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Theorem dvcnvre 19896
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 2436 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21tgioo2 18827 . 2  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3 reex 9074 . . . 4  |-  RR  e.  _V
43prid1 3905 . . 3  |-  RR  e.  { RR ,  CC }
54a1i 11 . 2  |-  ( ph  ->  RR  e.  { RR ,  CC } )
6 retop 18788 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
7 dvcnvre.1 . . . . . . 7  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1ofo 5674 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
9 forn 5649 . . . . . . 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 18916 . . . . . . 7  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
13 frn 5590 . . . . . . 7  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
1411, 12, 133syl 19 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1510, 14eqsstr3d 3376 . . . . 5  |-  ( ph  ->  Y  C_  RR )
16 uniretop 18789 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
1716ntrss2 17114 . . . . 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 6008 . . . . . . . 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 5680 . . . . . . . . . . . 12  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
22 f1of 5667 . . . . . . . . . . . 12  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
237, 21, 223syl 19 . . . . . . . . . . 11  |-  ( ph  ->  `' F : Y --> X )
2423ffvelrnda 5863 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  Y )  ->  ( `' F `  x )  e.  X )
25 dvcnvre.d . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  =  X )
26 dvbsss 19782 . . . . . . . . . . . . . . . 16  |-  dom  ( RR  _D  F )  C_  RR
2726a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
2825, 27eqsstr3d 3376 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  C_  RR )
2916ntrss2 17114 . . . . . . . . . . . . . 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 9040 . . . . . . . . . . . . . . . 16  |-  RR  C_  CC
3231a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  RR  C_  CC )
3311, 12syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
34 fss 5592 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
3533, 31, 34sylancl 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : X --> CC )
3632, 35, 28, 2, 1dvbssntr 19780 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
3725, 36eqsstr3d 3376 . . . . . . . . . . . . 13  |-  ( ph  ->  X  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  X
) )
3830, 37eqssd 3358 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  =  X )
3916isopn3 17123 . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4342rexmet 18815 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
44 eqid 2436 . . . . . . . . . . . . . 14  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
4542, 44tgioo 18820 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
4645mopni2 18516 . . . . . . . . . . . 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 10629 . . . . . . . . . . 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 3342 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( `' F `  x )  e.  RR )
6057rpred 10641 . . . . . . . . . . . . . . . . . . 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 9458 . . . . . . . . . . . . . . . . . 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 9108 . . . . . . . . . . . . . . . . . 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 10968 . . . . . . . . . . . . . . . . . 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 10641 . . . . . . . . . . . . . . . . 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 9458 . . . . . . . . . . . . . . . 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 10641 . . . . . . . . . . . . . . . . 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 10631 . . . . . . . . . . . . . . . . . 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 9632 . . . . . . . . . . . . . . . 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 9223 . . . . . . . . . . . . . . 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 9108 . . . . . . . . . . . . . . . 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 9222 . . . . . . . . . . . . . . . 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 9221 . . . . . . . . . . . . . . 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 9127 . . . . . . . . . . . . . . . 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 9127 . . . . . . . . . . . . . . . 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 10950 . . . . . . . . . . . . . . . 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 3347 . . . . . . . . . . . 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 10611 . . . . . . . . . . . . . 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 18816 . . . . . . . . . . . . 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 3378 . . . . . . . . . . 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 3351 . . . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
