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Theorem dvcnvrelem1 19901
Description: Lemma for dvcnvre 19903. (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 19789 . . . . . 6  |-  dom  ( RR  _D  F )  C_  RR
31, 2syl6eqssr 3399 . . . . 5  |-  ( ph  ->  X  C_  RR )
4 dvcnvre.c . . . . 5  |-  ( ph  ->  C  e.  X )
53, 4sseldd 3349 . . . 4  |-  ( ph  ->  C  e.  RR )
6 dvcnvre.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
76rpred 10648 . . . 4  |-  ( ph  ->  R  e.  RR )
85, 7resubcld 9465 . . 3  |-  ( ph  ->  ( C  -  R
)  e.  RR )
95, 7readdcld 9115 . . 3  |-  ( ph  ->  ( C  +  R
)  e.  RR )
105, 6ltsubrpd 10676 . . . . 5  |-  ( ph  ->  ( C  -  R
)  <  C )
115, 6ltaddrpd 10677 . . . . 5  |-  ( ph  ->  C  <  ( C  +  R ) )
128, 5, 9, 10, 11lttrd 9231 . . . 4  |-  ( ph  ->  ( C  -  R
)  <  ( C  +  R ) )
138, 9, 12ltled 9221 . . 3  |-  ( ph  ->  ( C  -  R
)  <_  ( C  +  R ) )
14 dvcnvre.s . . . 4  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  X )
15 dvcnvre.f . . . 4  |-  ( ph  ->  F  e.  ( X
-cn-> RR ) )
16 rescncf 18927 . . . 4  |-  ( ( ( C  -  R
) [,] ( C  +  R ) ) 
C_  X  ->  ( F  e.  ( X -cn->
RR )  ->  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) )  e.  ( ( ( C  -  R ) [,] ( C  +  R
) ) -cn-> RR ) ) )
1714, 15, 16sylc 58 . . 3  |-  ( ph  ->  ( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) )  e.  ( ( ( C  -  R
) [,] ( C  +  R ) )
-cn-> RR ) )
188, 9, 13, 17evthicc2 19357 . 2  |-  ( ph  ->  E. x  e.  RR  E. y  e.  RR  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) )  =  ( x [,] y ) )
19 cncff 18923 . . . . . . . . 9  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
2015, 19syl 16 . . . . . . . 8  |-  ( ph  ->  F : X --> RR )
2120, 4ffvelrnd 5871 . . . . . . 7  |-  ( ph  ->  ( F `  C
)  e.  RR )
2221adantr 452 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  C )  e.  RR )
238rexrd 9134 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  RR* )
249rexrd 9134 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  RR* )
25 lbicc2 11013 . . . . . . . . . . . 12  |-  ( ( ( C  -  R
)  e.  RR*  /\  ( C  +  R )  e.  RR*  /\  ( C  -  R )  <_ 
( C  +  R
) )  ->  ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
2623, 24, 13, 25syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( C  -  R
)  e.  ( ( C  -  R ) [,] ( C  +  R ) ) )
2726adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
288, 5, 10ltled 9221 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  <_  C )
295, 9, 11ltled 9221 . . . . . . . . . . . 12  |-  ( ph  ->  C  <_  ( C  +  R ) )
30 elicc2 10975 . . . . . . . . . . . . 13  |-  ( ( ( C  -  R
)  e.  RR  /\  ( C  +  R
)  e.  RR )  ->  ( C  e.  ( ( C  -  R ) [,] ( C  +  R )
)  <->  ( C  e.  RR  /\  ( C  -  R )  <_  C  /\  C  <_  ( C  +  R )
) ) )
318, 9, 30syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  e.  ( ( C  -  R
) [,] ( C  +  R ) )  <-> 
( C  e.  RR  /\  ( C  -  R
)  <_  C  /\  C  <_  ( C  +  R ) ) ) )
325, 28, 29, 31mpbir3and 1137 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  ( ( C  -  R ) [,] ( C  +  R ) ) )
3332adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  C  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
3410adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( C  -  R )  <  C )
35 isorel 6046 . . . . . . . . . . . . 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
) ) )
3635biimpd 199 . . . . . . . . . . . 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 ) ) )
3736exp32 589 . . . . . . . . . . 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 ) ) ) ) )
3837com4l 80 . . . . . . . . . 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 ) ) ) ) )
3927, 33, 34, 38syl3c 59 . . . . . . . . 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 ) ) )
40 fvres 5745 . . . . . . . . . . 11  |-  ( ( C  -  R )  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  -  R ) )  =  ( F `  ( C  -  R
) ) )
4127, 40syl 16 . . . . . . . . . 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
) ) )
42 fvres 5745 . . . . . . . . . . 11  |-  ( C  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C )  =  ( F `  C ) )
4333, 42syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  C )  =  ( F `  C ) )
4441, 43breq12d 4225 . . . . . . . . 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 )
) )
4539, 44sylibd 206 . . . . . . . 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 )
) )
4620adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  F : X --> RR )
47 ffun 5593 . . . . . . . . . . . . . . 15  |-  ( F : X --> RR  ->  Fun 
F )
4846, 47syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  Fun  F )
