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Theorem dvferm1 19857
Description: One-sided version of dvferm 19860. A point  U which is the local maximum of its right neighborhood has derivative at most 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 ) )
dvferm1.r  |-  ( ph  ->  A. y  e.  ( U (,) B ) ( F `  y
)  <_  ( F `  U ) )
Assertion
Ref Expression
dvferm1  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  <_  0 )
Distinct variable groups:    y, A    y, B    y, F    y, U    y, X    ph, y

Proof of Theorem dvferm1
Dummy variables  z  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvferm.a . . . . . . . 8  |-  ( ph  ->  F : X --> RR )
2 dvferm.b . . . . . . . 8  |-  ( ph  ->  X  C_  RR )
3 dvfre 19825 . . . . . . . 8  |-  ( ( F : X --> RR  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
41, 2, 3syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
5 dvferm.d . . . . . . 7  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
64, 5ffvelrnd 5862 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
76anim1i 552 . . . . 5  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( (
( RR  _D  F
) `  U )  e.  RR  /\  0  < 
( ( RR  _D  F ) `  U
) ) )
8 elrp 10603 . . . . 5  |-  ( ( ( RR  _D  F
) `  U )  e.  RR+  <->  ( ( ( RR  _D  F ) `
 U )  e.  RR  /\  0  < 
( ( RR  _D  F ) `  U
) ) )
97, 8sylibr 204 . . . 4  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( ( RR  _D  F ) `  U )  e.  RR+ )
10 dvf 19782 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
11 ffun 5584 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
12 funfvbrb 5834 . . . . . . . . . . 11  |-  ( Fun  ( RR  _D  F
)  ->  ( U  e.  dom  ( RR  _D  F )  <->  U ( RR  _D  F ) ( ( RR  _D  F
) `  U )
) )
1310, 11, 12mp2b 10 . . . . . . . . . 10  |-  ( U  e.  dom  ( RR 
_D  F )  <->  U ( RR  _D  F ) ( ( RR  _D  F
) `  U )
)
145, 13sylib 189 . . . . . . . . 9  |-  ( ph  ->  U ( RR  _D  F ) ( ( RR  _D  F ) `
 U ) )
15 eqid 2435 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
16 eqid 2435 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
17 eqid 2435 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
18 ax-resscn 9036 . . . . . . . . . . 11  |-  RR  C_  CC
1918a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
20 fss 5590 . . . . . . . . . . 11  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
211, 18, 20sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  F : X --> CC )
2215, 16, 17, 19, 21, 2eldv 19773 . . . . . . . . 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 ) ) ) )
2314, 22mpbid 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 ) ) )
2423simprd 450 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) lim CC  U ) )
2524adantr 452 . . . . . 6  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( ( RR  _D  F ) `  U )  e.  ( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) lim
CC  U ) )
262, 18syl6ss 3352 . . . . . . . . . 10  |-  ( ph  ->  X  C_  CC )
27 dvferm.s . . . . . . . . . . 11  |-  ( ph  ->  ( A (,) B
)  C_  X )
28 dvferm.u . . . . . . . . . . 11  |-  ( ph  ->  U  e.  ( A (,) B ) )
2927, 28sseldd 3341 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3021, 26, 29dvlem 19771 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3130, 17fmptd 5884 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3231adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3326adantr 452 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  X  C_  CC )
3433ssdifssd 3477 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( X  \  { U } ) 
C_  CC )
3526, 29sseldd 3341 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3635adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  U  e.  CC )
3732, 34, 36ellimc3 19754 . . . . . 6  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( (
( 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 ) ) ) )
3825, 37mpbid 202 . . . . 5  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  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 ) ) )
3938simprd 450 . . . 4  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  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 ) )
40 fveq2 5719 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4140oveq1d 6087 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
42 oveq1 6079 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4341, 42oveq12d 6090 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
