14  |-  0  e.  NN0
8887a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  NN0 )
8986, 88ffvelrnd 6038 . . . . . . . . . . . 12  |-  ( ph  ->  ( C `  0
)  e.  RR )
9085, 89syl5eqel 2521 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
91 signsply0.3 . . . . . . . . . . 11  |-  ( ph  ->  ( A  x.  B
)  <  0 )
9290, 43, 91mul2lt0rlt0 11398 . . . . . . . . . 10  |-  ( (
ph  /\  B  <  0 )  ->  0  <  A )
9392, 85syl6breq 4465 . . . . . . . . 9  |-  ( (
ph  /\  B  <  0 )  ->  0  <  ( C `  0
) )
9478, 84, 93syl2anc 665 . . . . . . . 8  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  ( C `  0 )
)
9538coefv0 23070 . . . . . . . . . 10  |-  ( F  e.  (Poly `  RR )  ->  ( F ` 
0 )  =  ( C `  0 ) )
9619, 95syl 17 . . . . . . . . 9  |-  ( ph  ->  ( F `  0
)  =  ( C `
 0 ) )
9796ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( F ` 
0 )  =  ( C `  0 ) )
9894, 97breqtrrd 4452 . . . . . . 7  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  ( F `  0 )
)
9977, 98jca 534 . . . . . 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 22287 . . . . 5  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  E. z  e.  ( 0 (,) d ) ( F `  z
)  =  0 )
101 0le0 10699 . . . . . . . 8  |-  0  <_  0
102 pnfge 11432 . . . . . . . . 9  |-  ( d  e.  RR*  ->  d  <_ +oo )
1033, 102syl 17 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  <_ +oo )
104 0xr 9686 . . . . . . . . 9  |-  0  e.  RR*
105 pnfxr 11412 . . . . . . . . 9  |- +oo  e.  RR*
106 ioossioo 11726 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <_  0  /\  d  <_ +oo ) )  -> 
( 0 (,) d
)  C_  ( 0 (,) +oo ) )
107104, 105, 106mpanl12 686 . . . . . . . 8  |-  ( ( 0  <_  0  /\  d  <_ +oo )  ->  (
0 (,) d ) 
C_  ( 0 (,) +oo ) )
108101, 103, 107sylancr 667 . . . . . . 7  |-  ( d  e.  RR+  ->  ( 0 (,) d )  C_  ( 0 (,) +oo ) )
109 ioorp 11712 . . . . . . 7  |-  ( 0 (,) +oo )  = 
RR+
110108, 109syl6sseq 3516 . . . . . 6  |-  ( d  e.  RR+  ->  ( 0 (,) d )  C_  RR+ )
111 ssrexv 3532 . . . . . 6  |-  ( ( 0 (,) d ) 
C_  RR+  ->  ( E. z  e.  ( 0 (,) d ) ( F `  z )  =  0  ->  E. z  e.  RR+  ( F `  z )  =  0 ) )
11264, 110, 1113syl 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 ) )
113100, 112mpd 15 . . . 4  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
11462, 113syldan 472 . . 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 23020 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
11619, 115syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : CC --> CC )
117 ffn 5746 . . . . . . . . . 10  |-  ( F : CC --> CC  ->  F  Fn  CC )
118116, 117syl 17 . . . . . . . . 9  |-  ( ph  ->  F  Fn  CC )
119 ovex 6333 . . . . . . . . . . 11  |-  ( x ^ D )  e. 
_V
120119rgenw 2793 . . . . . . . . . 10  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
121 eqid 2429 . . . . . . . . . . 11  |-  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( x  e.  RR+  |->  ( x ^ D ) )
122121fnmpt 5722 . . . . . . . . . 10  |-  ( A. x  e.  RR+  ( x ^ D )  e. 
_V  ->  ( x  e.  RR+  |->  ( x ^ D ) )  Fn  RR+ )
123120, 122mp1i 13 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  RR+  |->  ( x ^ D
) )  Fn  RR+ )
124 cnex 9619 . . . . . . . . . 10  |-  CC  e.  _V
125124a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
126 rpssre 11312 . . . . . . . . . . . 12  |-  RR+  C_  RR
127126, 69sstri 3479 . . . . . . . . . . 11  |-  RR+  C_  CC
128124, 127ssexi 4570 . . . . . . . . . 10  |-  RR+  e.  _V
129128a1i 11 . . . . . . . . 9  |-  ( ph  -> 
RR+  e.  _V )
130 dfss1 3673 . . . . . . . . . 10  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
131127, 130mpbi 211 . . . . . . . . 9  |-  ( CC 
i^i  RR+ )  =  RR+
132 eqidd 2430 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  CC )  ->  ( F `
 f )  =  ( F `  f
) )
133 eqidd 2430 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( x  e.  RR+  |->  ( x ^ D
) ) )
134 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  RR+ )  /\  x  =  f )  ->  x  =  f )
135134oveq1d 6320 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  RR+ )  /\  x  =  f )  -> 
( x ^ D
)  =  ( f ^ D ) )
136 simpr 462 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  e.  RR+ )
137 ovex 6333 . . . . . . . . . . 11  |-  ( f ^ D )  e. 
