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Theorem dvcnvre 19366
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 2283 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21tgioo2 18309 . 2  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3 reex 8828 . . . 4  |-  RR  e.  _V
43prid1 3734 . . 3  |-  RR  e.  { RR ,  CC }
54a1i 10 . 2  |-  ( ph  ->  RR  e.  { RR ,  CC } )
6 retop 18270 . . . . 5  |-  ( topGen ` 
ran  (,) )  e.  Top
7 dvcnvre.1 . . . . . . 7  |-  ( ph  ->  F : X -1-1-onto-> Y )
8 f1ofo 5479 . . . . . . 7  |-  ( F : X -1-1-onto-> Y  ->  F : X -onto-> Y )
9 forn 5454 . . . . . . 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 18397 . . . . . . 7  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
13 frn 5395 . . . . . . 7  |-  ( F : X --> RR  ->  ran 
F  C_  RR )
1411, 12, 133syl 18 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1510, 14eqsstr3d 3213 . . . . 5  |-  ( ph  ->  Y  C_  RR )
16 uniretop 18271 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
1716ntrss2 16794 . . . . 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 5793 . . . . . . . 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 5485 . . . . . . . . . . . 12  |-  ( F : X -1-1-onto-> Y  ->  `' F : Y -1-1-onto-> X )
22 f1of 5472 . . . . . . . . . . . 12  |-  ( `' F : Y -1-1-onto-> X  ->  `' F : Y --> X )
237, 21, 223syl 18 . . . . . . . . . . 11  |-  ( ph  ->  `' F : Y --> X )
24 ffvelrn 5663 . . . . . . . . . . 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 19252 . . . . . . . . . . . . . . . 16  |-  dom  ( RR  _D  F )  C_  RR
2827a1i 10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
2926, 28eqsstr3d 3213 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  C_  RR )
3016ntrss2 16794 . . . . . . . . . . . . . 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 8794 . . . . . . . . . . . . . . . 16  |-  RR  C_  CC
3332a1i 10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  RR  C_  CC )
3411, 12syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F : X --> RR )
35 fss 5397 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
3634, 32, 35sylancl 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : X --> CC )
3733, 36, 29, 2, 1dvbssntr 19250 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  F )  C_  (
( int `  ( topGen `
 ran  (,) )
) `  X )
)
3826, 37eqsstr3d 3213 . . . . . . . . . . . . 13  |-  ( ph  ->  X  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  X
) )
3931, 38eqssd 3196 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  X )  =  X )
4016isopn3 16803 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4443rexmet 18297 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
45 eqid 2283 . . . . . . . . . . . . . 14  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
4643, 45tgioo 18302 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
4746mopni2 18039 . . . . . . . . . . . 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 10378 . . . . . . . . . . . . 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 3181 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  Y )  /\  (
r  e.  RR+  /\  (
( `' F `  x ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  C_  X ) )  -> 
( `' F `  x )  e.  RR )
6158rpred 10390 . . . . . . . . . . . . . . . . . . . . 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 9211 . . . . . . . . . . . . . . . . . . . 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 8862 . . . . . . . . . . . . . . . . . . . 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 10715 . . . . . . . . . . . . . . . . . . . 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 10390 . . . . . . . . . . . . . . . . . . 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 9211 . . . . . . . . . . . . . . . . . 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 10390 . . . . . . . . . . . . . . . . . . 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 10380 . . . . . . . . . . . . . . . . . . . 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 9385 . . . . . . . . . . . . . . . . . 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 8976 . . . . . . . . . . . . . . . . 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 8862 . . . . . . . . . . . . . . . . . 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 8975 . . . . . . . . . . . . . . . . . 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 8974 . . . . . . . . . . . . . . . . 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 8881 . . . . . . . . . . . . . . . . . 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 8881 . . . . . . . . . . . . . . . . . 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 10697 . . . . . . . . . . . . . . . . . 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 3185 . . . . . . . . . . . . . 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 10360 . . . . . . . . . . . . . . . 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 18298 . . . . . . . . . . . . . . 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 3215 . . . . . . . . . . . . 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 3189 . . . . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
100 eqid 2283 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  X )  =  ( ( TopOpen ` fld )t  X )
101 eqid 2283 . . . . . . . . . . . 12  |-  ( (
TopOpen ` fld )t  Y )  =  ( ( TopOpen ` fld )t  Y )
10251, 52, 54, 55, 56, 58, 98, 99, 1, 100, 101dvcnvrelem2 19365 . . . . . . . . . . 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 2667 . . . . . . . . 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 2358 . . . . . 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 3185 . . . 4  |-  ( ph  ->  Y  C_  ( ( int `  ( topGen `  ran  (,) ) ) `  Y
) )
11018, 109eqssd 3196 . . 3  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  Y )  =  Y )
11116isopn3 16803 . . . 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 5529 . . . . . 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 2359 . . . . 5  |-  ( (
ph  /\  x  e.  Y )  ->  `' F  e.  ( (
( ( TopOpen ` fld )t  Y )  CnP  (
( TopOpen ` fld )t  X ) ) `  x ) )
117116ralrimiva 2626 . . . 4  |-  ( ph  ->  A. x  e.  Y  `' F  e.  (
( ( ( TopOpen ` fld )t  Y
)  CnP  ( ( TopOpen
` fld
)t 
X ) ) `  x ) )
1181cnfldtopon 18292 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
11915, 32syl6ss 3191 . . . . . 6  |-  ( ph  ->  Y  C_  CC )
120 resttopon 16892 . . . . . 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 3191 . . . . . 6  |-  ( ph  ->  X  C_  CC )
123 resttopon 16892 . . . . . 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 17009 . . . . 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 18413 . . . 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 2360 . 2  |-  ( ph  ->  `' F  e.  ( Y -cn-> X ) )
1311, 2, 5, 113, 7, 130, 26, 53dvcnv 19324 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   {cpr 3641   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   RR+crp 10354   (,)cioo 10656   [,]cicc 10659   abscabs 11719   ↾t crest 13325   TopOpenctopn 13326   topGenctg 13342   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372  ℂfldccnfld 16377   Topctop 16631  TopOnctopon 16632   intcnt 16754    Cn ccn 16954    CnP ccnp 16955   -cn->ccncf 18380    _D cdv 19213
This theorem is referenced by:  dvrelog  19984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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