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Theorem signsply0 29449
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 11317 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  e. 
RR* )
4 xrleid 11457 . . . . . . . 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 22 . . . . . . 7  |-  ( d  e.  RR+  ->  d  e.  RR+ )
8 simpr 462 . . . . . . . . 9  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
f  =  d )
98breq2d 4435 . . . . . . . 8  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( d  <_  f  <->  d  <_  d ) )
108fveq2d 5886 . . . . . . . . . . . 12  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( F `  f
)  =  ( F `
 d ) )
118oveq1d 6321 . . . . . . . . . . . 12  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( f ^ D
)  =  ( d ^ D ) )
1210, 11oveq12d 6324 . . . . . . . . . . 11  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( F `  f )  /  (
f ^ D ) )  =  ( ( F `  d )  /  ( d ^ D ) ) )
1312oveq1d 6321 . . . . . . . . . 10  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( ( ( F `
 f )  / 
( f ^ D
) )  -  B
)  =  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )
1413fveq2d 5886 . . . . . . . . 9  |-  ( ( d  e.  RR+  /\  f  =  d )  -> 
( abs `  (
( ( F `  f )  /  (
f ^ D ) )  -  B ) )  =  ( abs `  ( ( ( F `
 d )  / 
( d ^ D
) )  -  B
) ) )
1514breq1d 4433 . . . . . . . 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 3185 . . . . . 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 11349 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR )
2320, 22plyrecld 29447 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  ( F `  d )  e.  RR )
24 signsply0.d . . . . . . . . . . . . 13  |-  D  =  (deg `  F )
25 dgrcl 23186 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  RR )  ->  (deg `  F
)  e.  NN0 )
2619, 25syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  F )  e.  NN0 )
2724, 26syl5eqel 2511 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  NN0 )
2827ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  NN0 )
2922, 28reexpcld 12440 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR )
3021rpcnd 11351 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  CC )
3121rpne0d 11354 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  =/=  0 )
3227nn0zd 11046 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ZZ )
3332ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  D  e.  ZZ )
3430, 31, 33expne0d 12429 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  =/=  0 )
3523, 29, 34redivcld 10443 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  /  ( d ^ D ) )  e.  RR )
36 signsply0.b . . . . . . . . . . . 12  |-  B  =  ( C `  D
)
37 0re 9651 . . . . . . . . . . . . . 14  |-  0  e.  RR
38 signsply0.c . . . . . . . . . . . . . . 15  |-  C  =  (coeff `  F )
3938coef2 23184 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  C : NN0 --> RR )
4037, 39mpan2 675 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  RR )  ->  C : NN0 --> RR )
4140ffvelrnda 6038 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  RR )  /\  D  e. 
NN0 )  ->  ( C `  D )  e.  RR )
4236, 41syl5eqel 2511 . . . . . . . . . . 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 10054 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  -u B  e.  RR )
4635, 44, 45absdifltd 13496 . . . . . . . 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 9677 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
4948ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  B  e.  CC )
5049negidd 9984 . . . . . . . 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 4448 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  (
( ( F `  d )  /  (
d ^ D ) )  -  B ) )  <  -u B
)  ->  ( ( F `  d )  /  ( d ^ D ) )  <  0 )
5321, 33rpexpcld 12446 . . . . . . . . . 10  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR+ )
5423, 53ge0divd 11384 . . . . . . . . 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 9652 . . . . . . . . 9  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  0  e.  RR )
5723, 56ltnled 9790 . . . . . . . 8  |-  ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
( F `  d
)  <  0  <->  -.  0  <_  ( F `  d
) ) )
5835, 56ltnled 9790 . . . . . . . 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 9652 . . . . . 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 11349 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  d  e.  RR )
6664rpgt0d 11352 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  0  <  d
)
67 iccssre 11724 . . . . . . . 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 9604 . . . . . . 7  |-  RR  C_  CC
7068, 69syl6ss 3476 . . . . . 6  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  ( 0 [,] d )  C_  CC )
71 plycn 23214 . . . . . . . 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 3464 . . . . . . 7  |-  ( ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  /\  x  e.  ( 0 [,] d ) )  ->  x  e.  RR )
7674, 75plyrecld 29447 . . . . . 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 11341 . . . . . . . . . . 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 10892 . . . . . . . . . . . . . 14  |-  0  e.  NN0
8887a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  NN0 )
8986, 88ffvelrnd 6039 . . . . . . . . . . . 12  |-  ( ph  ->  ( C `  0
)  e.  RR )
9085, 89syl5eqel 2511 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
91 signsply0.3 . . . . . . . . . . 11  |-  ( ph  ->  ( A  x.  B
)  <  0 )
9290, 43, 91mul2lt0rlt0 11406 . . . . . . . . . 10  |-  ( (
ph  /\  B  <  0 )  ->  0  <  A )
9392, 85syl6breq 4463 . . . . . . . . 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 23201 . . . . . . . . . 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 4450 . . . . . . 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 22405 . . . . 5  |-  ( ( ( ( ph  /\  -u B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( F `  d
)  <  0 )  ->  E. z  e.  ( 0 (,) d ) ( F `  z
)  =  0 )
101 0le0 10707 . . . . . . . 8  |-  0  <_  0
102 pnfge 11440 . . . . . . . . 9  |-  ( d  e.  RR*  ->  d  <_ +oo )
1033, 102syl 17 . . . . . . . 8  |-  ( d  e.  RR+  ->  d  <_ +oo )
104 0xr 9695 . . . . . . . . 9  |-  0  e.  RR*
105 pnfxr 11420 . . . . . . . . 9  |- +oo  e.  RR*
106 ioossioo 11734 . . . . . . . . 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 11720 . . . . . . 7  |-  ( 0 (,) +oo )  = 
RR+
110108, 109syl6sseq 3510 . . . . . 6  |-  ( d  e.  RR+  ->  ( 0 (,) d )  C_  RR+ )
111 ssrexv 3526 . . . . . 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 23151 . . . . . . . . . . 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 6334 . . . . . . . . . . 11  |-  ( x ^ D )  e. 
