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Theorem dvcnvrelem1 19854
Description: Lemma for dvcnvre 19856. (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 19742 . . . . . 6  |-  dom  ( RR  _D  F )  C_  RR
31, 2syl6eqssr 3359 . . . . 5  |-  ( ph  ->  X  C_  RR )
4 dvcnvre.c . . . . 5  |-  ( ph  ->  C  e.  X )
53, 4sseldd 3309 . . . 4  |-  ( ph  ->  C  e.  RR )
6 dvcnvre.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
76rpred 10604 . . . 4  |-  ( ph  ->  R  e.  RR )
85, 7resubcld 9421 . . 3  |-  ( ph  ->  ( C  -  R
)  e.  RR )
95, 7readdcld 9071 . . 3  |-  ( ph  ->  ( C  +  R
)  e.  RR )
105, 6ltsubrpd 10632 . . . . 5  |-  ( ph  ->  ( C  -  R
)  <  C )
115, 6ltaddrpd 10633 . . . . 5  |-  ( ph  ->  C  <  ( C  +  R ) )
128, 5, 9, 10, 11lttrd 9187 . . . 4  |-  ( ph  ->  ( C  -  R
)  <  ( C  +  R ) )
138, 9, 12ltled 9177 . . 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 18880 . . . 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 19310 . 2  |-  ( ph  ->  E. x  e.  RR  E. y  e.  RR  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) )  =  ( x [,] y ) )
19 cncff 18876 . . . . . . . . 9  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
2015, 19syl 16 . . . . . . . 8  |-  ( ph  ->  F : X --> RR )
2120, 4ffvelrnd 5830 . . . . . . 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 9090 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  RR* )
249rexrd 9090 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  RR* )
25 lbicc2 10969 . . . . . . . . . . . 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 9177 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  <_  C )
295, 9, 11ltled 9177 . . . . . . . . . . . 12  |-  ( ph  ->  C  <_  ( C  +  R ) )
30 elicc2 10931 . . . . . . . . . . . . 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 6005 . . . . . . . . . . . . 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 5704 . . . . . . . . . . 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 5704 . . . . . . . . . . 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 4185 . . . . . . . . 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 5552 . . . . . . . . . . . . . . 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 5554 . . . . . . . . . . . . . . . 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 3345 . . . . . . . . . . . . . 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 5933 . . . . . . . . . . . . . 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 4850 . . . . . . . . . . . . 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 2448 . . . . . . . . . . . 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 2480 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  e.  ( x [,] y
) )
60 elicc2 10931 . . . . . . . . . . . 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 3309 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  X )
6620, 65ffvelrnd 5830 . . . . . . . . . . 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 9121 . . . . . . . . . 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 10970 . . . . . . . . . . . 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 6005 . . . . . . . . . . . . 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 5701 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C )  e.  _V
82 fvex 5701 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  +  R
) )  e.  _V
8381, 82brcnv 5014 . . . . . . . . . 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 5704 . . . . . . . . . . . 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 4185 . . . . . . . . . 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 5933 . . . . . . . . . . . . . 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 2480 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  e.  ( x [,] y
) )
93 elicc2 10931 . . . . . . . . . . . 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 3309 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  X )
9820, 97ffvelrnd 5830 . . . . . . . . . . 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 9121 . . . . . . . . . 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 9003 . . . . . . . . . . . . . 14  |-  RR  C_  CC
105104a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  C_  CC )
106 fss 5558 . . . . . . . . . . . . . 14  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
10720, 104, 106sylancl 644 . . . . . . . . . . . . 13  |-  ( ph  ->  F : X --> CC )
10814, 3sstrd 3318 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR )
109 eqid 2404 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
110109tgioo2 18787 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
111109, 110dvres 19751 . . . . . . . . . . . . 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 18805 . . . . . . . . . . . . . 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 5105 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) ) )
116112, 115eqtrd 2436 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) ) )
117116dmeqd 5031 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  dom  (
( RR  _D  F
)  |`  ( ( C  -  R ) (,) ( C  +  R
) ) ) )
118 dmres 5126 . . . . . . . . . . 11  |-  dom  (
( RR  _D  F
)  |`  ( ( C  -  R ) (,) ( C  +  R
) ) )  =  ( ( ( C  -  R ) (,) ( C  +  R
) )  i^i  dom  ( RR  _D  F
) )
119 ioossicc 10952 . . . . . . . . . . . . . 14  |-  ( ( C  -  R ) (,) ( C  +  R ) )  C_  ( ( C  -  R ) [,] ( C  +  R )
)
120119, 14syl5ss 3319 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) (,) ( C  +  R )
)  C_  X )
121120, 1sseqtr4d 3345 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( C  -  R ) (,) ( C  +  R )
)  C_  dom  ( RR 
_D  F ) )
122 df-ss 3294 . . . . . . . . . . . 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 2448 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) )  =  ( ( C  -  R
) (,) ( C  +  R ) ) )
125117, 124eqtrd 2436 . . . . . . . . 9  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  ( ( C  -  R ) (,) ( C  +  R ) ) )
126 resss 5129 . . . . . . . . . . . 12  |-  ( ( RR  _D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) )  C_  ( RR  _D  F )
127116, 126syl6eqss 3358 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) 
C_  ( RR  _D  F ) )
128 rnss 5057 . . . . . . . . . . 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 3311 . . . . . . . . 9  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) )
1328, 9, 17, 125, 131dvne0 19848 . . . . . . . 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 6005 . . . . . . . . . . . . 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 4185 . . . . . . . . 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 9122 . . . . . . . . . 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 6005 . . . . . . . . . . . . 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 5701 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) )  e.  _V
154153, 81brcnv 5014 . . . . . . . . . 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 4185 . . . . . . . . . 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 9122 . . . . . . . . . 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 9090 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  e.  RR* )
165143rexrd 9090 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  y  e.  RR* )
166 elioo2 10913 . . . . . . 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 5691 . . . . . 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 18805 . . . . . . 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 2436 . . . . 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 2481 . . . 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 2797 . 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 1649    e. wcel 1721   E.wrex 2667    i^i cin 3279    C_ wss 3280   class class class wbr 4172   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   RR+crp 10568   (,)cioo 10872   [,]cicc 10875   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   intcnt 17036   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  dvcnvrelem2  19855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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