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Theorem dvcnvre 19856
Description: The derivative rule for inverse functions. If  F is a continuous and differentiable bijective function from  X to  Y which never has derivative  0, then  `' F is also differentiable, and its derivative is the reciprocal of the derivative of  F. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
dvcnvre.f  |-  ( ph  ->  F  e.  ( X
-cn-> RR ) )
dvcnvre.d  |-  ( ph  ->  dom  ( RR  _D  F )  =  X )
dvcnvre.z  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F
) )
dvcnvre.1  |-  ( ph  ->  F : X -1-1-onto-> Y )
Assertion
Ref Expression
dvcnvre  |-  ( ph  ->  ( RR  _D  `' F )  =  ( x  e.  Y  |->  ( 1  /  ( ( RR  _D  F ) `
 ( `' F `  x ) ) ) ) )
Distinct variable groups:    x, F    ph, x    x, X    x, Y

Proof of Theorem dvcnvre
Dummy variables  y 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21tgioo2 18787 . 2  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3 reex 9037 . . . 4  |-  RR  e.  _V
43prid1 3872 . . 3  |-  RR  e.  { RR ,  CC }
54a1i 11 . 2  |-  ( ph  ->  RR  e.  { RR ,  CC } )
6 retop 18748 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
7 dvcnvre.1 . . . . . . 7  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1ofo 5640 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
9 forn 5615 . . . . . . 7  |-  ( F : X -onto-> Y  ->  ran  F  =  Y )
107, 8, 93syl 19 . . . . . 6  |-  ( ph  ->  ran  F  =  Y )
11 dvcnvre.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X
-cn-> RR ) )
12 cncff 18876 . . . . . . 7  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
13 frn 5556 . . . . . . 7  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
1411, 12, 133syl 19 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1510, 14eqsstr3d 3343 . . . . 5  |-  ( ph  ->  Y  C_  RR )
16 uniretop 18749 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
1716ntrss2 17076 . . . . 5  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  Y  C_  RR )  ->  ( ( int `  ( topGen ` 
ran  (,) ) ) `  Y )  C_  Y
)
186, 15, 17sylancr 645 . . . 4  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  C_  Y )
19 f1ocnvfv2 5974 . . . . . . . 8  |-  ( ( F : X -1-1-onto-> Y  /\  x  e.  Y )  ->  ( F `  ( `' F `  x ) )  =  x )
207, 19sylan 458 . . . . . . 7  |-  ( (
ph  /\  x  e.  Y )  ->  ( F `  ( `' F `  x )
)  =  x )
21 f1ocnv 5646 . . . . . . . . . . . 12  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
22 f1of 5633 . . . . . . . . . . . 12  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
237, 21, 223syl 19 . . . . . . . . . . 11  |-  ( ph  ->  `' F : Y --> X )
2423ffvelrnda 5829 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  Y )  ->  ( `' F `  x )  e.  X )
25 dvcnvre.d . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  =  X )
26 dvbsss 19742 . . . . . . . . . . . . . . . 16  |-  dom  ( RR  _D  F )  C_  RR
2726a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
2825, 27eqsstr3d 3343 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  C_  RR )
2916ntrss2 17076 . . . . . . . . . . . . . 14  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  X  C_  RR )  ->  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X )  C_  X
)
306, 28, 29sylancr 645 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  C_  X )
31 ax-resscn 9003 . . . . . . . . . . . . . . . 16  |-  RR  C_  CC
3231a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  RR  C_  CC )
3311, 12syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
34 fss 5558 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
3533, 31, 34sylancl 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : X --> CC )
3632, 35, 28, 2, 1dvbssntr 19740 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
3725, 36eqsstr3d 3343 . . . . . . . . . . . . 13  |-  ( ph  ->  X  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  X
) )
3830, 37eqssd 3325 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  =  X )
3916isopn3 17085 . . . . . . . . . . . . 13  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  X  C_  RR )  ->  ( X  e.  ( topGen `  ran  (,) )  <->  ( ( int `  ( topGen `  ran  (,) )
) `  X )  =  X ) )
406, 28, 39sylancr 645 . . . . . . . . . . . 12  |-  ( ph  ->  ( X  e.  (
topGen `  ran  (,) )  <->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  =  X ) )
4138, 40mpbird 224 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( topGen ` 
ran  (,) ) )
42 eqid 2404 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4342rexmet 18775 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
44 eqid 2404 . . . . . . . . . . . . . 14  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
4542, 44tgioo 18780 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