_V
120119rgenw 2783 . . . . . . . . . 10  |-  A. x  e.  RR+  ( x ^ D )  e.  _V
121 eqid 2422 . . . . . . . . . . 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 9628 . . . . . . . . . 10  |-  CC  e.  _V
125124a1i 11 . . . . . . . . 9  |-  ( ph  ->  CC  e.  _V )
126 rpssre 11320 . . . . . . . . . . . 12  |-  RR+  C_  RR
127126, 69sstri 3473 . . . . . . . . . . 11  |-  RR+  C_  CC
128124, 127ssexi 4569 . . . . . . . . . 10  |-  RR+  e.  _V
129128a1i 11 . . . . . . . . 9  |-  ( ph  -> 
RR+  e.  _V )
130 dfss1 3667 . . . . . . . . . 10  |-  ( RR+  C_  CC  <->  ( CC  i^i  RR+ )  =  RR+ )
131127, 130mpbi 211 . . . . . . . . 9  |-  ( CC 
i^i  RR+ )  =  RR+
132 eqidd 2423 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  CC )  ->  ( F `
 f )  =  ( F `  f
) )
133 eqidd 2423 . . . . . . . . . 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 6321 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  RR+ )  /\  x  =  f )  -> 
( x ^ D
)  =  ( f ^ D ) )
136 simpr 462 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  e.  RR+ )
137 ovex 6334 . . . . . . . . . . 11  |-  ( f ^ D )  e. 
_V
138137a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  e.  _V )
139133, 135, 136, 138fvmptd 5971 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( (
x  e.  RR+  |->  ( x ^ D ) ) `
 f )  =  ( f ^ D
) )
140118, 123, 125, 129, 131, 132, 139offval 6553 . . . . . . . 8  |-  ( ph  ->  ( F  oF  /  ( x  e.  RR+  |->  ( x ^ D ) ) )  =  ( f  e.  RR+  |->  ( ( F `
 f )  / 
( f ^ D
) ) ) )
141 oveq1 6313 . . . . . . . . . . 11  |-  ( x  =  f  ->  (
x ^ D )  =  ( f ^ D ) )
142141cbvmptv 4516 . . . . . . . . . 10  |-  ( x  e.  RR+  |->  ( x ^ D ) )  =  ( f  e.  RR+  |->  ( f ^ D ) )
14324, 38, 36, 142signsplypnf 29448 . . . . . . . . 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 4446 . . . . . . 7  |-  ( ph  ->  ( f  e.  RR+  |->  ( ( F `  f )  /  (
f ^ D ) ) )  ~~> r  B
)
146116adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  F : CC
--> CC )
147136rpcnd 11351 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  e.  CC )
148146, 147ffvelrnd 6039 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( F `  f )  e.  CC )
14927adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  D  e.  NN0 )
150147, 149expcld 12423 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  e.  CC )
151136rpne0d 11354 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  f  =/=  0 )
15232adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  RR+ )  ->  D  e.  ZZ )
153147, 151, 152expne0d 12429 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( f ^ D )  =/=  0
)
154148, 150, 153divcld 10391 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  RR+ )  ->  ( ( F `  f )  /  ( f ^ D ) )  e.  CC )
155154ralrimiva 2836 . . . . . . . 8  |-  ( ph  ->  A. f  e.  RR+  ( ( F `  f )  /  (
f ^ D ) )  e.  CC )
156126a1i 11 . . . . . . . 8  |-  ( ph  -> 
RR+  C_  RR )
157 1red 9666 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
158155, 156, 48, 157rlim3 13562 . . . . . . 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 10144 . . . . . . . . . 10  |-  0  <  1
161 pnfge 11440 . . . . . . . . . . 11  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
162105, 161ax-mp 5 . . . . . . . . . 10  |- +oo  <_ +oo
163 icossioo 11733 . . . . . . . . . 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 3496 . . . . . . . 8  |-  ( 1 [,) +oo )  C_  RR+
166 ssrexv 3526 . . . . . . . 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 2815 . . . . . 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 4435 . . . . . . 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 2944 . . . . 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 3185 . . . 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 2967 . 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 4433 . . . . . . . 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 3185 . . . . . 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 9982 . . . . . . . 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 3464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  d  e.  RR )
191188, 190plyrecld 29447 . . . . . . . . . 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 12440 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR )
194190recnd 9677 . . . . . . . . . . 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 11354 . . . . . . . . . . 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 12429 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  =/=  0 )
