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Theorem dvferm2 22807
Description: One-sided version of dvferm 22808. A point  U which is the local maximum of its left neighborhood has derivative at least 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 ) )
dvferm2.r  |-  ( ph  ->  A. y  e.  ( A (,) U ) ( F `  y
)  <_  ( F `  U ) )
Assertion
Ref Expression
dvferm2  |-  ( ph  ->  0  <_  ( ( RR  _D  F ) `  U ) )
Distinct variable groups:    y, A    y, B    y, F    y, U    y, X    ph, y

Proof of Theorem dvferm2
Dummy variables  z  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvferm.a . . . . . . . . 9  |-  ( ph  ->  F : X --> RR )
2 dvferm.b . . . . . . . . 9  |-  ( ph  ->  X  C_  RR )
3 dvfre 22773 . . . . . . . . 9  |-  ( ( F : X --> RR  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
41, 2, 3syl2anc 665 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
5 dvferm.d . . . . . . . 8  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
64, 5ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
76adantr 466 . . . . . 6  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( ( RR  _D  F ) `  U )  e.  RR )
87renegcld 10045 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR )
96lt0neg1d 10182 . . . . . 6  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  U )  <  0  <->  0  <  -u ( ( RR 
_D  F ) `  U ) ) )
109biimpa 486 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  0  <  -u ( ( RR  _D  F ) `  U
) )
118, 10elrpd 11338 . . . 4  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR+ )
12 dvf 22730 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
13 ffun 5748 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
14 funfvbrb 6010 . . . . . . . . . . 11  |-  ( Fun  ( RR  _D  F
)  ->  ( U  e.  dom  ( RR  _D  F )  <->  U ( RR  _D  F ) ( ( RR  _D  F
) `  U )
) )
1512, 13, 14mp2b 10 . . . . . . . . . 10  |-  ( U  e.  dom  ( RR 
_D  F )  <->  U ( RR  _D  F ) ( ( RR  _D  F
) `  U )
)
165, 15sylib 199 . . . . . . . . 9  |-  ( ph  ->  U ( RR  _D  F ) ( ( RR  _D  F ) `
 U ) )
17 eqid 2429 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
18 eqid 2429 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
19 eqid 2429 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
20 ax-resscn 9595 . . . . . . . . . . 11  |-  RR  C_  CC
2120a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
22 fss 5754 . . . . . . . . . . 11  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
231, 20, 22sylancl 666 . . . . . . . . . 10  |-  ( ph  ->  F : X --> CC )
2417, 18, 19, 21, 23, 2eldv 22721 . . . . . . . . 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 ) ) ) )
2516, 24mpbid 213 . . . . . . . 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 ) ) )
2625simprd 464 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) lim CC  U ) )
2726adantr 466 . . . . . 6  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( ( RR  _D  F ) `  U )  e.  ( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) lim
CC  U ) )
282, 20syl6ss 3482 . . . . . . . . . 10  |-  ( ph  ->  X  C_  CC )
29 dvferm.s . . . . . . . . . . 11  |-  ( ph  ->  ( A (,) B
)  C_  X )
30 dvferm.u . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( A (,) B ) )
3129, 30sseldd 3471 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3223, 28, 31dvlem 22719 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3332, 19fmptd 6061 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3433adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3528adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  X  C_  CC )
3635ssdifssd 3609 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( X  \  { U } ) 
C_  CC )
3728, 31sseldd 3471 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3837adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  U  e.  CC )
3934, 36, 38ellimc3 22702 . . . . . 6  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( (
( 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 ) ) ) )
4027, 39mpbid 213 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( (
( 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 464 . . . 4  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  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 5881 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4342oveq1d 6320 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
44 oveq1 6312 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4543, 44oveq12d 6323 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
46 ovex 6333 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4745, 19, 46fvmpt 5964 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4847oveq1d 6320 . . . . . . . . . 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 5885 . . . . . . . . 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 23 . . . . . . . . 9  |-  ( y  =  -u ( ( RR 
