MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvcnvrelem1 Unicode version

Theorem dvcnvrelem1 19380
Description: Lemma for dvcnvre 19382. (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 )
dvcnvre.c  |-  ( ph  ->  C  e.  X )
dvcnvre.r  |-  ( ph  ->  R  e.  RR+ )
dvcnvre.s  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  X )
Assertion
Ref Expression
dvcnvrelem1  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )

Proof of Theorem dvcnvrelem1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvcnvre.d . . . . . 6  |-  ( ph  ->  dom  ( RR  _D  F )  =  X )
2 dvbsss 19268 . . . . . . 7  |-  dom  ( RR  _D  F )  C_  RR
32a1i 10 . . . . . 6  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
41, 3eqsstr3d 3226 . . . . 5  |-  ( ph  ->  X  C_  RR )
5 dvcnvre.c . . . . 5  |-  ( ph  ->  C  e.  X )
64, 5sseldd 3194 . . . 4  |-  ( ph  ->  C  e.  RR )
7 dvcnvre.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
87rpred 10406 . . . 4  |-  ( ph  ->  R  e.  RR )
96, 8resubcld 9227 . . 3  |-  ( ph  ->  ( C  -  R
)  e.  RR )
106, 8readdcld 8878 . . 3  |-  ( ph  ->  ( C  +  R
)  e.  RR )
116, 7ltsubrpd 10434 . . . . 5  |-  ( ph  ->  ( C  -  R
)  <  C )
126, 7ltaddrpd 10435 . . . . 5  |-  ( ph  ->  C  <  ( C  +  R ) )
139, 6, 10, 11, 12lttrd 8993 . . . 4  |-  ( ph  ->  ( C  -  R
)  <  ( C  +  R ) )
149, 10, 13ltled 8983 . . 3  |-  ( ph  ->  ( C  -  R
)  <_  ( C  +  R ) )
15 dvcnvre.s . . . 4  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  X )
16 dvcnvre.f . . . 4  |-  ( ph  ->  F  e.  ( X
-cn-> RR ) )
17 rescncf 18417 . . . 4  |-  ( ( ( C  -  R
) [,] ( C  +  R ) ) 
C_  X  ->  ( F  e.  ( X -cn->
RR )  ->  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  e.  ( ( ( C  -  R ) [,] ( C  +  R
) ) -cn-> RR ) ) )
1815, 16, 17sylc 56 . . 3  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  e.  ( ( ( C  -  R
) [,] ( C  +  R ) )
-cn-> RR ) )
199, 10, 14, 18evthicc2 18836 . 2  |-  ( ph  ->  E. x  e.  RR  E. y  e.  RR  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) )  =  ( x [,] y ) )
20 cncff 18413 . . . . . . . . 9  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
2116, 20syl 15 . . . . . . . 8  |-  ( ph  ->  F : X --> RR )
22 ffvelrn 5679 . . . . . . . 8  |-  ( ( F : X --> RR  /\  C  e.  X )  ->  ( F `  C
)  e.  RR )
2321, 5, 22syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( F `  C
)  e.  RR )
2423adantr 451 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  C )  e.  RR )
259rexrd 8897 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  RR* )
2610rexrd 8897 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  RR* )
27 lbicc2 10768 . . . . . . . . . . . 12  |-  ( ( ( C  -  R
)  e.  RR*  /\  ( C  +  R )  e.  RR*  /\  ( C  -  R )  <_ 
( C  +  R
) )  ->  ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
2825, 26, 14, 27syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( C  -  R
)  e.  ( ( C  -  R ) [,] ( C  +  R ) ) )
2928adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
309, 6, 11ltled 8983 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  <_  C )
316, 10, 12ltled 8983 . . . . . . . . . . . 12  |-  ( ph  ->  C  <_  ( C  +  R ) )
32 elicc2 10731 . . . . . . . . . . . . 13  |-  ( ( ( C  -  R
)  e.  RR  /\  ( C  +  R
)  e.  RR )  ->  ( C  e.  ( ( C  -  R ) [,] ( C  +  R )
)  <->  ( C  e.  RR  /\  ( C  -  R )  <_  C  /\  C  <_  ( C  +  R )
) ) )
339, 10, 32syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  <-> 
( C  e.  RR  /\  ( C  -  R
)  <_  C  /\  C  <_  ( C  +  R ) ) ) )
346, 30, 31, 33mpbir3and 1135 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  ( ( C  -  R ) [,] ( C  +  R ) ) )
3534adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  C  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
3611adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( C  -  R )  <  C )
37 isorel 5839 . . . . . . . . . . . . 13  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  /\  ( ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
)  /\  C  e.  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( ( C  -  R )  <  C  <->  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  -  R ) )  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
) ) )
3837biimpd 198 . . . . . . . . . . . 12  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  /\  ( ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
)  /\  C  e.  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( ( C  -  R )  <  C  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  -  R )
