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Theorem dvferm1 19348
Description: One-sided version of dvferm 19351. A point  U which is the local maximum of its right neighborhood has derivative at most zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
Hypotheses
Ref Expression
dvferm.a  |-  ( ph  ->  F : X --> RR )
dvferm.b  |-  ( ph  ->  X  C_  RR )
dvferm.u  |-  ( ph  ->  U  e.  ( A (,) B ) )
dvferm.s  |-  ( ph  ->  ( A (,) B
)  C_  X )
dvferm.d  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
dvferm1.r  |-  ( ph  ->  A. y  e.  ( U (,) B ) ( F `  y
)  <_  ( F `  U ) )
Assertion
Ref Expression
dvferm1  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  <_  0 )
Distinct variable groups:    y, A    y, B    y, F    y, U    y, X    ph, y

Proof of Theorem dvferm1
Dummy variables  z  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvferm.a . . . . . . . 8  |-  ( ph  ->  F : X --> RR )
2 dvferm.b . . . . . . . 8  |-  ( ph  ->  X  C_  RR )
3 dvfre 19316 . . . . . . . 8  |-  ( ( F : X --> RR  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
41, 2, 3syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
5 dvferm.d . . . . . . 7  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
6 ffvelrn 5679 . . . . . . 7  |-  ( ( ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR 
/\  U  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  U )  e.  RR )
74, 5, 6syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
87anim1i 551 . . . . 5  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( (
( RR  _D  F
) `  U )  e.  RR  /\  0  < 
( ( RR  _D  F ) `  U
) ) )
9 elrp 10372 . . . . 5  |-  ( ( ( RR  _D  F
) `  U )  e.  RR+  <->  ( ( ( RR  _D  F ) `
 U )  e.  RR  /\  0  < 
( ( RR  _D  F ) `  U
) ) )
108, 9sylibr 203 . . . 4  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( ( RR  _D  F ) `  U )  e.  RR+ )
11 dvf 19273 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
12 ffun 5407 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
13 funfvbrb 5654 . . . . . . . . . . 11  |-  ( Fun  ( RR  _D  F
)  ->  ( U  e.  dom  ( RR  _D  F )  <->  U ( RR  _D  F ) ( ( RR  _D  F
) `  U )
) )
1411, 12, 13mp2b 9 . . . . . . . . . 10  |-  ( U  e.  dom  ( RR 
_D  F )  <->  U ( RR  _D  F ) ( ( RR  _D  F
) `  U )
)
155, 14sylib 188 . . . . . . . . 9  |-  ( ph  ->  U ( RR  _D  F ) ( ( RR  _D  F ) `
 U ) )
16 eqid 2296 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
17 eqid 2296 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
18 eqid 2296 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
19 ax-resscn 8810 . . . . . . . . . . 11  |-  RR  C_  CC
2019a1i 10 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
21 fss 5413 . . . . . . . . . . 11  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
221, 19, 21sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  F : X --> CC )
2316, 17, 18, 20, 22, 2eldv 19264 . . . . . . . . 9  |-  ( ph  ->  ( U ( RR 
_D  F ) ( ( RR  _D  F
) `  U )  <->  ( U  e.  ( ( int `  ( (
TopOpen ` fld )t  RR ) ) `  X )  /\  (
( RR  _D  F
) `  U )  e.  ( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) lim
CC  U ) ) ) )
2415, 23mpbid 201 . . . . . . . 8  |-  ( ph  ->  ( U  e.  ( ( int `  (
( TopOpen ` fld )t  RR ) ) `  X )  /\  (
( RR  _D  F
) `  U )  e.  ( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) lim
CC  U ) ) )
2524simprd 449 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) lim CC  U ) )
2625adantr 451 . . . . . 6  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( ( RR  _D  F ) `  U )  e.  ( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) lim
CC  U ) )
272, 19syl6ss 3204 . . . . . . . . . 10  |-  ( ph  ->  X  C_  CC )
28 dvferm.s . . . . . . . . . . 11  |-  ( ph  ->  ( A (,) B
