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Theorem dvferm2 19824
Description: One-sided version of dvferm 19825. 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 19790 . . . . . . . . 9  |-  ( ( F : X --> RR  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
41, 2, 3syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
5 dvferm.d . . . . . . . 8  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
64, 5ffvelrnd 5830 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
76adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( ( RR  _D  F ) `  U )  e.  RR )
87renegcld 9420 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR )
96lt0neg1d 9552 . . . . . 6  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  U )  <  0  <->  0  <  -u ( ( RR 
_D  F ) `  U ) ) )
109biimpa 471 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  0  <  -u ( ( RR  _D  F ) `  U
) )
118, 10elrpd 10602 . . . 4  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR+ )
12 dvf 19747 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
13 ffun 5552 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
14 funfvbrb 5802 . . . . . . . . . . 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 189 . . . . . . . . 9  |-  ( ph  ->  U ( RR  _D  F ) ( ( RR  _D  F ) `
 U ) )
17 eqid 2404 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
18 eqid 2404 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
19 eqid 2404 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
20 ax-resscn 9003 . . . . . . . . . . 11  |-  RR  C_  CC
2120a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
22 fss 5558 . . . . . . . . . . 11  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
231, 20, 22sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  F : X --> CC )
2417, 18, 19, 21, 23, 2eldv 19738 . . . . . . . . 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 202 . . . . . . . 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 450 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) lim CC  U ) )
2726adantr 452 . . . . . 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 3320 . . . . . . . . . 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 3309 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3223, 28, 31dvlem 19736 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3332, 19fmptd 5852 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3433adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3528adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  X  C_  CC )
3635ssdifssd 3445 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( X  \  { U } ) 
C_  CC )
3728, 31sseldd 3309 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3837adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  U  e.  CC )
3934, 36, 38ellimc3 19719 . . . . . 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 202 . . . . 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 450 . . . 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 5687 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4342oveq1d 6055 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
44 oveq1 6047 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4543, 44oveq12d 6058 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
46 ovex 6065 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4745, 19, 46fvmpt 5765 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4847oveq1d 6055 . . . . . . . . . 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 5691 . . . . . . . . 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 20 . . . . . . . . 9  |-  ( y  =  -u ( ( RR 
_D  F ) `  U )  ->  y  =  -u ( ( RR 
_D  F ) `  U ) )
5149, 50breqan12rd 4188 . . . . . . . 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 308 . . . . . . 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 2682 . . . . . 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 2687 . . . . 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 3008 . . . 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 58 . . 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 711 . . . . . 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 711 . . . . . 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 711 . . . . . 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 711 . . . . . 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 711 . . . . . 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 711 . . . . . 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 736 . . . . . 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 732 . . . . . 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 448 . . . . . 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 2404 . . . . . 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 19823 . . . . 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 413 . . . 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 2769 . . 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 560 . 2  |-  ( ph  ->  -.  ( ( RR 
_D  F ) `  U )  <  0
)
72 0re 9047 . . 3  |-  0  e.  RR
73 lenlt 9110 . . 3  |-  ( ( 0  e.  RR  /\  ( ( RR  _D  F ) `  U
)  e.  RR )  ->  ( 0  <_ 
( ( RR  _D  F ) `  U
)  <->  -.  ( ( RR  _D  F ) `  U )  <  0
) )
7472, 6, 73sylancr 645 . 2  |-  ( ph  ->  ( 0  <_  (
( RR  _D  F
) `  U )  <->  -.  ( ( RR  _D  F ) `  U
)  <  0 ) )
7571, 74mpbird 224 1  |-  ( ph  ->  0  <_  ( ( RR  _D  F ) `  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    \ cdif 3277    C_ wss 3280   ifcif 3699   {csn 3774   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   2c2 10005   RR+crp 10568   (,)cioo 10872   abscabs 11994   ↾t crest 13603   TopOpenctopn 13604  ℂfldccnfld 16658   intcnt 17036   lim CC climc 19702    _D cdv 19703
This theorem is referenced by:  dvferm  19825  dvivthlem1  19845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-cncf 18861  df-limc 19706  df-dv 19707
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