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C ) ) )
3938exp32 588 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R ) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( ( C  -  R )  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( C  e.  ( ( C  -  R ) [,] ( C  +  R )
)  ->  ( ( C  -  R )  <  C  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) )  <  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C ) ) ) ) )
4039com4l 78 . . . . . . . . . 10  |-  ( ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  ( C  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( C  -  R
)  <  C  ->  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  -  R )
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C ) ) ) ) )
4129, 35, 36, 40syl3c 57 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  -  R )
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C ) ) )
42 fvres 5558 . . . . . . . . . . 11  |-  ( ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  -  R ) )  =  ( F `  ( C  -  R
) ) )
4329, 42syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  -  R ) )  =  ( F `  ( C  -  R
) ) )
44 fvres 5558 . . . . . . . . . . 11  |-  ( C  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C )  =  ( F `  C ) )
4535, 44syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C )  =  ( F `  C ) )
4643, 45breq12d 4052 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  -  R )
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C )  <->  ( F `  ( C  -  R
) )  <  ( F `  C )
) )
4741, 46sylibd 205 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  ->  ( F `  ( C  -  R
) )  <  ( F `  C )
) )
4821adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  F : X --> RR )
49 ffun 5407 . . . . . . . . . . . . . . 15  |-  ( F : X --> RR  ->  Fun 
F )
5048, 49syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  Fun  F )
5115adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( C  -  R
) [,] ( C  +  R ) ) 
C_  X )
52 fdm 5409 . . . . . . . . . . . . . . . 16  |-  ( F : X --> RR  ->  dom 
F  =  X )
5348, 52syl 15 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  dom  F  =  X )
5451, 53sseqtr4d 3228 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( C  -  R
) [,] ( C  +  R ) ) 
C_  dom  F )
55 funfvima2 5770 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
( C  -  R
) [,] ( C  +  R ) ) 
C_  dom  F )  ->  ( ( C  -  R )  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( F `  ( C  -  R
) )  e.  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
5650, 54, 55syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( C  -  R
)  e.  ( ( C  -  R ) [,] ( C  +  R ) )  -> 
( F `  ( C  -  R )
)  e.  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
5729, 56mpd 14 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  e.  ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) )
58 df-ima 4718 . . . . . . . . . . . . 13  |-  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )
59 simprr 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) )  =  ( x [,] y ) )
6058, 59syl5eq 2340 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( x [,] y
) )
6157, 60eleqtrd 2372 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  e.  ( x [,] y
) )
62 elicc2 10731 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( F `  ( C  -  R
) )  e.  ( x [,] y )  <-> 
( ( F `  ( C  -  R
) )  e.  RR  /\  x  <_  ( F `  ( C  -  R
) )  /\  ( F `  ( C  -  R ) )  <_ 
y ) ) )
6362ad2antrl 708 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  ( C  -  R )
)  e.  ( x [,] y )  <->  ( ( F `  ( C  -  R ) )  e.  RR  /\  x  <_ 
( F `  ( C  -  R )
)  /\  ( F `  ( C  -  R
) )  <_  y
) ) )
6461, 63mpbid 201 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  ( C  -  R )
)  e.  RR  /\  x  <_  ( F `  ( C  -  R
) )  /\  ( F `  ( C  -  R ) )  <_ 
y ) )
6564simp2d 968 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  <_  ( F `  ( C  -  R )
) )
66 simprll 738 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  e.  RR )
6715, 28sseldd 3194 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  X )
68 ffvelrn 5679 . . . . . . . . . . . 12  |-  ( ( F : X --> RR  /\  ( C  -  R
)  e.  X )  ->  ( F `  ( C  -  R
) )  e.  RR )
6921, 67, 68syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  ( C  -  R )
)  e.  RR )
7069adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  e.  RR )
71 lelttr 8928 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  ( F `  ( C  -  R ) )  e.  RR  /\  ( F `  C )  e.  RR )  ->  (