)  C_  X )
29 dvferm.u . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( A (,) B ) )
3028, 29sseldd 3194 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3122, 27, 30dvlem 19262 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3231, 18fmptd 5700 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3332adantr 451 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
34 difss 3316 . . . . . . . 8  |-  ( X 
\  { U }
)  C_  X
3527adantr 451 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  X  C_  CC )
3634, 35syl5ss 3203 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( X  \  { U } ) 
C_  CC )
3727, 30sseldd 3194 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3837adantr 451 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  U  e.  CC )
3933, 36, 38ellimc3 19245 . . . . . 6  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( (
( RR  _D  F
) `  U )  e.  ( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) lim
CC  U )  <->  ( (
( RR  _D  F
) `  U )  e.  CC  /\  A. y  e.  RR+  E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  -  ( ( RR  _D  F ) `  U
) ) )  < 
y ) ) ) )
4026, 39mpbid 201 . . . . 5  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( (
( RR  _D  F
) `  U )  e.  CC  /\  A. y  e.  RR+  E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  -  ( ( RR  _D  F ) `  U
) ) )  < 
y ) ) )
4140simprd 449 . . . 4  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  A. y  e.  RR+  E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  -  ( ( RR  _D  F ) `  U
) ) )  < 
y ) )
42 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4342oveq1d 5889 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
44 oveq1 5881 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4543, 44oveq12d 5892 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
46 ovex 5899 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4745, 18, 46fvmpt 5618 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4847oveq1d 5889 . . . . . . . . . 10  |-  ( z  e.  ( X  \  { U } )  -> 
( ( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) ) `  z )  -  ( ( RR 
_D  F ) `  U ) )  =  ( ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) )  -  (
( RR  _D  F
) `  U )
) )
4948fveq2d 5545 . . . . . . . . 9  |-  ( z  e.  ( X  \  { U } )  -> 
( abs `  (
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  -  ( ( RR  _D  F ) `  U
) ) )  =  ( abs `  (
( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  ( ( RR 
_D  F ) `  U ) ) ) )
50 id 19 . . . . . . . . 9  |-  ( y  =  ( ( RR 
_D  F ) `  U )  ->  y  =  ( ( RR 
_D  F ) `  U ) )
5149, 50breqan12rd 4055 . . . . . . . 8  |-  ( ( y  =  ( ( RR  _D  F ) `
 U )  /\  z  e.  ( X  \  { U } ) )  ->  ( ( abs `  ( ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) `  z )  -  (
( RR  _D  F
) `  U )
) )  <  y  <->  ( abs `  ( ( ( ( F `  z )  -  ( F `  U )
)  /  ( z  -  U ) )  -  ( ( RR 
_D  F ) `  U ) ) )  <  ( ( RR 
_D  F ) `  U ) ) )
5251imbi2d 307 . . . . . . 7  |-  ( ( y  =  ( ( RR  _D  F ) `
 U )  /\  z  e.  ( X  \  { U } ) )  ->  ( (
( z  =/=  U  /\  ( abs `  (
z  -  U ) )  <  u )  ->  ( abs `  (
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  -  ( ( RR  _D  F ) `  U
) ) )  < 
y )  <->  ( (
z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  ( ( RR 
_D  F ) `  U ) ) )  <  ( ( RR 
_D  F ) `  U ) ) ) )
5352ralbidva 2572 . . . . . 6  |-  ( y  =  ( ( RR 
_D  F ) `  U )  ->  ( A. z  e.  ( X  \  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) `  z )  -  (
( RR  _D  F
) `  U )
) )  <  y
)  <->  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  ( ( RR 
_D  F ) `  U ) ) )  <  ( ( RR 
_D  F ) `  U ) ) ) )
5453rexbidv 2577 . . . . 5  |-  ( y  =  ( ( RR 