_V
138137a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  e.  _V )
139133, 135, 136, 138fvmptd 5970 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( (
x  e.  RR+  |->  ( x ^ D ) ) `
 f )  =  ( f ^ D
) )
140118, 123, 125, 129, 131, 132, 139offval 6552 . . . . . . . 8  |-  ( ph  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  =  ( f  e.  RR+  |->  ( ( F `
 f )  / 
( f ^ D
) ) ) )
141 oveq1 6312 . . . . . . . . . . 11  |-  ( x  =  f  ->  (
x ^ D )  =  ( f ^ D ) )
142141cbvmptv 4518 . . . . . . . . . 10  |-  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( f  e.  RR+  |->  ( f ^ D ) )
14324, 38, 36, 142signsplypnf 29227 . . . . . . . . 9  |-  ( F  e.  (Poly `  RR )  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  ~~> r  B )
14419, 143syl 17 . . . . . . . 8  |-  ( ph  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  ~~> r  B )
145140, 144eqbrtrrd 4448 . . . . . . 7  |-  ( ph  ->  ( f  e.  RR+  |->  ( ( F `  f )  /  (
f ^ D ) ) )  ~~> r  B
)
146116adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  F : CC
--> CC )
147136rpcnd 11343 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  e.  CC )
148146, 147ffvelrnd 6038 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( F `  f )  e.  CC )
14927adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  D  e.  NN0 )
150147, 149expcld 12413 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  e.  CC )
151136rpne0d 11346 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  =/=  0 )
15232adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  D  e.  ZZ )
153147, 151, 152expne0d 12419 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  =/=  0
)
154148, 150, 153divcld 10382 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( ( F `  f )  /  ( f ^ D ) )  e.  CC )
155154ralrimiva 2846 . . . . . . . 8  |-  ( ph  ->  A. f  e.  RR+  ( ( F `  f )  /  (
f ^ D ) )  e.  CC )
156126a1i 11 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
157 1red 9657 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
158155, 156, 48, 157rlim3 13540 . . . . . . 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 213 . . . . . 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 10135 . . . . . . . . . 10  |-  0  <  1
161 pnfge 11432 . . . . . . . . . . 11  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
162105, 161ax-mp 5 . . . . . . . . . 10  |- +oo  <_ +oo
163 icossioo 11725 . . . . . . . . . 10  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <  1  /\ +oo 
<_ +oo ) )  -> 
( 1 [,) +oo )  C_  ( 0 (,) +oo ) )
164104, 105, 160, 162, 163mp4an 677 . . . . . . . . 9  |-  ( 1 [,) +oo )  C_  ( 0 (,) +oo )
165164, 109sseqtri 3502 . . . . . . . 8  |-  ( 1 [,) +oo )  C_  RR+
166 ssrexv 3532 . . . . . . . 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 2825 . . . . . 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 17 . . . . 5  |-  ( ph  ->  A. e  e.  RR+  E. d  e.  RR+  A. f  e.  RR+  ( d  <_ 
f  ->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  e
) )
170169adantr 466 . . . 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 462 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  e  =  -u B )  -> 
e  =  -u B
)
172171breq2d 4438 . . . . . . 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 317 . . . . . 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 2954 . . . . 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 3191 . . . 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 2977 . 2  |-  ( (
ph  /\  -u B  e.  RR+ )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
178 simplr 760 . . . . . 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 462 . . . . . 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 731 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  A. f  e.  RR+  (
d  <_  f  ->  ( abs `  ( ( ( F `  f
)  /  ( f ^ D ) )  -  B ) )  <  B ) )  ->  d  <_  d
)
18114breq1d 4436 . . . . . . . 8  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  B  <->  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) )  <  B
) )
1829, 181imbi12d 321 . . . . . . 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 3191 . . . . . 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 63 . . . . 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 730 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  CC )
186185subidd 9973 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( B  -  B )  =  0 )
187186adantr 466 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
( B  -  B
)  =  0 )
18819ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  F  e.  (Poly `  RR )
)
189126a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  B  e.  RR+ )  ->  RR+  C_  RR )
190189sselda 3470 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR )
191188, 190plyrecld 29226 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( F `  d )  e.  RR )
19227ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  NN0 )
193190, 192reexpcld 12430 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR )
194190recnd 9668 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  CC )
195 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR+ )
196195rpne0d 11346 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  =/=  0 )
19732ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  ZZ )
198194, 196, 197expne0d 12419 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  =/=  0 )
199191, 193, 198redivcld 10434 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  /  ( d ^ D ) )  e.  RR )
20043ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  RR )
201199, 200, 200absdifltd 13474 . . . . . . . 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 627 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