( x  <_  ( F `  ( C  -  R ) )  /\  ( F `  ( C  -  R ) )  <  ( F `  C ) )  ->  x  <  ( F `  C ) ) )
7266, 70, 24, 71syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( x  <_  ( F `  ( C  -  R ) )  /\  ( F `  ( C  -  R ) )  <  ( F `  C ) )  ->  x  <  ( F `  C ) ) )
7365, 72mpand 656 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  ( C  -  R )
)  <  ( F `  C )  ->  x  <  ( F `  C
) ) )
7447, 73syld 40 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  ->  x  <  ( F `  C )
) )
75 ubicc2 10769 . . . . . . . . . . . 12  |-  ( ( ( C  -  R
)  e.  RR*  /\  ( C  +  R )  e.  RR*  /\  ( C  -  R )  <_ 
( C  +  R
) )  ->  ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
7625, 26, 14, 75syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( C  +  R
)  e.  ( ( C  -  R ) [,] ( C  +  R ) ) )
7776adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
7812adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  C  <  ( C  +  R
) )
79 isorel 5839 . . . . . . . . . . . . 13  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  /\  ( C  e.  ( ( C  -  R ) [,] ( C  +  R
) )  /\  ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( C  <  ( C  +  R )  <->  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C ) `'  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  +  R )
) ) )
8079biimpd 198 . . . . . . . . . . . 12  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  /\  ( C  e.  ( ( C  -  R ) [,] ( C  +  R
) )  /\  ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( C  <  ( C  +  R )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
) `'  <  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) ) ) )
8180exp32 588 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R
) [,] ( C  +  R ) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
)  ->  ( C  <  ( C  +  R
)  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C ) `'  <  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) ) ) ) ) )
8281com4l 78 . . . . . . . . . 10  |-  ( C  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( C  +  R
)  e.  ( ( C  -  R ) [,] ( C  +  R ) )  -> 
( C  <  ( C  +  R )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R ) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
) `'  <  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) ) ) ) ) )
8335, 77, 78, 82syl3c 57 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C ) `'  <  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) ) ) )
84 fvex 5555 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C )  e.  _V
85 fvex 5555 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  +  R
) )  e.  _V
8684, 85brcnv 4880 . . . . . . . . . 10  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C ) `'  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  +  R )
)  <->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  +  R )
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C ) )
87 fvres 5558 . . . . . . . . . . . 12  |-  ( ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) )  =  ( F `  ( C  +  R
) ) )
8877, 87syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) )  =  ( F `  ( C  +  R
) ) )
8988, 45breq12d 4052 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  +  R )
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C )  <->  ( F `  ( C  +  R
) )  <  ( F `  C )
) )
9086, 89syl5bb 248 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
) `'  <  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) )  <-> 
( F `  ( C  +  R )
)  <  ( F `  C ) ) )
9183, 90sylibd 205 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  ->  ( F `  ( C  +  R
) )  <  ( F `  C )
) )
92 funfvima2 5770 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
( C  -  R
) [,] ( C  +  R ) ) 
C_  dom  F )  ->  ( ( C  +  R )  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( F `  ( C  +  R
) )  e.  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
9350, 54, 92syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( C  +  R
)  e.  ( ( C  -  R ) [,] ( C  +  R ) )  -> 
( F `  ( C  +  R )
)  e.  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
9477, 93mpd 14 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  e.  ( F " (
( C  -  R
) [,] ( C  +  R ) ) ) )
9594, 60eleqtrd 2372 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  e.  ( x [,] y
) )
96 elicc2 10731 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( F `  ( C  +  R
) )  e.  ( x [,] y )  <-> 
( ( F `  ( C  +  R
) )  e.  RR  /\  x  <_  ( F `  ( C  +  R
