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Theorem dvferm1 22254
Description: One-sided version of dvferm 22257. 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 22222 . . . . . . . 8  |-  ( ( F : X --> RR  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
41, 2, 3syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
5 dvferm.d . . . . . . 7  |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )
64, 5ffvelrnd 6033 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  RR )
76anim1i 568 . . . . 5  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( (
( RR  _D  F
) `  U )  e.  RR  /\  0  < 
( ( RR  _D  F ) `  U
) ) )
8 elrp 11234 . . . . 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 22179 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
11 ffun 5739 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
12 funfvbrb 6001 . . . . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
16 eqid 2467 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
17 eqid 2467 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
18 ax-resscn 9561 . . . . . . . . . . 11  |-  RR  C_  CC
1918a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
20 fss 5745 . . . . . . . . . . 11  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
211, 18, 20sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  F : X --> CC )
2215, 16, 17, 19, 21, 2eldv 22170 . . . . . . . . 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 463 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  F ) `  U
)  e.  ( ( x  e.  ( X 
\  { U }
)  |->  ( ( ( F `  x )  -  ( F `  U ) )  / 
( x  -  U
) ) ) lim CC  U ) )
2524adantr 465 . . . . . 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 3521 . . . . . . . . . 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 3510 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3021, 26, 29dvlem 22168 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3130, 17fmptd 6056 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3231adantr 465 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) : ( X  \  { U } ) --> CC )
3326adantr 465 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  X  C_  CC )
3433ssdifssd 3647 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( X  \  { U } ) 
C_  CC )
3526, 29sseldd 3510 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3635adantr 465 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  U  e.  CC )
3732, 34, 36ellimc3 22151 . . . . . 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 463 . . . 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 5872 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4140oveq1d 6310 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
42 oveq1 6302 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4341, 42oveq12d 6313 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
44 ovex 6320 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4543, 17, 44fvmpt 5957 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4645oveq1d 6310 . . . . . . . . . 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 5876 . . . . . . . . 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 4469 . . . . . . . 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 2903 . . . . . 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 2978 . . . . 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 3215 . . . 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 729 . . . . . 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 729 . . . . . 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 729 . . . . . 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 729 . . . . . 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 729 . . . . . 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 729 . . . . . 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 758 . . . . . 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 754 . . . . . 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 461 . . . . . 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 2467 . . . . . 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 22253 . . . . 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 2923 . . 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 576 . 2  |-  ( ph  ->  -.  0  <  (
( RR  _D  F
) `  U )
)
70 0re 9608 . . 3  |-  0  e.  RR
71 lenlt 9675 . . 3  |-  ( ( ( ( RR  _D  F ) `  U
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  U )  <_  0  <->  -.  0  <  ( ( RR  _D  F ) `
 U ) ) )
726, 70, 71sylancl 662 . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    \ cdif 3478    C_ wss 3481   ifcif 3945   {csn 4033   class class class wbr 4453    |-> cmpt 4511   dom cdm 5005   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    + caddc 9507    < clt 9640    <_ cle 9641    - cmin 9817    / cdiv 10218   2c2 10597   RR+crp 11232   (,)cioo 11541   abscabs 13047   ↾t crest 14693   TopOpenctopn 14694  ℂfldccnfld 18290   intcnt 19386   lim CC climc 22134    _D cdv 22135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-icc 11548  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-cncf 21250  df-limc 22138  df-dv 22139
This theorem is referenced by:  dvferm  22257  dvivthlem1  22277
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