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Theorem dvcnvrelem1 19364
Description: Lemma for dvcnvre 19366. (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 19252 . . . . . . 7  |-  dom  ( RR  _D  F )  C_  RR
32a1i 10 . . . . . 6  |-  ( ph  ->  dom  ( RR  _D  F )  C_  RR )
41, 3eqsstr3d 3213 . . . . 5  |-  ( ph  ->  X  C_  RR )
5 dvcnvre.c . . . . 5  |-  ( ph  ->  C  e.  X )
64, 5sseldd 3181 . . . 4  |-  ( ph  ->  C  e.  RR )
7 dvcnvre.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
87rpred 10390 . . . 4  |-  ( ph  ->  R  e.  RR )
96, 8resubcld 9211 . . 3  |-  ( ph  ->  ( C  -  R
)  e.  RR )
106, 8readdcld 8862 . . 3  |-  ( ph  ->  ( C  +  R
)  e.  RR )
116, 7ltsubrpd 10418 . . . . 5  |-  ( ph  ->  ( C  -  R
)  <  C )
126, 7ltaddrpd 10419 . . . . 5  |-  ( ph  ->  C  <  ( C  +  R ) )
139, 6, 10, 11, 12lttrd 8977 . . . 4  |-  ( ph  ->  ( C  -  R
)  <  ( C  +  R ) )
149, 10, 13ltled 8967 . . 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 18401 . . . 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 18820 . 2  |-  ( ph  ->  E. x  e.  RR  E. y  e.  RR  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) )  =  ( x [,] y ) )
20 cncff 18397 . . . . . . . . 9  |-  ( F  e.  ( X -cn-> RR )  ->  F : X
--> RR )
2116, 20syl 15 . . . . . . . 8  |-  ( ph  ->  F : X --> RR )
22 ffvelrn 5663 . . . . . . . 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 8881 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  RR* )
2610rexrd 8881 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  RR* )
27 lbicc2 10752 . . . . . . . . . . . 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 8967 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  <_  C )
316, 10, 12ltled 8967 . . . . . . . . . . . 12  |-  ( ph  ->  C  <_  ( C  +  R ) )
32 elicc2 10715 . . . . . . . . . . . . 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 5823 . . . . . . . . . . . . 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 5542 . . . . . . . . . . 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 5542 . . . . . . . . . . 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 4036 . . . . . . . . 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 5391 . . . . . . . . . . . . . . 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 5393 . . . . . . . . . . . . . . . 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 3215 . . . . . . . . . . . . . 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 5754 . . . . . . . . . . . . . 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 4702 . . . . . . . . . . . . 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 2327 . . . . . . . . . . . 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 2359 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  -  R ) )  e.  ( x [,] y
) )
62 elicc2 10715 . . . . . . . . . . . 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 3181 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  -  R
)  e.  X )
68 ffvelrn 5663 . . . . . . . . . . . 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 8912 . . . . . . . . . 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 10753 . . . . . . . . . . . 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 5823 . . . . . . . . . . . . 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 5539 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  C )  e.  _V
85 fvex 5539 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  +  R
) )  e.  _V
8684, 85brcnv 4864 . . . . . . . . . 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 5542 . . . . . . . . . . . 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 4036 . . . . . . . . . 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 5754 . . . . . . . . . . . . . 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 2359 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  ( F `  ( C  +  R ) )  e.  ( x [,] y
) )
96 elicc2 10715 . . . . . . . . . . . 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 3181 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  +  R
)  e.  X )
101 ffvelrn 5663 . . . . . . . . . . . 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 8912 . . . . . . . . . 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 8794 . . . . . . . . . . . . . 14  |-  RR  C_  CC
109108a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  RR  C_  CC )
110 fss 5397 . . . . . . . . . . . . . 14  |-  ( ( F : X --> RR  /\  RR  C_  CC )  ->  F : X --> CC )
11121, 108, 110sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  F : X --> CC )
11215, 4sstrd 3189 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) [,] ( C  +  R )
)  C_  RR )
113 eqid 2283 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
114113tgioo2 18309 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
115113, 114dvres 19261 . . . . . . . . . . . . 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 18326 . . . . . . . . . . . . . 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 4955 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) ) )
120116, 119eqtrd 2315 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) )  =  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) ) )
121120dmeqd 4881 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R ) ) ) )  =  dom  (
( RR  _D  F
)  |`  ( ( C  -  R ) (,) ( C  +  R
) ) ) )
122 dmres 4976 . . . . . . . . . . 11  |-  dom  (
( RR  _D  F
)  |`  ( ( C  -  R ) (,) ( C  +  R
) ) )  =  ( ( ( C  -  R ) (,) ( C  +  R
) )  i^i  dom  ( RR  _D  F
) )
123 ioossicc 10735 . . . . . . . . . . . . . 14  |-  ( ( C  -  R ) (,) ( C  +  R ) )  C_  ( ( C  -  R ) [,] ( C  +  R )
)
124123, 15syl5ss 3190 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  -  R ) (,) ( C  +  R )
)  C_  X )
125124, 1sseqtr4d 3215 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( C  -  R ) (,) ( C  +  R )
)  C_  dom  ( RR 
_D  F ) )
126 df-ss 3166 . . . . . . . . . . . 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 2327 . . . . . . . . . 10  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( ( C  -  R ) (,) ( C  +  R )
) )  =  ( ( C  -  R
) (,) ( C  +  R ) ) )
129121, 128eqtrd 2315 . . . . . . . . 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 4979 . . . . . . . . . . . . . 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 3212 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) ) 
C_  ( RR  _D  F ) )
134 rnss 4907 . . . . . . . . . . . 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 3179 . . . . . . . . . 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 19358 . . . . . . . 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 5823 . . . . . . . . . . . . 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 4036 . . . . . . . . 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 8913 . . . . . . . . . 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 5823 . . . . . . . . . . . . 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 5539 . . . . . . . . . . 11  |-  ( ( F  |`  ( ( C  -  R ) [,] ( C  +  R
) ) ) `  ( C  -  R
) )  e.  _V
160159, 84brcnv 4864 . . . . . . . . . 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 4036 . . . . . . . . . 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 8913 . . . . . . . . . 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 8881 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  x  e.  RR* )
171149rexrd 8881 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  RR  /\  y  e.  RR )  /\  ran  ( F  |`  ( ( C  -  R ) [,] ( C  +  R )
) )  =  ( x [,] y ) ) )  ->  y  e.  RR* )
172 elioo2 10697 . . . . . . 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 5529 . . . . . 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 18326 . . . . . . 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 2315 . . . . 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 2360 . . . 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 2674 . 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 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   RR+crp 10354   (,)cioo 10656   [,]cicc 10659   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377   intcnt 16754   -cn->ccncf 18380    _D cdv 19213
This theorem is referenced by:  dvcnvrelem2  19365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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