) )  /\  ( F `  ( C  +  R ) )  <_ 
y ) ) )
9796ad2antrl 708 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  ( C  +  R )
)  e.  ( x [,] y )  <->  ( ( F `  ( C  +  R ) )  e.  RR  /\  x  <_ 
( F `  ( C  +  R )
)  /\  ( F `  ( C  +  R
) )  <_  y
) ) )
9895, 97mpbid 201 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  ( C  +  R )
)  e.  RR  /\  x  <_  ( F `  ( C  +  R
) )  /\  ( F `  ( C  +  R ) )  <_ 
y ) )
9998simp2d 968 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  <_  ( F `  ( C  +  R )
) )
10015, 76sseldd 3194 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  X )
101 ffvelrn 5679 . . . . . . . . . . . 12  |-  ( ( F : X --> RR  /\  ( C  +  R
)  e.  X )  ->  ( F `  ( C  +  R
) )  e.  RR )
10221, 100, 101syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  ( C  +  R )
)  e.  RR )
103102adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  e.  RR )
104 lelttr 8928 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  ( F `  ( C  +  R ) )  e.  RR  /\  ( F `  C )  e.  RR )  ->  (
( x  <_  ( F `  ( C  +  R ) )  /\  ( F `  ( C  +  R ) )  <  ( F `  C ) )  ->  x  <  ( F `  C ) ) )
10566, 103, 24, 104syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( x  <_  ( F `  ( C  +  R ) )  /\  ( F `  ( C  +  R ) )  <  ( F `  C ) )  ->  x  <  ( F `  C ) ) )
10699, 105mpand 656 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  ( C  +  R )
)  <  ( F `  C )  ->  x  <  ( F `  C
) ) )
10791, 106syld 40 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  ->  x  <  ( F `  C ) ) )
108 ax-resscn 8810 . . . . . . . . . . . . . 14  |-  RR  C_  CC
109108a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  C_  CC )
110 fss 5413 . . . . . . . . . . . . . 14  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
11121, 108, 110sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  F : X --> CC )
11215, 4sstrd 3202 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR )
113 eqid 2296 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
114113tgioo2 18325 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
115113, 114dvres 19277 . . . . . . . . . . . . 13  |-  ( ( ( RR  C_  CC  /\  F : X --> CC )  /\  ( X  C_  RR  /\  ( ( C  -  R ) [,] ( C  +  R
) )  C_  RR ) )  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
116109, 111, 4, 112, 115syl22anc 1183 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
117 iccntr 18342 . . . . . . . . . . . . . 14  |-  ( ( ( C  -  R
)  e.  RR  /\  ( C  +  R
)  e.  RR )  ->  ( ( int `  ( topGen `  ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( ( C  -  R ) (,) ( C  +  R )
) )
1189, 10, 117syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( ( C  -  R ) (,) ( C  +  R )
) )
119118reseq2d 4971 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) ) )
120116, 119eqtrd 2328 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) ) )
121120dmeqd 4897 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  dom  (
( RR  _D  F
)  |`  ( ( C  -  R ) (,) ( C  +  R
) ) ) )
122 dmres 4992 . . . . . . . . . . 11  |-  dom  (
( RR  _D  F
)  |`  ( ( C  -  R ) (,) ( C  +  R
) ) )  =  ( ( ( C  -  R ) (,) ( C  +  R
) )  i^i  dom  ( RR  _D  F
) )
123 ioossicc 10751 . . . . . . . . . . . . . 14  |-  ( ( C  -  R ) (,) ( C  +  R ) )  C_  ( ( C  -  R ) [,] ( C  +  R )
)
124123, 15syl5ss 3203 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) (,) ( C  +  R )
)  C_  X )
125124, 1sseqtr4d 3228 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( C  -  R ) (,) ( C  +  R )
)  C_  dom  ( RR 
_D  F ) )
126 df-ss 3179 . . . . . . . . . . . 12  |-  ( ( ( C  -  R
) (,) ( C  +  R ) ) 
C_  dom  ( RR  _D  F )  <->  ( (
( C  -  R
) (,) ( C  +  R ) )  i^i  dom  ( RR  _D  F ) )  =  ( ( C  -  R ) (,) ( C  +  R )
) )
127125, 126sylib 188 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( C  -  R ) (,) ( C  +  R
) )  i^i  dom  ( RR  _D  F
) )  =  ( ( C  -  R
) (,) ( C  +  R ) ) )
128122, 127syl5eq 2340 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) )  =  ( ( C  -  R
) (,) ( C  +  R ) ) )
129121, 128eqtrd 2328 . . . . . . . . 9  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( C  -  R ) (,) ( C  +  R ) ) )
130 dvcnvre.z . . . . . . . . . 10  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F
) )
131 resss 4995 . . . . . . . . . . . . . 14  |-  ( ( RR  _D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) )  C_  ( RR  _D  F )
132131a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( C  -  R
) (,) ( C  +  R ) ) )  C_  ( RR  _D  F ) )
