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Theorem dvferm1 21301
Description: One-sided version of dvferm 21304. 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 21269 . . . . . . . 8  |-  ( ( F : X --> RR  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
41, 2, 3syl2anc 656 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
5 dvferm.d . . . . . . 7  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
64, 5ffvelrnd 5834 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
76anim1i 565 . . . . 5  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( (
( RR  _D  F
) `  U )  e.  RR  /\  0  < 
( ( RR  _D  F ) `  U
) ) )
8 elrp 10983 . . . . 5  |-  ( ( ( RR  _D  F
) `  U )  e.  RR+  <->  ( ( ( RR  _D  F ) `
 U )  e.  RR  /\  0  < 
( ( RR  _D  F ) `  U
) ) )
97, 8sylibr 212 . . . 4  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( ( RR  _D  F ) `  U )  e.  RR+ )
10 dvf 21226 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
11 ffun 5551 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
12 funfvbrb 5806 . . . . . . . . . . 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 196 . . . . . . . . 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 9329 . . . . . . . . . . 11  |-  RR  C_  CC
1918a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
20 fss 5557 . . . . . . . . . . 11  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
211, 18, 20sylancl 657 . . . . . . . . . 10  |-  ( ph  ->  F : X --> CC )
2215, 16, 17, 19, 21, 2eldv 21217 . . . . . . . . 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 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 ) ) )
2423simprd 460 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) lim CC  U ) )
2524adantr 462 . . . . . 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 3358 . . . . . . . . . 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 3347 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3021, 26, 29dvlem 21215 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3130, 17fmptd 5857 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3231adantr 462 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3326adantr 462 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  X  C_  CC )
3433ssdifssd 3484 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( X  \  { U } ) 
C_  CC )
3526, 29sseldd 3347 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3635adantr 462 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  U  e.  CC )
3732, 34, 36ellimc3 21198 . . . . . 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 210 . . . . 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 460 . . . 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 5681 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4140oveq1d 6097 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
42 oveq1 6089 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4341, 42oveq12d 6100 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
44 ovex 6107 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4543, 17, 44fvmpt 5764 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4645oveq1d 6097 . . . . . . . . . 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 5685 . . . . . . . . 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 22 . . . . . . . . 9  |-  ( y  =  ( ( RR 
_D  F ) `  U )  ->  y  =  ( ( RR 
_D  F ) `  U ) )
4947, 48breqan12rd 4298 . . . . . . . 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 316 . . . . . . 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 2723 . . . . . 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 2728 . . . . 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 3060 . . . 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 60 . . 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 724 . . . . . 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 724 . . . . . 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 724 . . . . . 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 724 . . . . . 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 724 . . . . . 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 724 . . . . . 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 753 . . . . . 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 749 . . . . . 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 458 . . . . . 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 21300 . . . . 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 423 . . . 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 2811 . . 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 573 . 2  |-  ( ph  ->  -.  0  <  (
( RR  _D  F
) `  U )
)
70 0re 9376 . . 3  |-  0  e.  RR
71 lenlt 9443 . . 3  |-  ( ( ( ( RR  _D  F ) `  U
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  U )  <_  0  <->  -.  0  <  ( ( RR  _D  F ) `
 U ) ) )
726, 70, 71sylancl 657 . 2  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  U )  <_  0  <->  -.  0  <  ( ( RR  _D  F ) `
 U ) ) )
7369, 72mpbird 232 1  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  <_  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1757    =/= wne 2598   A.wral 2707   E.wrex 2708    \ cdif 3315    C_ wss 3318   ifcif 3781   {csn 3867   class class class wbr 4282    e. cmpt 4340   dom cdm 4829   Fun wfun 5402   -->wf 5404   ` cfv 5408  (class class class)co 6082   CCcc 9270   RRcr 9271   0cc0 9272    + caddc 9275    < clt 9408    <_ cle 9409    - cmin 9585    / cdiv 9983   2c2 10361   RR+crp 10981   (,)cioo 11290   abscabs 12709   ↾t crest 14344   TopOpenctopn 14345  ℂfldccnfld 17664   intcnt 18465   lim CC climc 21181    _D cdv 21182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349  ax-pre-sup 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-iin 4164  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-om 6468  df-1st 6568  df-2nd 6569  df-recs 6820  df-rdg 6854  df-1o 6910  df-oadd 6914  df-er 7091  df-map 7206  df-pm 7207  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-fi 7651  df-sup 7681  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-nn 10313  df-2 10370  df-3 10371  df-4 10372  df-5 10373  df-6 10374  df-7 10375  df-8 10376  df-9 10377  df-10 10378  df-n0 10570  df-z 10637  df-dec 10746  df-uz 10852  df-q 10944  df-rp 10982  df-xneg 11079  df-xadd 11080  df-xmul 11081  df-ioo 11294  df-icc 11297  df-fz 11427  df-seq 11793  df-exp 11852  df-cj 12574  df-re 12575  df-im 12576  df-sqr 12710  df-abs 12711  df-struct 14161  df-ndx 14162  df-slot 14163  df-base 14164  df-plusg 14236  df-mulr 14237  df-starv 14238  df-tset 14242  df-ple 14243  df-ds 14245  df-unif 14246  df-rest 14346  df-topn 14347  df-topgen 14367  df-psmet 17655  df-xmet 17656  df-met 17657  df-bl 17658  df-mopn 17659  df-fbas 17660  df-fg 17661  df-cnfld 17665  df-top 18347  df-bases 18349  df-topon 18350  df-topsp 18351  df-cld 18467  df-ntr 18468  df-cls 18469  df-nei 18546  df-lp 18584  df-perf 18585  df-cn 18675  df-cnp 18676  df-haus 18763  df-fil 19263  df-fm 19355  df-flim 19356  df-flf 19357  df-xms 19739  df-ms 19740  df-cncf 20298  df-limc 21185  df-dv 21186
This theorem is referenced by:  dvferm  21304  dvivthlem1  21324
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