_D  F ) `  U )  ->  ( E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  -  ( ( RR  _D  F ) `  U
) ) )  < 
y )  <->  E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  ( ( RR 
_D  F ) `  U ) ) )  <  ( ( RR 
_D  F ) `  U ) ) ) )
5554rspcv 2893 . . . 4  |-  ( ( ( RR  _D  F
) `  U )  e.  RR+  ->  ( A. y  e.  RR+  E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  -  ( ( RR  _D  F ) `  U
) ) )  < 
y )  ->  E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  ( ( RR 
_D  F ) `  U ) ) )  <  ( ( RR 
_D  F ) `  U ) ) ) )
5610, 41, 55sylc 56 . . 3  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  ( ( RR 
_D  F ) `  U ) ) )  <  ( ( RR 
_D  F ) `  U ) ) )
571ad3antrrr 710 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  F : X --> RR )
582ad3antrrr 710 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  X  C_  RR )
5929ad3antrrr 710 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  U  e.  ( A (,) B ) )
6028ad3antrrr 710 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  -> 
( A (,) B
)  C_  X )
615ad3antrrr 710 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  U  e.  dom  ( RR 
_D  F ) )
62 dvferm1.r . . . . . . 7  |-  ( ph  ->  A. y  e.  ( U (,) B ) ( F `  y
)  <_  ( F `  U ) )
6362ad3antrrr 710 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  A. y  e.  ( U (,) B ) ( F `  y )  <_  ( F `  U ) )
64 simpllr 735 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  -> 
0  <  ( ( RR  _D  F ) `  U ) )
65 simplr 731 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  u  e.  RR+ )
66 simpr 447 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  A. z  e.  ( X  \  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )
67 eqid 2296 . . . . . 6  |-  ( ( U  +  if ( B  <_  ( U  +  u ) ,  B ,  ( U  +  u ) ) )  /  2 )  =  ( ( U  +  if ( B  <_  ( U  +  u ) ,  B ,  ( U  +  u ) ) )  /  2 )
6857, 58, 59, 60, 61, 63, 64, 65, 66, 67dvferm1lem 19347 . . . . 5  |-  -.  (
( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  u  e.  RR+ )  /\  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )
6968imnani 412 . . . 4  |-  ( ( ( ph  /\  0  <  ( ( RR  _D  F ) `  U
) )  /\  u  e.  RR+ )  ->  -.  A. z  e.  ( X 
\  { U }
) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  < 
u )  ->  ( abs `  ( ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  -  ( ( RR  _D  F ) `  U
) ) )  < 
( ( RR  _D  F ) `  U
) ) )
7069nrexdv 2659 . . 3  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  -.  E. u  e.  RR+  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  u )  -> 
( abs `  (
( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  ( ( RR 
_D  F ) `  U ) ) )  <  ( ( RR 
_D  F ) `  U ) ) )
7156, 70pm2.65da 559 . 2  |-  ( ph  ->  -.  0  <  (
( RR  _D  F
) `  U )
)
72 0re 8854 . . 3  |-  0  e.  RR
73 lenlt 8917 . . 3  |-  ( ( ( ( RR  _D  F ) `  U
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  U )  <_  0  <->  -.  0  <  ( ( RR  _D  F ) `
 U ) ) )
747, 72, 73sylancl 643 . 2  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  U )  <_  0  <->  -.  0  <  ( ( RR  _D  F ) `
 U ) ) )
7571, 74mpbird 223 1  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  <_  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    \ cdif 3162    C_ wss 3165   ifcif 3578   {csn 3653   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   RR+crp 10370   (,)cioo 10672   abscabs 11735   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393   intcnt 16770   lim CC climc 19228    _D cdv 19229
This theorem is referenced by:  dvferm  19351  dvivthlem1  19371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-cncf 18398  df-limc 19232  df-dv 19233
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