4914adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( C  -  R
) [,] ( C  +  R ) ) 
C_  X )
50 fdm 5595 . . . . . . . . . . . . . . . 16  |-  ( F : X --> RR  ->  dom 
F  =  X )
5146, 50syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  dom  F  =  X )
5249, 51sseqtr4d 3385 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( C  -  R
) [,] ( C  +  R ) ) 
C_  dom  F )
53 funfvima2 5974 . . . . . . . . . . . . . 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 ) ) ) ) )
5448, 52, 53syl2anc 643 . . . . . . . . . . . . 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
) ) ) ) )
5527, 54mpd 15 . . . . . . . . . . . 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 ) ) ) )
56 df-ima 4891 . . . . . . . . . . . . 13  |-  ( F
" ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )
57 simprr 734 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) )  =  ( x [,] y ) )
5856, 57syl5eq 2480 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F " ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( x [,] y
) )
5955, 58eleqtrd 2512 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  e.  ( x [,] y
) )
60 elicc2 10975 . . . . . . . . . . . 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 ) ) )
6160ad2antrl 709 . . . . . . . . . . 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
) ) )
6259, 61mpbid 202 . . . . . . . . . 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 ) )
6362simp2d 970 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  <_  ( F `  ( C  -  R )
) )
64 simprll 739 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  e.  RR )
6514, 26sseldd 3349 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  X )
6620, 65ffvelrnd 5871 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  ( C  -  R )
)  e.  RR )
6766adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  e.  RR )
68 lelttr 9165 . . . . . . . . . 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 ) ) )
6964, 67, 22, 68syl3anc 1184 . . . . . . . . 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 ) ) )
7063, 69mpand 657 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  ( C  -  R )
)  <  ( F `  C )  ->  x  <  ( F `  C
) ) )
7145, 70syld 42 . . . . . . 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 )
) )
72 ubicc2 11014 . . . . . . . . . . . 12  |-  ( ( ( C  -  R
)  e.  RR*  /\  ( C  +  R )  e.  RR*  /\  ( C  -  R )  <_ 
( C  +  R
) )  ->  ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
7323, 24, 13, 72syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( C  +  R
)  e.  ( ( C  -  R ) [,] ( C  +  R ) ) )
7473adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R )
) )
7511adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  C  <  ( C  +  R
) )
76 isorel 6046 . . . . . . . . . . . . 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 )
) ) )
7776biimpd 199 . . . . . . . . . . . 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 ) ) ) )
7877exp32 589 . . . . . . . . . . 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 ) ) ) ) ) )
7978com4l 80 . . . . . . . . . 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 ) ) ) ) ) )
8033, 74, 75, 79syl3c 59 . . . . . . . . 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 ) ) ) )
81 fvex 5742 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C )  e.  _V
82 fvex 5742 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  +  R
) )  e.  _V
8381, 82brcnv 5055 . . . . . . . . . 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 ) )
84 fvres 5745 . . . . . . . . . . . 12  |-  ( ( C  +  R )  e.  ( ( C  -  R ) [,] ( C  +  R
) )  ->  (
( F  |`  (
( C  -  R
) [,] ( C  +  R ) ) ) `  ( C  +  R ) )  =  ( F `  ( C  +  R
) ) )
8574, 84syl 16 . . . . . . . . . . 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
) ) )
8685, 43breq12d 4225 . . . . . . . . . 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 )
) )
8783, 86syl5bb 249 . . . . . . . . 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 ) ) )
8880, 87sylibd 206 . . . . . . . 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 )
) )
89 funfvima2 5974 . . . . . . . . . . . . . 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 ) ) ) ) )
9048, 52, 89syl2anc 643 . . . . . . . . . . . . 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
) ) ) ) )
9174, 90mpd 15 . . . . . . . . . . . 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 ) ) ) )
9291, 58eleqtrd 2512 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  e.  ( x [,] y
) )
93 elicc2 10975 . . . . . . . . . . . 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 ) ) )
9493ad2antrl 709 . . . . . . . . . . 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