44 ovex 6097 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4543, 17, 44fvmpt 5797 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4645oveq1d 6087 . . . . . . . . . 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 )
) )
4746fveq2d 5723 . . . . . . . . 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 ) ) ) )
48 id 20 . . . . . . . . 9  |-  ( y  =  ( ( RR 
_D  F ) `  U )  ->  y  =  ( ( RR 
_D  F ) `  U ) )
4947, 48breqan12rd 4220 . . . . . . . 8  |-  ( ( y  =  ( ( 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 ) ) )  <  ( ( RR 
_D  F ) `  U ) ) )
5049imbi2d 308 . . . . . . 7  |-  ( ( y  =  ( ( 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 ) ) )  <  ( ( RR 
_D  F ) `  U ) ) ) )
5150ralbidva 2713 . . . . . 6  |-  ( y  =  ( ( 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 ) ) )  <  ( ( RR 
_D  F ) `  U ) ) ) )
5251rexbidv 2718 . . . . 5  |-  ( y  =  ( ( 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 ) ) )  <  ( ( RR 
_D  F ) `  U ) ) ) )
5352rspcv 3040 . . . 4  |-  ( ( ( 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 ) ) )  <  ( ( RR 
_D  F ) `  U ) ) ) )
549, 39, 53sylc 58 . . 3  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  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 ) ) )  <  ( ( RR 
_D  F ) `  U ) ) )
551ad3antrrr 711 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  F : X --> RR )
562ad3antrrr 711 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  X  C_  RR )
5728ad3antrrr 711 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  U  e.  ( A (,) B ) )
5827ad3antrrr 711 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  -> 
( A (,) B
)  C_  X )
595ad3antrrr 711 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  U  e.  dom  ( RR 
_D  F ) )
60 dvferm1.r . . . . . . 7  |-  ( ph  ->  A. y  e.  ( U (,) B ) ( F `  y
)  <_  ( F `  U ) )
6160ad3antrrr 711 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  A. y  e.  ( U (,) B ) ( F `  y )  <_  ( F `  U ) )
62 simpllr 736 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  -> 
0  <  ( ( RR  _D  F ) `  U ) )
63 simplr 732 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )  ->  u  e.  RR+ )
64 simpr 448 . . . . . 6  |-  ( ( ( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( 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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )
65 eqid 2435 . . . . . 6  |-  ( ( U  +  if ( B  <_  ( U  +  u ) ,  B ,  ( U  +  u ) ) )  /  2 )  =  ( ( U  +  if ( B  <_  ( U  +  u ) ,  B ,  ( U  +  u ) ) )  /  2 )
6655, 56, 57, 58, 59, 61, 62, 63, 64, 65dvferm1lem 19856 . . . . 5  |-  -.  (
( ( ph  /\  0  <  ( ( RR 
_D  F ) `  U ) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )
6766imnani 413 . . . 4  |-  ( ( ( ph  /\  0  <  ( ( RR  _D  F ) `  U
) )  /\  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
) ) )  < 
( ( RR  _D  F ) `  U
) ) )
6867nrexdv 2801 . . 3  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  -.  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 ) ) )  <  ( ( RR 
_D  F ) `  U ) ) )
6954, 68pm2.65da 560 . 2  |-  ( ph  ->  -.  0  <  (
( RR  _D  F
) `  U )
)
70 0re 9080 . . 3  |-  0  e.  RR
71 lenlt 9143 . . 3  |-  ( ( ( ( RR  _D  F ) `  U
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  U )  <_  0  <->  -.  0  <  ( ( RR  _D  F ) `
 U ) ) )
726, 70, 71sylancl 644 . 2  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  U )  <_  0  <->  -.  0  <  ( ( RR  _D  F ) `
 U ) ) )
7369, 72mpbird 224 1  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  <_  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    \ cdif 3309    C_ wss 3312   ifcif 3731   {csn 3806   class class class wbr 4204    e. cmpt 4258   dom cdm 4869   Fun wfun 5439   -->wf 5441   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979    + caddc 8982    < clt 9109    <_ cle 9110    - cmin 9280    / cdiv 9666   2c2 10038   RR+crp 10601   (,)cioo 10905   abscabs 12027   ↾t crest 13636   TopOpenctopn 13637  ℂfldccnfld 16691   intcnt 17069   lim CC climc 19737    _D cdv 19738
This theorem is referenced by:  dvferm  19860  dvivthlem1  19880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-icc 10912  df-fz 11033  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-plusg 13530  df-mulr 13531  df-starv 13532  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-rest 13638  df-topn 13639  df-topgen 13655  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-cncf 18896  df-limc 19741  df-dv 19742
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