99 eqid 2436 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  X )  =  ( ( TopOpen ` fld )t  X )
100 eqid 2436 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  Y )  =  ( ( TopOpen ` fld )t  Y )
10150, 51, 53, 54, 55, 57, 97, 98, 1, 99, 100dvcnvrelem2 19895 . . . . . . . . 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 2827 . . . . . . . 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 2511 . . . . . 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 3347 . . . 4  |-  ( ph  ->  Y  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  Y
) )
10718, 106eqssd 3358 . . 3  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  =  Y )
10816isopn3 17123 . . . 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 5725 . . . . . 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 2512 . . . . 5  |-  ( (
ph  /\  x  e.  Y )  ->  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  x ) )
114113ralrimiva 2782 . . . 4  |-  ( ph  ->  A. x  e.  Y  `' F  e.  (
( ( ( TopOpen ` fld )t  Y
)  CnP  ( ( TopOpen
` fld
)t 
X ) ) `  x ) )
1151cnfldtopon 18810 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
11615, 31syl6ss 3353 . . . . . 6  |-  ( ph  ->  Y  C_  CC )
117 resttopon 17218 . . . . . 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 3353 . . . . . 6  |-  ( ph  ->  X  C_  CC )
120 resttopon 17218 . . . . . 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 17337 . . . . 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 18932 . . . 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 2513 . 2  |-  ( ph  ->  `' F  e.  ( Y -cn-> X ) )
1281, 2, 5, 110, 7, 127, 25, 52dvcnv 19854 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 1652    e. wcel 1725   A.wral 2698   E.wrex 2699    C_ wss 3313   {cpr 3808   class class class wbr 4205    e. cmpt 4259    X. cxp 4869   `'ccnv 4870   dom cdm 4871   ran crn 4872    |` cres 4873    o. ccom 4875   -->wf 5443   -onto->wfo 5445   -1-1-onto->wf1o 5446   ` cfv 5447  (class class class)co 6074   CCcc 8981   RRcr 8982   0cc0 8983   1c1 8984    + caddc 8986   RR*cxr 9112    < clt 9113    <_ cle 9114    - cmin 9284    / cdiv 9670   2c2 10042   RR+crp 10605   (,)cioo 10909   [,]cicc 10912   abscabs 12032   ↾t crest 13641   TopOpenctopn 13642   topGenctg 13658   * Metcxmt 16679   ballcbl 16681   MetOpencmopn 16684  ℂfldccnfld 16696   Topctop 16951  TopOnctopon 16952   intcnt 17074    Cn ccn 17281    CnP ccnp 17282   -cn->ccncf 18899    _D cdv 19743
This theorem is referenced by:  dvrelog  20521
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061  ax-addf 9062  ax-mulf 9063
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-of 6298  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-2o 6718  df-oadd 6721  df-er 6898  df-map 7013  df-pm 7014  df-ixp 7057  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-fi 7409  df-sup 7439  df-oi 7472  df-card 7819  df-cda 8041  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-uz 10482  df-q 10568  df-rp 10606  df-xneg 10703  df-xadd 10704  df-xmul 10705  df-ioo 10913  df-ico 10915  df-icc 10916  df-fz 11037  df-fzo 11129  df-seq 11317  df-exp 11376  df-hash 11612  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-starv 13537  df-sca 13538  df-vsca 13539  df-tset 13541  df-ple 13542  df-ds 13544  df-unif 13545  df-hom 13546  df-cco 13547  df-rest 13643  df-topn 13644  df-topgen 13660  df-pt 13661  df-prds 13664  df-xrs 13719  df-0g 13720  df-gsum 13721  df-qtop 13726  df-imas 13727  df-xps 13729  df-mre 13804  df-mrc 13805  df-acs 13807  df-mnd 14683  df-submnd 14732  df-mulg 14808  df-cntz 15109  df-cmn 15407  df-psmet 16687  df-xmet 16688  df-met 16689  df-bl 16690  df-mopn 16691  df-fbas 16692  df-fg 16693  df-cnfld 16697  df-top 16956  df-bases 16958  df-topon 16959  df-topsp 16960  df-cld 17076  df-ntr 17077  df-cls 17078  df-nei 17155  df-lp 17193  df-perf 17194  df-cn 17284  df-cnp 17285  df-haus 17372  df-cmp 17443  df-tx 17587  df-hmeo 17780  df-fil 17871  df-fm 17963  df-flim 17964  df-flf 17965  df-xms 18343  df-ms 18344  df-tms 18345  df-cncf 18901  df-limc 19746  df-dv 19747
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