) ) )
9592, 94mpbid 202 . . . . . . . . . 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 ) )
9695simp2d 970 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  <_  ( F `  ( C  +  R )
) )
9714, 73sseldd 3349 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  X )
9820, 97ffvelrnd 5871 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  ( C  +  R )
)  e.  RR )
9998adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  e.  RR )
100 lelttr 9165 . . . . . . . . . 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 ) ) )
10164, 99, 22, 100syl3anc 1184 . . . . . . . . 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 ) ) )
10296, 101mpand 657 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  ( C  +  R )
)  <  ( F `  C )  ->  x  <  ( F `  C
) ) )
10388, 102syld 42 . . . . . . 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 ) ) )
104 ax-resscn 9047 . . . . . . . . . . . . . 14  |-  RR  C_  CC
105104a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  C_  CC )
106 fss 5599 . . . . . . . . . . . . . 14  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
10720, 104, 106sylancl 644 . . . . . . . . . . . . 13  |-  ( ph  ->  F : X --> CC )
10814, 3sstrd 3358 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR )
109 eqid 2436 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
110109tgioo2 18834 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
111109, 110dvres 19798 . . . . . . . . . . . . 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
) ) ) ) )
112105, 107, 3, 108, 111syl22anc 1185 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
113 iccntr 18852 . . . . . . . . . . . . . 14  |-  ( ( ( C  -  R
)  e.  RR  /\  ( C  +  R
)  e.  RR )  ->  ( ( int `  ( topGen `  ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( ( C  -  R ) (,) ( C  +  R )
) )
1148, 9, 113syl2anc 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) )  =  ( ( C  -  R ) (,) ( C  +  R )
) )
115114reseq2d 5146 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) ) )
116112, 115eqtrd 2468 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) ) )
117116dmeqd 5072 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  dom  (
( RR  _D  F
)  |`  ( ( C  -  R ) (,) ( C  +  R
) ) ) )
118 dmres 5167 . . . . . . . . . . 11  |-  dom  (
( RR  _D  F
)  |`  ( ( C  -  R ) (,) ( C  +  R
) ) )  =  ( ( ( C  -  R ) (,) ( C  +  R
) )  i^i  dom  ( RR  _D  F
) )
119 ioossicc 10996 . . . . . . . . . . . . . 14  |-  ( ( C  -  R ) (,) ( C  +  R ) )  C_  ( ( C  -  R ) [,] ( C  +  R )
)
120119, 14syl5ss 3359 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) (,) ( C  +  R )
)  C_  X )
121120, 1sseqtr4d 3385 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( C  -  R ) (,) ( C  +  R )
)  C_  dom  ( RR 
_D  F ) )
122 df-ss 3334 . . . . . . . . . . . 12  |-  ( ( ( C  -  R
) (,) ( C  +  R ) ) 
C_  dom  ( RR  _D  F )  <->  ( (
( C  -  R
) (,) ( C  +  R ) )  i^i  dom  ( RR  _D  F ) )  =  ( ( C  -  R ) (,) ( C  +  R )
) )
123121, 122sylib 189 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( C  -  R ) (,) ( C  +  R
) )  i^i  dom  ( RR  _D  F
) )  =  ( ( C  -  R
) (,) ( C  +  R ) ) )
124118, 123syl5eq 2480 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) )  =  ( ( C  -  R
) (,) ( C  +  R ) ) )
125117, 124eqtrd 2468 . . . . . . . . 9  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( C  -  R ) (,) ( C  +  R ) ) )
126 resss 5170 . . . . . . . . . . . 12  |-  ( ( RR  _D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) )  C_  ( RR  _D  F )
127116, 126syl6eqss 3398 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) 
C_  ( RR  _D  F ) )
128 rnss 5098 . . . . . . . . . . 11  |-  ( ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  C_  ( RR  _D  F
)  ->  ran  ( RR 
_D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) ) )  C_  ran  ( RR  _D  F
) )
129127, 128syl 16 . . . . . . . . . 10  |-  ( ph  ->  ran  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  C_  ran  ( RR 
_D  F ) )
130 dvcnvre.z . . . . . . . . . 10  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F
) )
131129, 130ssneldd 3351 . . . . . . . . 9  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
1328, 9, 17, 125, 131dvne0 19895 . . . . . . . 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 ) ) ) ) ) )
133132adantr 452 . . . . . . 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 )
) ) ) ) )
13471, 103, 133mpjaod 371 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  <  ( F `  C
) )
135 isorel 6046 . . . . . . . . . . . . 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 ) ) ) )
