MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvferm2 Unicode version

Theorem dvferm2 19432
Description: One-sided version of dvferm 19433. 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 19398 . . . . . . . . 9  |-  ( ( F : X --> RR  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
41, 2, 3syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
5 dvferm.d . . . . . . . 8  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
6 ffvelrn 5743 . . . . . . . 8  |-  ( ( ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR 
/\  U  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  U )  e.  RR )
74, 5, 6syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
87adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( ( RR  _D  F ) `  U )  e.  RR )
98renegcld 9297 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR )
107lt0neg1d 9429 . . . . . 6  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  U )  <  0  <->  0  <  -u ( ( RR 
_D  F ) `  U ) ) )
1110biimpa 470 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  0  <  -u ( ( RR  _D  F ) `  U
) )
129, 11elrpd 10477 . . . 4  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR+ )
13 dvf 19355 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
14 ffun 5471 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
15 funfvbrb 5718 . . . . . . . . . . 11  |-  ( Fun  ( RR  _D  F
)  ->  ( U  e.  dom  ( RR  _D  F )  <->  U ( RR  _D  F ) ( ( RR  _D  F
) `  U )
) )
1613, 14, 15mp2b 9 . . . . . . . . . 10  |-  ( U  e.  dom  ( RR 
_D  F )  <->  U ( RR  _D  F ) ( ( RR  _D  F
) `  U )
)
175, 16sylib 188 . . . . . . . . 9  |-  ( ph  ->  U ( RR  _D  F ) ( ( RR  _D  F ) `
 U ) )
18 eqid 2358 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
19 eqid 2358 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
20 eqid 2358 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
21 ax-resscn 8881 . . . . . . . . . . 11  |-  RR  C_  CC
2221a1i 10 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
23 fss 5477 . . . . . . . . . . 11  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
241, 21, 23sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  F : X --> CC )
2518, 19, 20, 22, 24, 2eldv 19346 . . . . . . . . 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 ) ) ) )
2617, 25mpbid 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 ) ) )
2726simprd 449 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) lim CC  U ) )
2827adantr 451 . . . . . 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 ) )
292, 21syl6ss 3267 . . . . . . . . . 10  |-  ( ph  ->  X  C_  CC )
30 dvferm.s . . . . . . . . . . 11  |-  ( ph  ->  ( A (,) B
)  C_  X )
31 dvferm.u . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( A (,) B ) )
3230, 31sseldd 3257 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3324, 29, 32dvlem 19344 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3433, 20fmptd 5764 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3534adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
36 difss 3379 . . . . . . . 8  |-  ( X 
\  { U }
)  C_  X
3729adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  X  C_  CC )
3836, 37syl5ss 3266 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( X  \  { U } ) 
C_  CC )
3929, 32sseldd 3257 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
4039adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  U  e.  CC )
4135, 38, 40ellimc3 19327 . . . . . 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 ) ) ) )
4228, 41mpbid 201 . . . . 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 ) ) )
4342simprd 449 . . . 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 ) )
44 fveq2 5605 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4544oveq1d 5957 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
46 oveq1 5949 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4745, 46oveq12d 5960 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
48 ovex 5967 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4947, 20, 48fvmpt 5682 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
5049oveq1d 5957 . . . . . . . . . 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 )
) )
5150fveq2d 5609 . . . . . . . . 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 ) ) ) )
52 id 19 . . . . . . . . 9  |-  ( y  =  -u ( ( RR 
_D  F ) `  U )  ->  y  =  -u ( ( RR 
_D  F ) `  U ) )
5351, 52breqan12rd 4118 . . . . . . . 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 ) ) )
5453imbi2d 307 . . . . . . 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 ) ) ) )
5554ralbidva 2635 . . . . . 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 ) ) ) )
5655rexbidv 2640 . . . . 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 ) ) ) )
5756rspcv 2956 . . . 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 ) ) ) )
5812, 43, 57sylc 56 . . 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 ) ) )
591ad3antrrr 710 . . . . . 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 )
602ad3antrrr 710 . . . . . 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 )
6131ad3antrrr 710 . . . . . 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 ) )
6230ad3antrrr 710 . . . . . 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
)
635ad3antrrr 710 . . . . . 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
) )
64 dvferm2.r . . . . . . 7  |-  ( ph  ->  A. y  e.  ( A (,) U ) ( F `  y
)  <_  ( F `  U ) )
6564ad3antrrr 710 . . . . . 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 ) )
66 simpllr 735 . . . . . 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
)
67 simplr 731 . . . . . 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+ )
68 simpr 447 . . . . . 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 ) ) )
69 eqid 2358 . . . . . 6  |-  ( ( if ( A  <_ 
( U  -  u
) ,  ( U  -  u ) ,  A )  +  U
)  /  2 )  =  ( ( if ( A  <_  ( U  -  u ) ,  ( U  -  u ) ,  A
)  +  U )  /  2 )
7059, 60, 61, 62, 63, 65, 66, 67, 68, 69dvferm2lem 19431 . . . . 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 ) ) )
7170imnani 412 . . . 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
) ) )
7271nrexdv 2722 . . 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 ) ) )
7358, 72pm2.65da 559 . 2  |-  ( ph  ->  -.  ( ( RR 
_D  F ) `  U )  <  0
)
74 0re 8925 . . 3  |-  0  e.  RR
75 lenlt 8988 . . 3  |-  ( ( 0  e.  RR  /\  ( ( RR  _D  F ) `  U
)  e.  RR )  ->  ( 0  <_ 
( ( RR  _D  F ) `  U
)  <->  -.  ( ( RR  _D  F ) `  U )  <  0
) )
7674, 7, 75sylancr 644 . 2  |-  ( ph  ->  ( 0  <_  (
( RR  _D  F
) `  U )  <->  -.  ( ( RR  _D  F ) `  U
)  <  0 ) )
7773, 76mpbird 223 1  |-  ( ph  ->  0  <_  ( ( RR  _D  F ) `  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620    \ cdif 3225    C_ wss 3228   ifcif 3641   {csn 3716   class class class wbr 4102    e. cmpt 4156   dom cdm 4768   Fun wfun 5328   -->wf 5330   ` cfv 5334  (class class class)co 5942   CCcc 8822   RRcr 8823   0cc0 8824    + caddc 8827    < clt 8954    <_ cle 8955    - cmin 9124   -ucneg 9125    / cdiv 9510   2c2 9882   RR+crp 10443   (,)cioo 10745   abscabs 11809   ↾t crest 13418   TopOpenctopn 13419  ℂfldccnfld 16476   intcnt 16854   lim CC climc 19310    _D cdv 19311
This theorem is referenced by:  dvferm  19433  dvivthlem1  19453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-icc 10752  df-fz 10872  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-plusg 13312  df-mulr 13313  df-starv 13314  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-rest 13420  df-topn 13421  df-topgen 13437  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-cnfld 16477  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lp 16968  df-perf 16969  df-cn 17057  df-cnp 17058  df-haus 17143  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-xms 17981  df-ms 17982  df-cncf 18479  df-limc 19314  df-dv 19315
  Copyright terms: Public domain W3C validator