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Theorem dvcnvre 19763
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 2380 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21tgioo2 18698 . 2  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3 reex 9007 . . . 4  |-  RR  e.  _V
43prid1 3848 . . 3  |-  RR  e.  { RR ,  CC }
54a1i 11 . 2  |-  ( ph  ->  RR  e.  { RR ,  CC } )
6 retop 18659 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
7 dvcnvre.1 . . . . . . 7  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1ofo 5614 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
9 forn 5589 . . . . . . 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 18787 . . . . . . 7  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
13 frn 5530 . . . . . . 7  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
1411, 12, 133syl 19 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1510, 14eqsstr3d 3319 . . . . 5  |-  ( ph  ->  Y  C_  RR )
16 uniretop 18660 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
1716ntrss2 17037 . . . . 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 5947 . . . . . . . 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 5620 . . . . . . . . . . . 12  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
22 f1of 5607 . . . . . . . . . . . 12  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
237, 21, 223syl 19 . . . . . . . . . . 11  |-  ( ph  ->  `' F : Y --> X )
2423ffvelrnda 5802 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  Y )  ->  ( `' F `  x )  e.  X )
25 dvcnvre.d . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  =  X )
26 dvbsss 19649 . . . . . . . . . . . . . . . 16  |-  dom  ( RR  _D  F )  C_  RR
2726a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
2825, 27eqsstr3d 3319 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  C_  RR )
2916ntrss2 17037 . . . . . . . . . . . . . 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 8973 . . . . . . . . . . . . . . . 16  |-  RR  C_  CC
3231a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  RR  C_  CC )
3311, 12syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
34 fss 5532 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
3533, 31, 34sylancl 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : X --> CC )
3632, 35, 28, 2, 1dvbssntr 19647 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
3725, 36eqsstr3d 3319 . . . . . . . . . . . . 13  |-  ( ph  ->  X  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  X
) )
3830, 37eqssd 3301 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  =  X )
3916isopn3 17046 . . . . . . . . . . . . 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 2380 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4342rexmet 18686 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
44 eqid 2380 . . . . . . . . . . . . . 14  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
4542, 44tgioo 18691 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
4645mopni2 18406 . . . . . . . . . . . 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 10561 . . . . . . . . . . 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 3285 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( `' F `  x )  e.  RR )
6057rpred 10573 . . . . . . . . . . . . . . . . . . 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 9390 . . . . . . . . . . . . . . . . . 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 9041 . . . . . . . . . . . . . . . . . 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 10900 . . . . . . . . . . . . . . . . . 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 10573 . . . . . . . . . . . . . . . . 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 9390 . . . . . . . . . . . . . . . 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 10573 . . . . . . . . . . . . . . . . 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 10563 . . . . . . . . . . . . . . . . . 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 9564 . . . . . . . . . . . . . . . 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 9155 . . . . . . . . . . . . . . 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 9041 . . . . . . . . . . . . . . . 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 9154 . . . . . . . . . . . . . . . 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 9153 . . . . . . . . . . . . . . 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 9060 . . . . . . . . . . . . . . . 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 9060 . . . . . . . . . . . . . . . 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 10882 . . . . . . . . . . . . . . . 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 3290 . . . . . . . . . . . 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 10543 . . . . . . . . . . . . . 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 18687 . . . . . . . . . . . . 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 3321 . . . . . . . . . . 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 3294 . . . . . . . . . 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 2380 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
99 eqid 2380 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  X )  =  ( ( TopOpen ` fld )t  X )
100 eqid 2380 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  Y )  =  ( ( TopOpen ` fld )t  Y )
10150, 51, 53, 54, 55, 57, 97, 98, 1, 99, 100dvcnvrelem2 19762 . . . . . . . . 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 2770 . . . . . . . 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 2455 . . . . . 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 3290 . . . 4  |-  ( ph  ->  Y  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  Y
) )
10718, 106eqssd 3301 . . 3  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  =  Y )
10816isopn3 17046 . . . 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 5665 . . . . . 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 2456 . . . . 5  |-  ( (
ph  /\  x  e.  Y )  ->  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  x ) )
114113ralrimiva 2725 . . . 4  |-  ( ph  ->  A. x  e.  Y  `' F  e.  (
( ( ( TopOpen ` fld )t  Y
)  CnP  ( ( TopOpen
` fld
)t 
X ) ) `  x ) )
1151cnfldtopon 18681 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
11615, 31syl6ss 3296 . . . . . 6  |-  ( ph  ->  Y  C_  CC )
117 resttopon 17140 . . . . . 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 3296 . . . . . 6  |-  ( ph  ->  X  C_  CC )
120 resttopon 17140 . . . . . 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 17259 . . . . 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 18803 . . . 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 2457 . 2  |-  ( ph  ->  `' F  e.  ( Y -cn-> X ) )
1281, 2, 5, 110, 7, 127, 25, 52dvcnv 19721 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 1717   A.wral 2642   E.wrex 2643    C_ wss 3256   {cpr 3751   class class class wbr 4146    e. cmpt 4200    X. cxp 4809   `'ccnv 4810   dom cdm 4811   ran crn 4812    |` cres 4813    o. ccom 4815   -->wf 5383   -onto->wfo 5385   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917    + caddc 8919   RR*cxr 9045    < clt 9046    <_ cle 9047    - cmin 9216    / cdiv 9602   2c2 9974   RR+crp 10537   (,)cioo 10841   [,]cicc 10844   abscabs 11959   ↾t crest 13568   TopOpenctopn 13569   topGenctg 13585   * Metcxmt 16605   ballcbl 16607   MetOpencmopn 16610  ℂfldccnfld 16619   Topctop 16874  TopOnctopon 16875   intcnt 16997    Cn ccn 17203    CnP ccnp 17204   -cn->ccncf 18770    _D cdv 19610
This theorem is referenced by:  dvrelog  20388
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-fi 7344  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ioo 10845  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-hom 13473  df-cco 13474  df-rest 13570  df-topn 13571  df-topgen 13587  df-pt 13588  df-prds 13591  df-xrs 13646  df-0g 13647  df-gsum 13648  df-qtop 13653  df-imas 13654  df-xps 13656  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-submnd 14659  df-mulg 14735  df-cntz 15036  df-cmn 15334  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-fbas 16616  df-fg 16617  df-cnfld 16620  df-top 16879  df-bases 16881  df-topon 16882  df-topsp 16883  df-cld 16999  df-ntr 17000  df-cls 17001  df-nei 17078  df-lp 17116  df-perf 17117  df-cn 17206  df-cnp 17207  df-haus 17294  df-cmp 17365  df-tx 17508  df-hmeo 17701  df-fil 17792  df-fm 17884  df-flim 17885  df-flf 17886  df-xms 18252  df-ms 18253  df-tms 18254  df-cncf 18772  df-limc 19613  df-dv 19614
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