199191, 193, 198redivcld 10443 . . . . . . . . 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 13496 . . . . . . . 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 4446 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  ( abs `  ( ( ( F `  d
)  /  ( d ^ D ) )  -  B ) )  <  B )  -> 
0  <  ( ( F `  d )  /  ( d ^ D ) ) )
204195, 197rpexpcld 12446 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  ->  (
d ^ D )  e.  RR+ )
205191, 204gt0divd 11383 . . . . . . 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 9652 . . . . . 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 11349 . . . . . 6  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
d  e.  RR )
212210rpgt0d 11352 . . . . . 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 3476 . . . . . 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 3464 . . . . . . 7  |-  ( ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `
 d ) )  /\  x  e.  ( 0 [,] d ) )  ->  x  e.  RR )
218216, 217plyrecld 29447 . . . . . 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 11352 . . . . . . . . . . 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 11407 . . . . . . . . . 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 4458 . . . . . . . 8  |-  ( ( ( ( ph  /\  B  e.  RR+ )  /\  d  e.  RR+ )  /\  0  <  ( F `  d ) )  -> 
( C `  0
)  <  0 )
227219, 226eqbrtrd 4444 . . . . . . 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 22404 . . . . 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 4435 . . . . . . 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 2944 . . . . 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 3185 . . . 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 2967 . 2  |-  ( (
ph  /\  B  e.  RR+ )  ->  E. z  e.  RR+  ( F `  z )  =  0 )
243 signsply0.2 . . . . 5  |-  ( ph  ->  F  =/=  0p )
24424, 38dgreq0 23218 . . . . . . 7  |-  ( F  e.  (Poly `  RR )  ->  ( F  =  0p  <->  ( C `  D )  =  0 ) )
24519, 244syl 17 . . . . . 6  |-  ( ph  ->  ( F  =  0p  <->  ( C `  D )  =  0 ) )
246245necon3bid 2678 . . . . 5  |-  ( ph  ->  ( F  =/=  0p 
<->  ( C `  D
)  =/=  0 ) )
247243, 246mpbid 213 . . . 4  |-  ( ph  ->  ( C `  D
)  =/=  0 )
24836neeq1i 2705 . . . 4  |-  ( B  =/=  0  <->  ( C `  D )  =/=  0
)
249247, 248sylibr 215 . . 3  |-  ( ph  ->  B  =/=  0 )
250 rpneg 11340 . . . . 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 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080    i^i cin 3435    C_ wss 3436   class class class wbr 4423    |-> cmpt 4482    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6306    oFcof 6544   CCcc 9545   RRcr 9546   0cc0 9547   1c1 9548    + caddc 9550    x. cmul 9552   +oocpnf 9680   RR*cxr 9682    < clt 9683    <_ cle 9684    - cmin 9868   -ucneg 9869    / cdiv 10277   NN0cn0 10877   ZZcz 10945   RR+crp 11310   (,)cioo 11643   [,)cico 11645   [,]cicc 11646   ^cexp 12279   abscabs 13298    ~~> r crli 13549   -cn->ccncf 21907   0pc0p 22626  Polycply 23137  coeffccoe 23139  degcdgr 23140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625  ax-addf 9626  ax-mulf 9627
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-of 6546  df-om 6708  df-1st 6808  df-2nd 6809  df-supp 6927  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-er 7375  df-map 7486  df-pm 7487  df-ixp 7535  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-fsupp 7894  df-fi 7935  df-sup 7966  df-inf 7967  df-oi 8035  df-card 8382  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-4 10678  df-5 10679  df-6 10680  df-7 10681  df-8 10682  df-9 10683  df-10 10684  df-n0 10878  df-z 10946  df-dec 11060  df-uz 11168  df-q 11273  df-rp 11311  df-xneg 11417  df-xadd 11418  df-xmul 11419  df-ioo 11647  df-ioc 11648  df-ico 11649  df-icc 11650  df-fz 11793  df-fzo 11924  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-shft 13131  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19920  df-bases 19921  df-topon 19922  df-topsp 19923  df-cld 20033  df-ntr 20034  df-cls 20035  df-nei 20113  df-lp 20151  df-perf 20152  df-cn 20242  df-cnp 20243  df-haus 20330  df-tx 20576  df-hmeo 20769  df-fil 20860  df-fm 20952  df-flim 20953  df-flf 20954  df-xms 21334  df-ms 21335  df-tms 21336  df-cncf 21909  df-0p 22627  df-limc 22820  df-dv 22821  df-ply 23141  df-coe 23143  df-dgr 23144  df-log 23505  df-cxp 23506
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator