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Theorem dvcnvre 19382
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 2296 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21tgioo2 18325 . 2  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3 reex 8844 . . . 4  |-  RR  e.  _V
43prid1 3747 . . 3  |-  RR  e.  { RR ,  CC }
54a1i 10 . 2  |-  ( ph  ->  RR  e.  { RR ,  CC } )
6 retop 18286 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
7 dvcnvre.1 . . . . . . 7  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1ofo 5495 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
9 forn 5470 . . . . . . 7  |-  ( F : X -onto-> Y  ->  ran  F  =  Y )
107, 8, 93syl 18 . . . . . 6  |-  ( ph  ->  ran  F  =  Y )
11 dvcnvre.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X
-cn-> RR ) )
12 cncff 18413 . . . . . . 7  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
13 frn 5411 . . . . . . 7  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
1411, 12, 133syl 18 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1510, 14eqsstr3d 3226 . . . . 5  |-  ( ph  ->  Y  C_  RR )
16 uniretop 18287 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
1716ntrss2 16810 . . . . 5  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  Y  C_  RR )  ->  ( ( int `  ( topGen ` 
ran  (,) ) ) `  Y )  C_  Y
)
186, 15, 17sylancr 644 . . . 4  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  C_  Y )
19 f1ocnvfv2 5809 . . . . . . . 8  |-  ( ( F : X -1-1-onto-> Y  /\  x  e.  Y )  ->  ( F `  ( `' F `  x ) )  =  x )
207, 19sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  Y )  ->  ( F `  ( `' F `  x )
)  =  x )
21 f1ocnv 5501 . . . . . . . . . . . 12  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
22 f1of 5488 . . . . . . . . . . . 12  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
237, 21, 223syl 18 . . . . . . . . . . 11  |-  ( ph  ->  `' F : Y --> X )
24 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( `' F : Y --> X  /\  x  e.  Y )  ->  ( `' F `  x )  e.  X
)
2523, 24sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  Y )  ->  ( `' F `  x )  e.  X )
26 dvcnvre.d . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  =  X )
27 dvbsss 19268 . . . . . . . . . . . . . . . 16  |-  dom  ( RR  _D  F )  C_  RR
2827a1i 10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
2926, 28eqsstr3d 3226 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  C_  RR )
3016ntrss2 16810 . . . . . . . . . . . . . 14  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  X  C_  RR )  ->  ( ( int `  ( topGen ` 
ran  (,) ) ) `  X )  C_  X
)
316, 29, 30sylancr 644 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  C_  X )
32 ax-resscn 8810 . . . . . . . . . . . . . . . 16  |-  RR  C_  CC
3332a1i 10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  RR  C_  CC )
3411, 12syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
35 fss 5413 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
3634, 32, 35sylancl 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : X --> CC )
3733, 36, 29, 2, 1dvbssntr 19266 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
3826, 37eqsstr3d 3226 . . . . . . . . . . . . 13  |-  ( ph  ->  X  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  X
) )
3931, 38eqssd 3209 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  =  X )
4016isopn3 16819 . . . . . . . . . . . . 13  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  X  C_  RR )  ->  ( X  e.  ( topGen `  ran  (,) )  <->  ( ( int `  ( topGen `  ran  (,) )
) `  X )  =  X ) )
416, 29, 40sylancr 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( X  e.  (
topGen `  ran  (,) )  <->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  =  X ) )
4239, 41mpbird 223 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( topGen ` 
ran  (,) ) )
43 eqid 2296 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4443rexmet 18313 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
45 eqid 2296 . . . . . . . . . . . . . 14  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
4643, 45tgioo 18318 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
4746mopni2 18055 . . . . . . . . . . . 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 )
4844, 47mp3an1 1264 . . . . . . . . . . 11  |-  ( ( X  e.  ( topGen ` 
ran  (,) )  /\  ( `' F `  x )  e.  X )  ->  E. r  e.  RR+  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X )
4942, 48sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' F `  x )  e.  X )  ->  E. r  e.  RR+  ( ( `' F `  x ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  X )
5025, 49syldan 456 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  Y )  ->  E. r  e.  RR+  ( ( `' F `  x ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  X )
5111ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  F  e.  ( X -cn->
RR ) )
5226ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  dom  ( RR  _D  F
)  =  X )
53 dvcnvre.z . . . . . . . . . . . . 13  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F
) )
5453ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  -.  0  e.  ran  ( RR  _D  F
) )
557ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  F : X -1-1-onto-> Y )
5625adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( `' F `  x )  e.  X
)
57 rphalfcl 10394 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR+ )
5857ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( r  /  2
)  e.  RR+ )
5929ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  ->  X  C_  RR )
6059, 56sseldd 3194 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( `' F `  x )  e.  RR )
6158rpred 10406 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( r  /  2
)  e.  RR )
6260, 61resubcld 9227 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( `' F `  x )  -  (
r  /  2 ) )  e.  RR )
6360, 61readdcld 8878 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( ( `' F `  x )  +  ( r  /  2 ) )  e.  RR )
64 elicc2 10731 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( `' 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 ) ) ) ) )
6562, 63, 64syl2anc 642 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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 ) ) ) ) )
6665biimpa 470 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( y  e.  RR  /\  ( ( `' F `  x )  -  ( r  / 
2 ) )  <_ 
y  /\  y  <_  ( ( `' F `  x )  +  ( r  /  2 ) ) ) )
6766simp1d 967 . . . . . . . . . . . . . . . . 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 )
6860adantr 451 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 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 )
69 simplrl 736 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( 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+ )
7069rpred 10406 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 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 )
7168, 70resubcld 9227 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  -  r )  e.  RR )
7262adantr 451 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  -  ( r  / 
2 ) )  e.  RR )
7369, 57syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( 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+ )
7473rpred 10406 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 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 )
75 rphalflt 10396 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
7669, 75syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 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 )
7774, 70, 68, 76ltsub2dd 9401 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  -  r )  < 
( ( `' F `  x )  -  (
r  /  2 ) ) )
7866simp2d 968 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  -  ( r  / 
2 ) )  <_ 
y )
7971, 72, 67, 77, 78ltletrd 8992 . . . . . . . . . . . . . . . . 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 )  -  r )  < 
y )
8063adantr 451 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  +  ( r  / 
2 ) )  e.  RR )
8168, 70readdcld 8878 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  +  r )  e.  RR )
8266simp3d 969 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  y  <_  ( ( `' F `  x )  +  ( r  /  2 ) ) )
8374, 70, 68, 76ltadd2dd 8991 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  +  ( r  / 
2 ) )  < 
( ( `' F `  x )  +  r ) )
8467, 80, 81, 82, 83lelttrd 8990 . . . . . . . . . . . . . . . . 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  <  ( ( `' F `  x )  +  r ) )
8571rexrd 8897 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  -  r )  e. 
RR* )
8681rexrd 8897 . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( `' F `  x )  +  r )  e. 
RR* )
87 elioo2 10713 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( `' 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 ) ) ) )
8885, 86, 87syl2anc 642 . . . . . . . . . . . . . . . . 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.  ( ( ( `' F `  x )  -  r ) (,) ( ( `' F `  x )  +  r ) )  <->  ( y  e.  RR  /\  ( ( `' F `  x )  -  r )  < 
y  /\  y  <  ( ( `' F `  x )  +  r ) ) ) )
8967, 79, 84, 88mpbir3and 1135 . . . . . . . . . . . . . . . 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.  ( ( ( `' F `  x )  -  r ) (,) ( ( `' F `  x )  +  r ) ) )
9089ex 423 . . . . . . . . . . . . . . 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 ) ) ) )
9190ssrdv 3198 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )
92 rpre 10376 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  r  e.  RR )
9392ad2antrl 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
r  e.  RR )
9443bl2ioo 18314 . . . . . . . . . . . . . . 15  |-  ( ( ( `' F `  x )  e.  RR  /\  r  e.  RR )  ->  ( ( `' F `  x ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  =  ( ( ( `' F `  x )  -  r
) (,) ( ( `' F `  x )  +  r ) ) )
9560, 93, 94syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )
9691, 95sseqtr4d 3228 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) )
97 simprr 733 . . . . . . . . . . . . 13  |-  ( ( ( 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 )
9896, 97sstrd 3202 . . . . . . . . . . . 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_  X
)
99 eqid 2296 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
100 eqid 2296 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  X )  =  ( ( TopOpen ` fld )t  X )
101 eqid 2296 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  Y )  =  ( ( TopOpen ` fld )t  Y )
10251, 52, 54, 55, 56, 58, 98, 99, 1, 100, 101dvcnvrelem2 19381 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) ) ) )
103102expr 598 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) ) ) )
104103rexlimdva 2680 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  Y )  ->  ( E. 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 ) ) ) ) ) )
10550, 104mpd 14 . . . . . . . 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 ) ) ) ) )
106105simpld 445 . . . . . . 7  |-  ( (
ph  /\  x  e.  Y )  ->  ( F `  ( `' F `  x )
)  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  Y ) )
10720, 106eqeltrrd 2371 . . . . . 6  |-  ( (
ph  /\  x  e.  Y )  ->  x  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )
)
108107ex 423 . . . . 5  |-  ( ph  ->  ( x  e.  Y  ->  x  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  Y ) ) )
109108ssrdv 3198 . . . 4  |-  ( ph  ->  Y  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  Y
) )
11018, 109eqssd 3209 . . 3  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  =  Y )
11116isopn3 16819 . . . 4  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  Y  C_  RR )  ->  ( Y  e.  ( topGen `  ran  (,) )  <->  ( ( int `  ( topGen `  ran  (,) )
) `  Y )  =  Y ) )
1126, 15, 111sylancr 644 . . 3  |-  ( ph  ->  ( Y  e.  (
topGen `  ran  (,) )  <->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  =  Y ) )
113110, 112mpbird 223 . 2  |-  ( ph  ->  Y  e.  ( topGen ` 
ran  (,) ) )
114105simprd 449 . . . . . 6  |-  ( (
ph  /\  x  e.  Y )  ->  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  ( F `  ( `' F `  x ) ) ) )
11520fveq2d 5545 . . . . . 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 ) )
116114, 115eleqtrd 2372 . . . . 5  |-  ( (
ph  /\  x  e.  Y )  ->  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  x ) )
117116ralrimiva 2639 . . . 4  |-  ( ph  ->  A. x  e.  Y  `' F  e.  (
( ( ( TopOpen ` fld )t  Y
)  CnP  ( ( TopOpen
` fld
)t 
X ) ) `  x ) )
1181cnfldtopon 18308 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
11915, 32syl6ss 3204 . . . . . 6  |-  ( ph  ->  Y  C_  CC )
120 resttopon 16908 . . . . . 6  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  Y  C_  CC )  ->  (
( TopOpen ` fld )t  Y )  e.  (TopOn `  Y ) )
121118, 119, 120sylancr 644 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  Y )  e.  (TopOn `  Y ) )
12229, 32syl6ss 3204 . . . . . 6  |-  ( ph  ->  X  C_  CC )
123 resttopon 16908 . . . . . 6  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  X  C_  CC )  ->  (
( TopOpen ` fld )t  X )  e.  (TopOn `  X ) )
124118, 122, 123sylancr 644 . . . . 5  |-  ( ph  ->  ( ( TopOpen ` fld )t  X )  e.  (TopOn `  X ) )
125 cncnp 17025 . . . . 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 ) ) ) )
126121, 124, 125syl2anc 642 . . . 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 ) ) ) )
12723, 117, 126mpbir2and 888 . . 3  |-  ( ph  ->  `' F  e.  (
( ( TopOpen ` fld )t  Y )  Cn  (
( TopOpen ` fld )t  X ) ) )
1281, 101, 100cncfcn 18429 . . . 4  |-  ( ( Y  C_  CC  /\  X  C_  CC )  ->  ( Y -cn-> X )  =  ( ( ( TopOpen ` fld )t  Y
)  Cn  ( (
TopOpen ` fld )t  X ) ) )
129119, 122, 128syl2anc 642 . . 3  |-  ( ph  ->  ( Y -cn-> X )  =  ( ( (
TopOpen ` fld )t  Y )  Cn  (
( TopOpen ` fld )t  X ) ) )
130127, 129eleqtrrd 2373 . 2  |-  ( ph  ->  `' F  e.  ( Y -cn-> X ) )
1311, 2, 5, 113, 7, 130, 26, 53dvcnv 19340 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   {cpr 3654   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707    o. ccom 4709   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   abscabs 11735   ↾t crest 13341   TopOpenctopn 13342   topGenctg 13358   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388  ℂfldccnfld 16393   Topctop 16647  TopOnctopon 16648   intcnt 16770    Cn ccn 16970    CnP ccnp 16971   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  dvrelog  20000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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