133120, 132eqsstrd 3225 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) 
C_  ( RR  _D  F ) )
134 rnss 4923 . . . . . . . . . . . 12  |-  ( ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  C_  ( RR  _D  F
)  ->  ran  ( RR 
_D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  C_  ran  ( RR  _D  F
) )
135133, 134syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  C_  ran  ( RR 
_D  F ) )
136135sseld 3192 . . . . . . . . . 10  |-  ( ph  ->  ( 0  e.  ran  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  ->  0  e.  ran  ( RR  _D  F
) ) )
137130, 136mtod 168 . . . . . . . . 9  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
1389, 10, 18, 129, 137dvne0 19374 . . . . . . . 8  |-  ( ph  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R ) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  \/  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) ) ) )
139138adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  \/  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R ) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) ) ) )
14074, 107, 139mpjaod 370 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  <  ( F `  C
) )
141 isorel 5839 . . . . . . . . . . . . 13  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  /\  ( C  e.  ( ( C  -  R ) [,] ( C  +  R )
)  /\  ( C  +  R )  e.  ( ( C  -  R
) [,] ( C  +  R ) ) ) )  ->  ( C  <  ( C  +  R )  <->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C )  <  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) ) ) )
142141biimpd 198 . . . . . . . . . . . 12  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  /\  ( C  e.  ( ( C  -  R ) [,] ( C  +  R )
)  /\  ( C  +  R )  e.  ( ( C  -  R
) [,] ( C  +  R ) ) ) )  ->  ( C  <  ( C  +  R )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C )  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  +  R )
) ) )
143142exp32 588 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R ) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
)  ->  ( C  <  ( C  +  R
)  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C )  <  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) ) ) ) ) )
144143com4l 78 . . . . . . . . . 10  |-  ( C  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( C  +  R
)  e.  ( ( C  -  R ) [,] ( C  +  R ) )  -> 
( C  <  ( C  +  R )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R ) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  +  R
) ) ) ) ) )
14535, 77, 78, 144syl3c 57 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  +  R
) ) ) )
14645, 88breq12d 4052 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  +  R
) )  <->  ( F `  C )  <  ( F `  ( C  +  R ) ) ) )
147145, 146sylibd 205 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  ->  ( F `  C )  <  ( F `  ( C  +  R ) ) ) )
14898simp3d 969 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  <_ 
y )
149 simprlr 739 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  y  e.  RR )
150 ltletr 8929 . . . . . . . . . 10  |-  ( ( ( F `  C
)  e.  RR  /\  ( F `  ( C  +  R ) )  e.  RR  /\  y  e.  RR )  ->  (
( ( F `  C )  <  ( F `  ( C  +  R ) )  /\  ( F `  ( C  +  R ) )  <_  y )  -> 
( F `  C
)  <  y )
)
15124, 103, 149, 150syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( ( F `  C )  <  ( F `  ( C  +  R ) )  /\  ( F `  ( C  +  R ) )  <_  y )  -> 
( F `  C
)  <  y )
)
152148, 151mpan2d 655 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  C
)  <  ( F `  ( C  +  R
) )  ->  ( F `  C )  <  y ) )
153147, 152syld 40 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  <  ( ( ( C  -  R ) [,] ( C  +  R )
) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  ->  ( F `  C )  <  y
) )
154 isorel 5839 . . . . . . . . . . . . 13  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  /\  ( ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R
) )  /\  C  e.  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( ( C  -  R )  <  C  <->  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  -  R ) ) `'  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
) ) )
155154biimpd 198 . . . . . . . . . . . 12  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  /\  ( ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R
) )  /\  C  e.  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( ( C  -  R )  <  C  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  -  R )
) `'  <  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C ) ) )
156155exp32 588 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R