4645mopni2 18476 . . . . . . . . . . . 12  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR )  /\  X  e.  ( topGen `  ran  (,) )  /\  ( `' F `  x )  e.  X
)  ->  E. r  e.  RR+  ( ( `' F `  x ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  X )
4743, 46mp3an1 1266 . . . . . . . . . . 11  |-  ( ( X  e.  ( topGen ` 
ran  (,) )  /\  ( `' F `  x )  e.  X )  ->  E. r  e.  RR+  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X )
4841, 47sylan 458 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' F `  x )  e.  X )  ->  E. r  e.  RR+  ( ( `' F `  x ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  X )
4924, 48syldan 457 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  Y )  ->  E. r  e.  RR+  ( ( `' F `  x ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  X )
5011ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  F  e.  ( X -cn->
RR ) )
5125ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  dom  ( RR  _D  F
)  =  X )
52 dvcnvre.z . . . . . . . . . . 11  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F
) )
5352ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  -.  0  e.  ran  ( RR  _D  F
) )
547ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  F : X -1-1-onto-> Y )
5524adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( `' F `  x )  e.  X
)
56 rphalfcl 10592 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR+ )
5756ad2antrl 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( r  /  2
)  e.  RR+ )
5828ad2antrr 707 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  X  C_  RR )
5958, 55sseldd 3309 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( `' F `  x )  e.  RR )
6057rpred 10604 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( r  /  2
)  e.  RR )
6159, 60resubcld 9421 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( `' F `  x )  -  (
r  /  2 ) )  e.  RR )
6259, 60readdcld 9071 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( `' F `  x )  +  ( r  /  2 ) )  e.  RR )
63 elicc2 10931 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( `' F `  x )  -  (
r  /  2 ) )  e.  RR  /\  ( ( `' F `  x )  +  ( r  /  2 ) )  e.  RR )  ->  ( y  e.  ( ( ( `' F `  x )  -  ( r  / 
2 ) ) [,] ( ( `' F `  x )  +  ( r  /  2 ) ) )  <->  ( y  e.  RR  /\  ( ( `' F `  x )  -  ( r  / 
2 ) )  <_ 
y  /\  y  <_  ( ( `' F `  x )  +  ( r  /  2 ) ) ) ) )
6461, 62, 63syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( y  e.  ( ( ( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) )  <-> 
( y  e.  RR  /\  ( ( `' F `  x )  -  (
r  /  2 ) )  <_  y  /\  y  <_  ( ( `' F `  x )  +  ( r  / 
2 ) ) ) ) )
6564biimpa 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( y  e.  RR  /\  ( ( `' F `  x )  -  ( r  / 
2 ) )  <_ 
y  /\  y  <_  ( ( `' F `  x )  +  ( r  /  2 ) ) ) )
6665simp1d 969 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  y  e.  RR )
6759adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( `' F `  x )  e.  RR )
68 simplrl 737 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  r  e.  RR+ )
6968rpred 10604 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  r  e.  RR )
7067, 69resubcld 9421 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  -  r )  e.  RR )
7161adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  -  ( r  / 
2 ) )  e.  RR )
7268, 56syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( r  /  2 )  e.  RR+ )
7372rpred 10604 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( r  /  2 )  e.  RR )
74 rphalflt 10594 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
7568, 74syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( r  /  2 )  < 
r )
7673, 69, 67, 75ltsub2dd 9595 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  -  r )  < 
( ( `' F `  x )  -  (
r  /  2 ) ) )
7765simp2d 970 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  -  ( r  / 
2 ) )  <_ 
y )
7870, 71, 66, 76, 77ltletrd 9186 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  -  r )  < 
y )
7962adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  +  ( r  / 
2 ) )  e.  RR )
8067, 69readdcld 9071 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  +  r )  e.  RR )
8165simp3d 971 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  y  <_  ( ( `' F `  x )  +  ( r  /  2 ) ) )
8273, 69, 67, 75ltadd2dd 9185 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  +  ( r  / 
2 ) )  < 
( ( `' F `  x )  +  r ) )
8366, 79, 80, 81, 82lelttrd 9184 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  y  <  ( ( `' F `  x )  +  r ) )
8470rexrd 9090 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  -  r )  e. 
RR* )
8580rexrd 9090 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( ( `' F `  x )  +  r )  e. 