_D  F ) `  U )  ->  y  =  -u ( ( RR 
_D  F ) `  U ) )
5149, 50breqan12rd 4442 . . . . . . . 8  |-  ( ( y  =  -u (
( 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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )
5251imbi2d 317 . . . . . . 7  |-  ( ( y  =  -u (
( 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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) ) )
5352ralbidva 2868 . . . . . 6  |-  ( y  =  -u ( ( 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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) ) )
5453rexbidv 2946 . . . . 5  |-  ( y  =  -u ( ( 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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) ) )
5554rspcv 3184 . . . 4  |-  ( -u ( ( 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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) ) )
5611, 41, 55sylc 62 . . 3  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )
571ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )  ->  F : X --> RR )
582ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )  ->  X  C_  RR )
5930ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )  ->  U  e.  ( A (,) B ) )
6029ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )  ->  ( A (,) B )  C_  X
)
615ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )  ->  U  e.  dom  ( RR  _D  F
) )
62 dvferm2.r . . . . . . 7  |-  ( ph  ->  A. y  e.  ( A (,) U ) ( F `  y
)  <_  ( F `  U ) )
6362ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )  ->  A. y  e.  ( A (,) U ) ( F `  y
)  <_  ( F `  U ) )
64 simpllr 767 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )  ->  ( ( RR 
_D  F ) `  U )  <  0
)
65 simplr 760 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )  ->  u  e.  RR+ )
66 simpr 462 . . . . . 6  |-  ( ( ( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )
67 eqid 2429 . . . . . 6  |-  ( ( if ( A  <_ 
( U  -  u
) ,  ( U  -  u ) ,  A )  +  U
)  /  2 )  =  ( ( if ( A  <_  ( U  -  u ) ,  ( U  -  u ) ,  A
)  +  U )  /  2 )
6857, 58, 59, 60, 61, 63, 64, 65, 66, 67dvferm2lem 22806 . . . . 5  |-  -.  (
( ( ph  /\  ( ( RR  _D  F ) `  U
)  <  0 )  /\  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )
6968imnani 424 . . . 4  |-  ( ( ( ph  /\  (
( RR  _D  F
) `  U )  <  0 )  /\  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
) ) )  <  -u ( ( RR  _D  F ) `  U
) ) )
7069nrexdv 2888 . . 3  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -.  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 ) ) )  <  -u ( ( RR 
_D  F ) `  U ) ) )
7156, 70pm2.65da 578 . 2  |-  ( ph  ->  -.  ( ( RR 
_D  F ) `  U )  <  0
)
72 0re 9642 . . 3  |-  0  e.  RR
73 lenlt 9711 . . 3  |-  ( ( 0  e.  RR  /\  ( ( RR  _D  F ) `  U
)  e.  RR )  ->  ( 0  <_ 
( ( RR  _D  F ) `  U
)  <->  -.  ( ( RR  _D  F ) `  U )  <  0
) )
7472, 6, 73sylancr 667 . 2  |-  ( ph  ->  ( 0  <_  (
( RR  _D  F
) `  U )  <->  -.  ( ( RR  _D  F ) `  U
)  <  0 ) )
7571, 74mpbird 235 1  |-  ( ph  ->  0  <_  ( ( RR  _D  F ) `  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783    \ cdif 3439    C_ wss 3442   ifcif 3915   {csn 4002   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538    + caddc 9541    < clt 9674    <_ cle 9675    - cmin 9859   -ucneg 9860    / cdiv 10268   2c2 10659   RR+crp 11302   (,)cioo 11635   abscabs 13276   ↾t crest 15269   TopOpenctopn 15270  ℂfldccnfld 18896   intcnt 19954   lim CC climc 22685    _D cdv 22686
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-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
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  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-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-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  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-icc 11642  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15077  df-ndx 15078  df-slot 15079  df-base 15080  df-plusg 15156  df-mulr 15157  df-starv 15158  df-tset 15162  df-ple 15163  df-ds 15165  df-unif 15166  df-rest 15271  df-topn 15272  df-topgen 15292  df-psmet 18888  df-xmet 18889  df-met 18890  df-bl 18891  df-mopn 18892  df-fbas 18893  df-fg 18894  df-cnfld 18897  df-top 19843  df-bases 19844  df-topon 19845  df-topsp 19846  df-cld 19956  df-ntr 19957  df-cls 19958  df-nei 20036  df-lp 20074  df-perf 20075  df-cn 20165  df-cnp 20166  df-haus 20253  df-fil 20783  df-fm 20875  df-flim 20876  df-flf 20877  df-xms 21257  df-ms 21258  df-cncf 21797  df-limc 22689  df-dv 22690
This theorem is referenced by:  dvferm  22808  dvivthlem1  22828
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