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Theorem dvferm1 21460
Description: One-sided version of dvferm 21463. 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 21428 . . . . . . . 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 5847 . . . . . 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 10996 . . . . 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 21385 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
11 ffun 5564 . . . . . . . . . . 11  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
12 funfvbrb 5819 . . . . . . . . . . 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 2443 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
16 eqid 2443 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
17 eqid 2443 . . . . . . . . . 10  |-  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )  =  ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) ) )
18 ax-resscn 9342 . . . . . . . . . . 11  |-  RR  C_  CC
1918a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
20 fss 5570 . . . . . . . . . . 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 21376 . . . . . . . . 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 3371 . . . . . . . . . 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 3360 . . . . . . . . . 10  |-  ( ph  ->  U  e.  X )
3021, 26, 29dvlem 21374 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X  \  { U } ) )  -> 
( ( ( F `
 x )  -  ( F `  U ) )  /  ( x  -  U ) )  e.  CC )
3130, 17fmptd 5870 . . . . . . . 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 3497 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  ( X  \  { U } ) 
C_  CC )
3526, 29sseldd 3360 . . . . . . . 8  |-  ( ph  ->  U  e.  CC )
3635adantr 465 . . . . . . 7  |-  ( (
ph  /\  0  <  ( ( RR  _D  F
) `  U )
)  ->  U  e.  CC )
3732, 34, 36ellimc3 21357 . . . . . 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 5694 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
4140oveq1d 6109 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( F `  x
)  -  ( F `
 U ) )  =  ( ( F `
 z )  -  ( F `  U ) ) )
42 oveq1 6101 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
x  -  U )  =  ( z  -  U ) )
4341, 42oveq12d 6112 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( F `  x )  -  ( F `  U )
)  /  ( x  -  U ) )  =  ( ( ( F `  z )  -  ( F `  U ) )  / 
( z  -  U
) ) )
44 ovex 6119 . . . . . . . . . . . 12  |-  ( ( ( F `  z
)  -  ( F `
 U ) )  /  ( z  -  U ) )  e. 
_V
4543, 17, 44fvmpt 5777 . . . . . . . . . . 11  |-  ( z  e.  ( X  \  { U } )  -> 
( ( x  e.  ( X  \  { U } )  |->  ( ( ( F `  x
)  -  ( F `
 U ) )  /  ( x  -  U ) ) ) `
 z )  =  ( ( ( F `
 z )  -  ( F `  U ) )  /  ( z  -  U ) ) )
4645oveq1d 6109 . . . . . . . . . 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 5698 . . . . . . . . 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 4311 . . . . . . . 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 2734 . . . . . 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 2739 . . . . 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 3072 . . . 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 2443 . . . . . 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 21459 . . . . 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 2822 . . 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 9389 . . 3  |-  0  e.  RR
71 lenlt 9456 . . 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 1369    e. wcel 1756    =/= wne 2609   A.wral 2718   E.wrex 2719    \ cdif 3328    C_ wss 3331   ifcif 3794   {csn 3880   class class class wbr 4295    e. cmpt 4353   dom cdm 4843   Fun wfun 5415   -->wf 5417   ` cfv 5421  (class class class)co 6094   CCcc 9283   RRcr 9284   0cc0 9285    + caddc 9288    < clt 9421    <_ cle 9422    - cmin 9598    / cdiv 9996   2c2 10374   RR+crp 10994   (,)cioo 11303   abscabs 12726   ↾t crest 14362   TopOpenctopn 14363  ℂfldccnfld 17821   intcnt 18624   lim CC climc 21340    _D cdv 21341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fi 7664  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-q 10957  df-rp 10995  df-xneg 11092  df-xadd 11093  df-xmul 11094  df-ioo 11307  df-icc 11310  df-fz 11441  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-plusg 14254  df-mulr 14255  df-starv 14256  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-rest 14364  df-topn 14365  df-topgen 14385  df-psmet 17812  df-xmet 17813  df-met 17814  df-bl 17815  df-mopn 17816  df-fbas 17817  df-fg 17818  df-cnfld 17822  df-top 18506  df-bases 18508  df-topon 18509  df-topsp 18510  df-cld 18626  df-ntr 18627  df-cls 18628  df-nei 18705  df-lp 18743  df-perf 18744  df-cn 18834  df-cnp 18835  df-haus 18922  df-fil 19422  df-fm 19514  df-flim 19515  df-flf 19516  df-xms 19898  df-ms 19899  df-cncf 20457  df-limc 21344  df-dv 21345
This theorem is referenced by:  dvferm  21463  dvivthlem1  21483
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