RR* )
86 elioo2 10913 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( `' F `  x )  -  r
)  e.  RR*  /\  (
( `' F `  x )  +  r )  e.  RR* )  ->  ( y  e.  ( ( ( `' F `  x )  -  r
) (,) ( ( `' F `  x )  +  r ) )  <-> 
( y  e.  RR  /\  ( ( `' F `  x )  -  r
)  <  y  /\  y  <  ( ( `' F `  x )  +  r ) ) ) )
8784, 85, 86syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  ( y  e.  ( ( ( `' F `  x )  -  r ) (,) ( ( `' F `  x )  +  r ) )  <->  ( y  e.  RR  /\  ( ( `' F `  x )  -  r )  < 
y  /\  y  <  ( ( `' F `  x )  +  r ) ) ) )
8866, 78, 83, 87mpbir3and 1137 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  Y )  /\  ( r  e.  RR+  /\  ( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  /\  y  e.  ( (
( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) ) )  ->  y  e.  ( ( ( `' F `  x )  -  r ) (,) ( ( `' F `  x )  +  r ) ) )
8988ex 424 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( y  e.  ( ( ( `' F `  x )  -  (
r  /  2 ) ) [,] ( ( `' F `  x )  +  ( r  / 
2 ) ) )  ->  y  e.  ( ( ( `' F `  x )  -  r
) (,) ( ( `' F `  x )  +  r ) ) ) )
9089ssrdv 3314 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( ( `' F `  x )  -  ( r  / 
2 ) ) [,] ( ( `' F `  x )  +  ( r  /  2 ) ) )  C_  (
( ( `' F `  x )  -  r
) (,) ( ( `' F `  x )  +  r ) ) )
91 rpre 10574 . . . . . . . . . . . . . 14  |-  ( r  e.  RR+  ->  r  e.  RR )
9291ad2antrl 709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
r  e.  RR )
9342bl2ioo 18776 . . . . . . . . . . . . 13  |-  ( ( ( `' F `  x )  e.  RR  /\  r  e.  RR )  ->  ( ( `' F `  x ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  =  ( ( ( `' F `  x )  -  r
) (,) ( ( `' F `  x )  +  r ) ) )
9459, 92, 93syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  =  ( ( ( `' F `  x )  -  r ) (,) ( ( `' F `  x )  +  r ) ) )
9590, 94sseqtr4d 3345 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( ( `' F `  x )  -  ( r  / 
2 ) ) [,] ( ( `' F `  x )  +  ( r  /  2 ) ) )  C_  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r ) )
96 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X )
9795, 96sstrd 3318 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( ( `' F `  x )  -  ( r  / 
2 ) ) [,] ( ( `' F `  x )  +  ( r  /  2 ) ) )  C_  X
)
98 eqid 2404 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
99 eqid 2404 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  X )  =  ( ( TopOpen ` fld )t  X )
100 eqid 2404 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  Y )  =  ( ( TopOpen ` fld )t  Y )
10150, 51, 53, 54, 55, 57, 97, 98, 1, 99, 100dvcnvrelem2 19855 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( F `  ( `' F `  x ) )  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  Y )  /\  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  ( F `  ( `' F `  x ) ) ) ) )
10249, 101rexlimddv 2794 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Y )  ->  (
( F `  ( `' F `  x ) )  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  Y )  /\  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  ( F `  ( `' F `  x ) ) ) ) )
103102simpld 446 . . . . . . 7  |-  ( (
ph  /\  x  e.  Y )  ->  ( F `  ( `' F `  x )
)  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  Y ) )
10420, 103eqeltrrd 2479 . . . . . 6  |-  ( (
ph  /\  x  e.  Y )  ->  x  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )
)
105104ex 424 . . . . 5  |-  ( ph  ->  ( x  e.  Y  ->  x  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  Y ) ) )
106105ssrdv 3314 . . . 4  |-  ( ph  ->  Y  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  Y
) )
10718, 106eqssd 3325 . . 3  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  =  Y )