136135biimpd 199 . . . . . . . . . . . 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 )
) ) )
137136exp32 589 . . . . . . . . . . 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 ) ) ) ) ) )
138137com4l 80 . . . . . . . . . 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
) ) ) ) ) )
13933, 74, 75, 138syl3c 59 . . . . . . . . 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
) ) ) )
14043, 85breq12d 4225 . . . . . . . . 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 ) ) ) )
141139, 140sylibd 206 . . . . . . . 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 ) ) ) )
14295simp3d 971 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  <_ 
y )
143 simprlr 740 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  y  e.  RR )
144 ltletr 9166 . . . . . . . . . 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 )
)
14522, 99, 143, 144syl3anc 1184 . . . . . . . . 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 )
)
146142, 145mpan2d 656 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  C
)  <  ( F `  ( C  +  R
) )  ->  ( F `  C )  <  y ) )
147141, 146syld 42 . . . . . . 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
) )
148 isorel 6046 . . . . . . . . . . . . 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
) ) )
149148biimpd 199 . . . . . . . . . . . 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 ) ) )
150149exp32 589 . . . . . . . . . . 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 ) ) ) ) )
151150com4l 80 . . . . . . . . . 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 ) ) ) ) )
15227, 33, 34, 151syl3c 59 . . . . . . . . 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 ) ) )
153 fvex 5742 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) )  e.  _V
154153, 81brcnv 5055 . . . . . . . . . 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
) ) )
15543, 41breq12d 4225 . . . . . . . . . 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 ) ) ) )
156154, 155syl5bb 249 . . . . . . . . 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
) ) ) )
157152, 156sylibd 206 . . . . . . . 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 ) ) ) )
15862simp3d 971 . . . . . . . . 9  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  <_ 
y )
159 ltletr 9166 . . . . . . . . . 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 )
)
16022, 67, 143, 159syl3anc 1184 . . . . . . . . 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 )
)
161158, 160mpan2d 656 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( F `  C
)  <  ( F `  ( C  -  R
) )  ->  ( F `  C )  <  y ) )
162157, 161syld 42 . . . . . . 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
) )
163147, 162, 133mpjaod 371 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  C )  <  y )
16464rexrd 9134 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  e.  RR* )
165143rexrd 9134 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  y  e.  RR* )
166 elioo2 10957 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( F `  C
)  e.  ( x (,) y )  <->  ( ( F `  C )  e.  RR  /\  x  < 
( F `  C
)  /\  ( F `  C )  <  y
) ) )
167164, 165, 166syl2anc 643 . . . . . 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
) ) )
16822, 134, 163, 167mpbir3and 1137 . . . . 5  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  C )  e.  ( x (,) y
) )
16958fveq2d 5732 . . . . . 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 ) ) )
170 iccntr 18852 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
171170ad2antrl 709 . . . . . 6  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  (
( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
172169, 171eqtrd 2468 . . . . 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
) )
173168, 172eleqtrrd 2513 . . . 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 )
) ) ) )
174173expr 599 . . 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 )
) ) ) ) )
175174rexlimdvva 2837 . 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 )
) ) ) ) )
17618, 175mpd 15 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319    C_ wss 3320   class class class wbr 4212   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990    + caddc 8993   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291   RR+crp 10612   (,)cioo 10916   [,]cicc 10919   TopOpenctopn 13649   topGenctg 13665  ℂfldccnfld 16703   intcnt 17081   -cn->ccncf 18906    _D cdv 19750
This theorem is referenced by:  dvcnvrelem2  19902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-cmp 17450  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754
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