) [,] ( C  +  R ) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  -> 
( ( C  -  R )  e.  ( ( C  -  R
) [,] ( C  +  R ) )  ->  ( C  e.  ( ( C  -  R ) [,] ( C  +  R )
)  ->  ( ( C  -  R )  <  C  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) ) `'  <  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C ) ) ) ) )
157156com4l 78 . . . . . . . . . 10  |-  ( ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  ( C  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( C  -  R
)  <  C  ->  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) ) `'  <  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C ) ) ) ) )
15829, 35, 36, 157syl3c 57 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  ->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) ) `'  <  ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C ) ) )
159 fvex 5555 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) )  e.  _V
160159, 84brcnv 4880 . . . . . . . . . 10  |-  ( ( ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  -  R ) ) `'  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
)  <->  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) ) )
16145, 43breq12d 4052 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  C
)  <  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) )  <->  ( F `  C )  <  ( F `  ( C  -  R ) ) ) )
162160, 161syl5bb 248 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) `  ( C  -  R )
) `'  <  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C )  <-> 
( F `  C
)  <  ( F `  ( C  -  R
) ) ) )
163158, 162sylibd 205 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  ->  ( F `  C )  <  ( F `  ( C  -  R ) ) ) )
16464simp3d 969 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  <_ 
y )
165 ltletr 8929 . . . . . . . . . 10  |-  ( ( ( F `  C
)  e.  RR  /\  ( F `  ( C  -  R ) )  e.  RR  /\  y  e.  RR )  ->  (
( ( F `  C )  <  ( F `  ( C  -  R ) )  /\  ( F `  ( C  -  R ) )  <_  y )  -> 
( F `  C
)  <  y )
)
16624, 70, 149, 165syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( ( F `  C )  <  ( F `  ( C  -  R ) )  /\  ( F `  ( C  -  R ) )  <_  y )  -> 
( F `  C
)  <  y )
)
167164, 166mpan2d 655 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  C
)  <  ( F `  ( C  -  R
) )  ->  ( F `  C )  <  y ) )
168163, 167syld 40 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  Isom  <  ,  `'  <  ( ( ( C  -  R ) [,] ( C  +  R
) ) ,  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  ->  ( F `  C )  <  y
) )
169153, 168, 139mpjaod 370 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  C )  <  y )
17066rexrd 8897 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  e.  RR* )
171149rexrd 8897 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  y  e.  RR* )
172 elioo2 10713 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( F `  C
)  e.  ( x (,) y )  <->  ( ( F `  C )  e.  RR  /\  x  < 
( F `  C
)  /\  ( F `  C )  <  y
) ) )
173170, 171, 172syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  C
)  e.  ( x (,) y )  <->  ( ( F `  C )  e.  RR  /\  x  < 
( F `  C
)  /\  ( F `  C )  <  y
) ) )
17424, 140, 169, 173mpbir3and 1135 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  C )  e.  ( x (,) y
) )
17560fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) ) )
176 iccntr 18342 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
177176ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
178175, 177eqtrd 2328 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) )  =  ( x (,) y
) )
179174, 178eleqtrrd 2373 . . . 4  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  C )  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) ) )
180179expr 598 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y )  ->  ( F `  C )  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) ) ) )
181180rexlimdvva 2687 . 2  |-  ( ph  ->  ( E. x  e.  RR  E. y  e.  RR  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y )  ->  ( F `  C )  e.  ( ( int `  ( topGen `
 ran  (,) )
) `  ( F " ( ( C  -  R ) [,] ( C  +  R )
) ) ) ) )
18219, 181mpd 14 1  |-  ( ph  ->  ( F `  C
)  e.  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( F " ( ( C  -  R ) [,] ( C  +  R ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164    C_ wss 3165   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   intcnt 16770   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  dvcnvrelem2  19381
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
  Copyright terms: Public domain W3C validator