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Theorem dvferm2 21577
Description: One-sided version of dvferm 21578. 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 21543 . . . . . . . . 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 5945 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( ( RR  _D  F ) `  U )  e.  RR )
87renegcld 9878 . . . . 5  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR )
96lt0neg1d 10012 . . . . . 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 11128 . . . 4  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  -u ( ( RR  _D  F ) `
 U )  e.  RR+ )
12 dvf 21500 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
13 ffun 5661 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
14 funfvbrb 5917 . . . . . . . . . . 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 2451 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
18 eqid 2451 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
19 eqid 2451 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
20 ax-resscn 9442 . . . . . . . . . . 11  |-  RR  C_  CC
2120a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
22 fss 5667 . . . . . . . . . . 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 21491 . . . . . . . . 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 3468 . . . . . . . . . 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 3457 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3223, 28, 31dvlem 21489 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3332, 19fmptd 5968 . . . . . . . 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 3594 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  ( X  \  { U } ) 
C_  CC )
3728, 31sseldd 3457 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3837adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( RR  _D  F ) `  U )  <  0
)  ->  U  e.  CC )
3934, 36, 38ellimc3 21472 . . . . . 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 5791 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4342oveq1d 6207 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
44 oveq1 6199 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4543, 44oveq12d 6210 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
46 ovex 6217 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4745, 19, 46fvmpt 5875 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4847oveq1d 6207 . . . . . . . . . 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 5795 . . . . . . . . 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 4408 . . . . . . . 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 2836 . . . . . 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 2848 . . . . 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 3167 . . . 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 2451 . . . . . 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 21576 . . . . 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 2917 . . 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 9489 . . 3  |-  0  e.  RR
73 lenlt 9556 . . 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 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796    \ cdif 3425    C_ wss 3428   ifcif 3891   {csn 3977   class class class wbr 4392    |-> cmpt 4450   dom cdm 4940   Fun wfun 5512   -->wf 5514   ` cfv 5518  (class class class)co 6192   CCcc 9383   RRcr 9384   0cc0 9385    + caddc 9388    < clt 9521    <_ cle 9522    - cmin 9698   -ucneg 9699    / cdiv 10096   2c2 10474   RR+crp 11094   (,)cioo 11403   abscabs 12827   ↾t crest 14463   TopOpenctopn 14464  ℂfldccnfld 17929   intcnt 18739   lim CC climc 21455    _D cdv 21456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fi 7764  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-q 11057  df-rp 11095  df-xneg 11192  df-xadd 11193  df-xmul 11194  df-ioo 11407  df-icc 11410  df-fz 11541  df-seq 11910  df-exp 11969  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-plusg 14355  df-mulr 14356  df-starv 14357  df-tset 14361  df-ple 14362  df-ds 14364  df-unif 14365  df-rest 14465  df-topn 14466  df-topgen 14486  df-psmet 17920  df-xmet 17921  df-met 17922  df-bl 17923  df-mopn 17924  df-fbas 17925  df-fg 17926  df-cnfld 17930  df-top 18621  df-bases 18623  df-topon 18624  df-topsp 18625  df-cld 18741  df-ntr 18742  df-cls 18743  df-nei 18820  df-lp 18858  df-perf 18859  df-cn 18949  df-cnp 18950  df-haus 19037  df-fil 19537  df-fm 19629  df-flim 19630  df-flf 19631  df-xms 20013  df-ms 20014  df-cncf 20572  df-limc 21459  df-dv 21460
This theorem is referenced by:  dvferm  21578  dvivthlem1  21598
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