10816isopn3 17085 . . . 4  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  Y  C_  RR )  ->  ( Y  e.  ( topGen `  ran  (,) )  <->  ( ( int `  ( topGen `  ran  (,) )
) `  Y )  =  Y ) )
1096, 15, 108sylancr 645 . . 3  |-  ( ph  ->  ( Y  e.  (
topGen `  ran  (,) )  <->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  =  Y ) )
110107, 109mpbird 224 . 2  |-  ( ph  ->  Y  e.  ( topGen ` 
ran  (,) ) )
111102simprd 450 . . . . . 6  |-  ( (
ph  /\  x  e.  Y )  ->  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  ( F `  ( `' F `  x ) ) ) )
11220fveq2d 5691 . . . . . 6  |-  ( (
ph  /\  x  e.  Y )  ->  (
( ( ( TopOpen ` fld )t  Y
)  CnP  ( ( TopOpen
` fld
)t 
X ) ) `  ( F `  ( `' F `  x ) ) )  =  ( ( ( ( TopOpen ` fld )t  Y
)  CnP  ( ( TopOpen
` fld
)t 
X ) ) `  x ) )
113111, 112eleqtrd 2480 . . . . 5  |-  ( (
ph  /\  x  e.  Y )  ->  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  x ) )
114113ralrimiva 2749 . . . 4  |-  ( ph  ->  A. x  e.  Y  `' F  e.  (
( ( ( TopOpen ` fld )t  Y
)  CnP  ( ( TopOpen
` fld
)t 
X ) ) `  x ) )
1151cnfldtopon 18770 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
11615, 31syl6ss 3320 . . . . . 6  |-  ( ph  ->  Y  C_  CC )
117 resttopon 17179 . . . . . 6  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  Y  C_  CC )  ->  (
( TopOpen ` fld )t  Y )  e.  (TopOn `  Y ) )
118115, 116, 117sylancr 645 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  Y )  e.  (TopOn `  Y ) )
11928, 31syl6ss 3320 . . . . . 6  |-  ( ph  ->  X  C_  CC )
120 resttopon 17179 . . . . . 6  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  X  C_  CC )  ->  (
( TopOpen ` fld )t  X )  e.  (TopOn `  X ) )
121115, 119, 120sylancr 645 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  X )  e.  (TopOn `  X ) )
122 cncnp 17298 . . . . 5  |-  ( ( ( ( TopOpen ` fld )t  Y )  e.  (TopOn `  Y )  /\  (
( TopOpen ` fld )t  X )  e.  (TopOn `  X ) )  -> 
( `' F  e.  ( ( ( TopOpen ` fld )t  Y
)  Cn  ( (
TopOpen ` fld )t  X ) )  <->  ( `' F : Y --> X  /\  A. x  e.  Y  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  x ) ) ) )
123118, 121, 122syl2anc 643 . . . 4  |-  ( ph  ->  ( `' F  e.  ( ( ( TopOpen ` fld )t  Y
)  Cn  ( (
TopOpen ` fld )t  X ) )  <->  ( `' F : Y --> X  /\  A. x  e.  Y  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  x ) ) ) )
12423, 114, 123mpbir2and 889 . . 3  |-  ( ph  ->  `' F  e.  (
( ( TopOpen ` fld )t  Y )  Cn  (
( TopOpen ` fld )t  X ) ) )
1251, 100, 99cncfcn 18892 . . . 4  |-  ( ( Y  C_  CC  /\  X  C_  CC )  ->  ( Y -cn-> X )  =  ( ( ( TopOpen ` fld )t  Y
)  Cn  ( (
TopOpen ` fld )t  X ) ) )
126116, 119, 125syl2anc 643 . . 3  |-  ( ph  ->  ( Y -cn-> X )  =  ( ( (
TopOpen ` fld )t  Y )  Cn  (
( TopOpen ` fld )t  X ) ) )
127124, 126eleqtrrd 2481 . 2  |-  ( ph  ->  `' F  e.  ( Y -cn-> X ) )
1281, 2, 5, 110, 7, 127, 25, 52dvcnv 19814 1  |-  ( ph  ->  ( RR  _D  `' F )  =  ( x  e.  Y  |->  ( 1  /  ( ( RR  _D  F ) `
 ( `' F `  x ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    C_ wss 3280   {cpr 3775   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839    o. ccom 4841   -->wf 5409   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   RR+crp 10568   (,)cioo 10872   [,]cicc 10875   abscabs 11994   ↾t crest 13603   TopOpenctopn 13604   topGenctg 13620   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646  ℂfldccnfld 16658   Topctop 16913  TopOnctopon 16914   intcnt 17036    Cn ccn 17242    CnP ccnp 17243   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  dvrelog  20481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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