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Theorem dvferm2 22123
Description: One-sided version of dvferm 22124. 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 22089 . . . . . . . . 9  |-  ( ( F : X --> RR  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
41, 2, 3syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
5 dvferm.d . . . . . . . 8  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
64, 5ffvelrnd 6020 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( ( RR  _D  F ) `  U )  e.  RR )
87renegcld 9982 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR )
96lt0neg1d 10118 . . . . . 6  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  U )  <  0  <->  0  <  -u ( ( RR 
_D  F ) `  U ) ) )
109biimpa 484 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  0  <  -u ( ( RR  _D  F ) `  U
) )
118, 10elrpd 11250 . . . 4  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR+ )
12 dvf 22046 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
13 ffun 5731 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
14 funfvbrb 5992 . . . . . . . . . . 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 196 . . . . . . . . 9  |-  ( ph  ->  U ( RR  _D  F ) ( ( RR  _D  F ) `
 U ) )
17 eqid 2467 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
18 eqid 2467 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
19 eqid 2467 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
20 ax-resscn 9545 . . . . . . . . . . 11  |-  RR  C_  CC
2120a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
22 fss 5737 . . . . . . . . . . 11  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
231, 20, 22sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  F : X --> CC )
2417, 18, 19, 21, 23, 2eldv 22037 . . . . . . . . 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 210 . . . . . . . 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 463 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) lim CC  U ) )
2726adantr 465 . . . . . 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 3516 . . . . . . . . . 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 3505 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3223, 28, 31dvlem 22035 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3332, 19fmptd 6043 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3433adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3528adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  X  C_  CC )
3635ssdifssd 3642 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( X  \  { U } ) 
C_  CC )
3728, 31sseldd 3505 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3837adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  U  e.  CC )
3934, 36, 38ellimc3 22018 . . . . . 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 210 . . . . 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 463 . . . 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 5864 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4342oveq1d 6297 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
44 oveq1 6289 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4543, 44oveq12d 6300 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
46 ovex 6307 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4745, 19, 46fvmpt 5948 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4847oveq1d 6297 . . . . . . . . . 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 5868 . . . . . . . . 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 22 . . . . . . . . 9  |-  ( y  =  -u ( ( RR 
_D  F ) `  U )  ->  y  =  -u ( ( RR 
_D  F ) `  U ) )
5149, 50breqan12rd 4463 . . . . . . . 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 316 . . . . . . 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 2900 . . . . . 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 2973 . . . . 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 3210 . . . 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 60 . . 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 729 . . . . . 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 729 . . . . . 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 729 . . . . . 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 729 . . . . . 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 729 . . . . . 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 729 . . . . . 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 758 . . . . . 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 754 . . . . . 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 461 . . . . . 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 2467 . . . . . 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 22122 . . . . 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 423 . . . 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 2920 . . 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 576 . 2  |-  ( ph  ->  -.  ( ( RR 
_D  F ) `  U )  <  0
)
72 0re 9592 . . 3  |-  0  e.  RR
73 lenlt 9659 . . 3  |-  ( ( 0  e.  RR  /\  ( ( RR  _D  F ) `  U
)  e.  RR )  ->  ( 0  <_ 
( ( RR  _D  F ) `  U
)  <->  -.  ( ( RR  _D  F ) `  U )  <  0
) )
7472, 6, 73sylancr 663 . 2  |-  ( ph  ->  ( 0  <_  (
( RR  _D  F
) `  U )  <->  -.  ( ( RR  _D  F ) `  U
)  <  0 ) )
7571, 74mpbird 232 1  |-  ( ph  ->  0  <_  ( ( RR  _D  F ) `  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    \ cdif 3473    C_ wss 3476   ifcif 3939   {csn 4027   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488    + caddc 9491    < clt 9624    <_ cle 9625    - cmin 9801   -ucneg 9802    / cdiv 10202   2c2 10581   RR+crp 11216   (,)cioo 11525   abscabs 13026   ↾t crest 14672   TopOpenctopn 14673  ℂfldccnfld 18191   intcnt 19284   lim CC climc 22001    _D cdv 22002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-icc 11532  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-rest 14674  df-topn 14675  df-topgen 14695  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-cncf 21117  df-limc 22005  df-dv 22006
This theorem is referenced by:  dvferm  22124  dvivthlem1  22144
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