( B  -  B
)  <  ( ( F `  d )  /  ( d ^ D ) ) )
203187, 202eqbrtrrd 4448 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
0  <  ( ( F `  d )  /  ( d ^ D ) ) )
204195, 197rpexpcld 12436 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR+ )
205191, 204gt0divd 11375 . . . . . . 7  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
0  <  ( F `  d )  <->  0  <  ( ( F `  d
)  /  ( d ^ D ) ) ) )
206205adantr 466 . . . . . 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 235 . . . . 5  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
0  <  ( F `  d ) )
208184, 207syldan 472 . . . 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 9643 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  e.  RR )
210 simplr 760 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
d  e.  RR+ )
211210rpred 11341 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
d  e.  RR )
212210rpgt0d 11344 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  d )
21337, 211, 67sylancr 667 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( 0 [,] d
)  C_  RR )
214213, 69syl6ss 3482 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( 0 [,] d
)  C_  CC )
21572ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  F  e.  ( CC -cn-> CC ) )
21619ad4antr 736 . . . . . . 7  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  F  e.  (Poly `  RR ) )
217213sselda 3470 . . . . . . 7  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  x  e.  RR )
218216, 217plyrecld 29226 . . . . . 6  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  ( F `  x )  e.  RR )
21996ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( F `  0
)  =  ( C `
 0 ) )
220 simplll 766 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  ph )
221 simpr1 1011 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( B  e.  RR+  /\  d  e.  RR+  /\  0  <  ( F `  d )
) )  ->  B  e.  RR+ )
222221rpgt0d 11344 . . . . . . . . . . 11  |-  ( (
ph  /\  ( B  e.  RR+  /\  d  e.  RR+  /\  0  <  ( F `  d )
) )  ->  0  <  B )
2232223anassrs 1228 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  B )
22490, 43, 91mul2lt0rgt0 11399 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  B )  ->  A  <  0 )
225220, 223, 224syl2anc 665 . . . . . . . . 9  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  A  <  0 )
22685, 225syl5eqbrr 4460 . . . . . . . 8  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( C `  0
)  <  0 )
227219, 226eqbrtrd 4446 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( F `  0
)  <  0 )
228 simpr 462 . . . . . . 7  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
0  <  ( F `  d ) )
229227, 228jca 534 . . . . . 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 22286 . . . . 5  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  E. z  e.  (
0 (,) d ) ( F `  z
)  =  0 )
231210, 110, 1113syl 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 ) )
232230, 231mpd 15 . . . 4  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
233208, 232syldan 472 . . 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 466 . . . 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 462 . . . . 5  |-  ( (
ph  /\  B  e.  RR+ )  ->  B  e.  RR+ )
236 simpr 462 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  e  =  B )  ->  e  =  B )
237236breq2d 4438 . . . . . . 7  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  e  =  B )  ->  (
( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  <  e  <->  ( abs `  ( ( ( F `
 f )  / 
( f ^ D
) )  -  B
) )  <  B
) )
238237imbi2d 317 . . . . . 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 2954 . . . . 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 3191 . . . 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 2977 . 2  |-  ( (
ph  /\  B  e.  RR+ )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
243 signsply0.2 . . . . 5  |-  ( ph  ->  F  =/=  0p )
24424, 38dgreq0 23087 . . . . . . 7  |-  ( F  e.  (Poly `  RR )  ->  ( F  =  0p  <->  ( C `  D )  =  0 ) )
24519, 244syl 17 . . . . . 6  |-  ( ph  ->  ( F  =  0p  <->  ( C `  D )  =  0 ) )
246245necon3bid 2689 . . . . 5  |-  ( ph  ->  ( F  =/=  0p 
<->  ( C `  D
)  =/=  0 ) )
247243, 246mpbid 213 . . . 4  |-  ( ph  ->  ( C `  D
)  =/=  0 )
24836neeq1i 2716 . . . 4  |-  ( B  =/=  0  <->  ( C `  D )  =/=  0
)
249247, 248sylibr 215 . . 3  |-  ( ph  ->  B  =/=  0 )
250 rpneg 11332 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( B  e.  RR+  <->  -.  -u B  e.  RR+ )
)
251250biimprd 226 . . . 4  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( -.  -u B  e.  RR+  ->  B  e.  RR+ ) )
252251orrd 379 . . 3  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( -u B  e.  RR+  \/  B  e.  RR+ )
)
25343, 249, 252syl2anc 665 . 2  |-  ( ph  ->  ( -u B  e.  RR+  \/  B  e.  RR+ ) )
254177, 242, 253mpjaodan 793 1  |-  ( ph  ->  E. z  e.  RR+  ( F `  z )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783   _Vcvv 3087    i^i cin 3441    C_ wss 3442   class class class wbr 4426    |-> cmpt 4484    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543   +oocpnf 9671   RR*cxr 9673    < clt 9674    <_ cle 9675    - cmin 9859   -ucneg 9860    / cdiv 10268   NN0cn0 10869   ZZcz 10937   RR+crp 11302   (,)cioo 11635   [,)cico 11637   [,]cicc 11638   ^cexp 12269   abscabs 13276    ~~> r crli 13527   -cn->ccncf 21804   0pc0p 22504  Polycply 23006  coeffccoe 23008  degcdgr 23009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-pi 14104  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-0p 22505  df-limc 22698  df-dv 22699  df-ply 23010  df-coe 23012  df-dgr 23013  df-log 23371  df-cxp 23372
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator