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Theorem areacirclem3 26182
Description: Continuity of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
Assertion
Ref Expression
areacirclem3  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )  e.  ( ( -u R (,) R ) -cn-> CC ) )
Distinct variable group:    t, R

Proof of Theorem areacirclem3
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21mulcn 18850 . . 3  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
32a1i 11 . 2  |-  ( R  e.  RR+  ->  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
4 2cn 10026 . . . 4  |-  2  e.  CC
5 ioossre 10928 . . . . 5  |-  ( -u R (,) R )  C_  RR
6 ax-resscn 9003 . . . . 5  |-  RR  C_  CC
75, 6sstri 3317 . . . 4  |-  ( -u R (,) R )  C_  CC
8 ssid 3327 . . . 4  |-  CC  C_  CC
9 cncfmptc 18894 . . . 4  |-  ( ( 2  e.  CC  /\  ( -u R (,) R
)  C_  CC  /\  CC  C_  CC )  ->  (
t  e.  ( -u R (,) R )  |->  2 )  e.  ( (
-u R (,) R
) -cn-> CC ) )
104, 7, 8, 9mp3an 1279 . . 3  |-  ( t  e.  ( -u R (,) R )  |->  2 )  e.  ( ( -u R (,) R ) -cn-> CC )
1110a1i 11 . 2  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  |->  2 )  e.  ( ( -u R (,) R ) -cn-> CC ) )
126a1i 11 . . 3  |-  ( R  e.  RR+  ->  RR  C_  CC )
13 rpcn 10576 . . . . . . . 8  |-  ( R  e.  RR+  ->  R  e.  CC )
1413sqcld 11476 . . . . . . 7  |-  ( R  e.  RR+  ->  ( R ^ 2 )  e.  CC )
1514adantr 452 . . . . . 6  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( R ^ 2 )  e.  CC )
16 elioore 10902 . . . . . . . . 9  |-  ( t  e.  ( -u R (,) R )  ->  t  e.  RR )
1716recnd 9070 . . . . . . . 8  |-  ( t  e.  ( -u R (,) R )  ->  t  e.  CC )
1817sqcld 11476 . . . . . . 7  |-  ( t  e.  ( -u R (,) R )  ->  (
t ^ 2 )  e.  CC )
1918adantl 453 . . . . . 6  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( t ^ 2 )  e.  CC )
2015, 19subcld 9367 . . . . 5  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( ( R ^
2 )  -  (
t ^ 2 ) )  e.  CC )
2120sqrcld 12194 . . . 4  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  CC )
22 eqid 2404 . . . 4  |-  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )  =  ( t  e.  (
-u R (,) R
)  |->  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
2321, 22fmptd 5852 . . 3  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) ) : ( -u R (,) R ) --> CC )
245a1i 11 . . 3  |-  ( R  e.  RR+  ->  ( -u R (,) R )  C_  RR )
25 reex 9037 . . . . . . . 8  |-  RR  e.  _V
2625prid1 3872 . . . . . . 7  |-  RR  e.  { RR ,  CC }
2726a1i 11 . . . . . 6  |-  ( R  e.  RR+  ->  RR  e.  { RR ,  CC }
)
28 rpre 10574 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  e.  RR )
2928resqcld 11504 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( R ^ 2 )  e.  RR )
3029adantr 452 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( R ^ 2 )  e.  RR )
3116resqcld 11504 . . . . . . . . 9  |-  ( t  e.  ( -u R (,) R )  ->  (
t ^ 2 )  e.  RR )
3231adantl 453 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( t ^ 2 )  e.  RR )
3330, 32resubcld 9421 . . . . . . 7  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( ( R ^
2 )  -  (
t ^ 2 ) )  e.  RR )
3428renegcld 9420 . . . . . . . . . . 11  |-  ( R  e.  RR+  ->  -u R  e.  RR )
3534rexrd 9090 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  -u R  e.  RR* )
36 rpxr 10575 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  e. 
RR* )
37 elioo2 10913 . . . . . . . . . 10  |-  ( (
-u R  e.  RR*  /\  R  e.  RR* )  ->  ( t  e.  (
-u R (,) R
)  <->  ( t  e.  RR  /\  -u R  <  t  /\  t  < 
R ) ) )
3835, 36, 37syl2anc 643 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  <->  ( t  e.  RR  /\  -u R  <  t  /\  t  < 
R ) ) )
39 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  t  e.  RR )
4028adantr 452 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  R  e.  RR )
4139, 40absltd 12187 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( abs `  t
)  <  R  <->  ( -u R  <  t  /\  t  < 
R ) ) )
42 recn 9036 . . . . . . . . . . . . . . . . 17  |-  ( t  e.  RR  ->  t  e.  CC )
4342abscld 12193 . . . . . . . . . . . . . . . 16  |-  ( t  e.  RR  ->  ( abs `  t )  e.  RR )
4443adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  ( abs `  t )  e.  RR )
4542absge0d 12201 . . . . . . . . . . . . . . . 16  |-  ( t  e.  RR  ->  0  <_  ( abs `  t
) )
4645adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  0  <_  ( abs `  t
) )
47 rpge0 10580 . . . . . . . . . . . . . . . 16  |-  ( R  e.  RR+  ->  0  <_  R )
4847adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  0  <_  R )
4944, 40, 46, 48lt2sqd 11512 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( abs `  t
)  <  R  <->  ( ( abs `  t ) ^
2 )  <  ( R ^ 2 ) ) )
50 absresq 12062 . . . . . . . . . . . . . . . 16  |-  ( t  e.  RR  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
5150adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
5251breq1d 4182 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( ( abs `  t
) ^ 2 )  <  ( R ^
2 )  <->  ( t ^ 2 )  < 
( R ^ 2 ) ) )
53 resqcl 11404 . . . . . . . . . . . . . . . 16  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  RR )
5453adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  RR )
5529adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  RR )
5654, 55posdifd 9569 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( t ^ 2 )  <  ( R ^ 2 )  <->  0  <  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
5749, 52, 563bitrd 271 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( abs `  t
)  <  R  <->  0  <  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
5841, 57bitr3d 247 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( -u R  <  t  /\  t  <  R )  <->  0  <  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )
5958biimpd 199 . . . . . . . . . . 11  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( -u R  <  t  /\  t  <  R )  ->  0  <  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
6059exp4b 591 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  ( t  e.  RR  ->  ( -u R  <  t  -> 
( t  <  R  ->  0  <  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) ) ) )
61603impd 1167 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( ( t  e.  RR  /\  -u R  <  t  /\  t  <  R )  -> 
0  <  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
6238, 61sylbid 207 . . . . . . . 8  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  ->  0  <  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
6362imp 419 . . . . . . 7  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
0  <  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )
6433, 63elrpd 10602 . . . . . 6  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( ( R ^
2 )  -  (
t ^ 2 ) )  e.  RR+ )
65 ovex 6065 . . . . . . 7  |-  ( 0  -  ( 2  x.  ( t ^ (
2  -  1 ) ) ) )  e. 
_V
6665a1i 11 . . . . . 6  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( 0  -  (
2  x.  ( t ^ ( 2  -  1 ) ) ) )  e.  _V )
67 rpcn 10576 . . . . . . . 8  |-  ( u  e.  RR+  ->  u  e.  CC )
6867sqrcld 12194 . . . . . . 7  |-  ( u  e.  RR+  ->  ( sqr `  u )  e.  CC )
6968adantl 453 . . . . . 6  |-  ( ( R  e.  RR+  /\  u  e.  RR+ )  ->  ( sqr `  u )  e.  CC )
70 ovex 6065 . . . . . . 7  |-  ( 1  /  ( 2  x.  ( sqr `  u
) ) )  e. 
_V
7170a1i 11 . . . . . 6  |-  ( ( R  e.  RR+  /\  u  e.  RR+ )  ->  (
1  /  ( 2  x.  ( sqr `  u
) ) )  e. 
_V )
7214adantr 452 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  CC )
7342sqcld 11476 . . . . . . . . 9  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  CC )
7473adantl 453 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  CC )
7572, 74subcld 9367 . . . . . . 7  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( R ^ 2 )  -  ( t ^ 2 ) )  e.  CC )
7665a1i 11 . . . . . . 7  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
0  -  ( 2  x.  ( t ^
( 2  -  1 ) ) ) )  e.  _V )
77 0re 9047 . . . . . . . . 9  |-  0  e.  RR
7877a1i 11 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  0  e.  RR )
7927, 14dvmptc 19797 . . . . . . . 8  |-  ( R  e.  RR+  ->  ( RR 
_D  ( t  e.  RR  |->  ( R ^
2 ) ) )  =  ( t  e.  RR  |->  0 ) )
80 ovex 6065 . . . . . . . . 9  |-  ( 2  x.  ( t ^
( 2  -  1 ) ) )  e. 
_V
8180a1i 11 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
2  x.  ( t ^ ( 2  -  1 ) ) )  e.  _V )
821cnfldtopon 18770 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
83 toponmax 16948 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
8482, 83mp1i 12 . . . . . . . . 9  |-  ( R  e.  RR+  ->  CC  e.  ( TopOpen ` fld ) )
85 df-ss 3294 . . . . . . . . . . 11  |-  ( RR  C_  CC  <->  ( RR  i^i  CC )  =  RR )
866, 85mpbi 200 . . . . . . . . . 10  |-  ( RR 
i^i  CC )  =  RR
8786a1i 11 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( RR 
i^i  CC )  =  RR )
88 sqcl 11399 . . . . . . . . . 10  |-  ( t  e.  CC  ->  (
t ^ 2 )  e.  CC )
8988adantl 453 . . . . . . . . 9  |-  ( ( R  e.  RR+  /\  t  e.  CC )  ->  (
t ^ 2 )  e.  CC )
9080a1i 11 . . . . . . . . 9  |-  ( ( R  e.  RR+  /\  t  e.  CC )  ->  (
2  x.  ( t ^ ( 2  -  1 ) ) )  e.  _V )
91 2nn 10089 . . . . . . . . . 10  |-  2  e.  NN
92 dvexp 19792 . . . . . . . . . 10  |-  ( 2  e.  NN  ->  ( CC  _D  ( t  e.  CC  |->  ( t ^
2 ) ) )  =  ( t  e.  CC  |->  ( 2  x.  ( t ^ (
2  -  1 ) ) ) ) )
9391, 92mp1i 12 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( CC 
_D  ( t  e.  CC  |->  ( t ^
2 ) ) )  =  ( t  e.  CC  |->  ( 2  x.  ( t ^ (
2  -  1 ) ) ) ) )
941, 27, 84, 87, 89, 90, 93dvmptres3 19795 . . . . . . . 8  |-  ( R  e.  RR+  ->  ( RR 
_D  ( t  e.  RR  |->  ( t ^
2 ) ) )  =  ( t  e.  RR  |->  ( 2  x.  ( t ^ (
2  -  1 ) ) ) ) )
9527, 72, 78, 79, 74, 81, 94dvmptsub 19806 . . . . . . 7  |-  ( R  e.  RR+  ->  ( RR 
_D  ( t  e.  RR  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )  =  ( t  e.  RR  |->  ( 0  -  ( 2  x.  (
t ^ ( 2  -  1 ) ) ) ) ) )
961tgioo2 18787 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
97 iooretop 18753 . . . . . . . 8  |-  ( -u R (,) R )  e.  ( topGen `  ran  (,) )
9897a1i 11 . . . . . . 7  |-  ( R  e.  RR+  ->  ( -u R (,) R )  e.  ( topGen `  ran  (,) )
)
9927, 75, 76, 95, 24, 96, 1, 98dvmptres 19802 . . . . . 6  |-  ( R  e.  RR+  ->  ( RR 
_D  ( t  e.  ( -u R (,) R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )  =  ( t  e.  ( -u R (,) R )  |->  ( 0  -  ( 2  x.  ( t ^ (
2  -  1 ) ) ) ) ) )
100 dvsqr 20581 . . . . . . 7  |-  ( RR 
_D  ( u  e.  RR+  |->  ( sqr `  u
) ) )  =  ( u  e.  RR+  |->  ( 1  /  (
2  x.  ( sqr `  u ) ) ) )
101100a1i 11 . . . . . 6  |-  ( R  e.  RR+  ->  ( RR 
_D  ( u  e.  RR+  |->  ( sqr `  u
) ) )  =  ( u  e.  RR+  |->  ( 1  /  (
2  x.  ( sqr `  u ) ) ) ) )
102 fveq2 5687 . . . . . 6  |-  ( u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) )  ->  ( sqr `  u )  =  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
103102oveq2d 6056 . . . . . . 7  |-  ( u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) )  ->  (
2  x.  ( sqr `  u ) )  =  ( 2  x.  ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )
104103oveq2d 6056 . . . . . 6  |-  ( u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) )  ->  (
1  /  ( 2  x.  ( sqr `  u
) ) )  =  ( 1  /  (
2  x.  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) ) ) )
10527, 27, 64, 66, 69, 71, 99, 101, 102, 104dvmptco 19811 . . . . 5  |-  ( R  e.  RR+  ->  ( RR 
_D  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) ) )  =  ( t  e.  ( -u R (,) R )  |->  ( ( 1  /  ( 2  x.  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )  x.  ( 0  -  (
2  x.  ( t ^ ( 2  -  1 ) ) ) ) ) ) )
106105dmeqd 5031 . . . 4  |-  ( R  e.  RR+  ->  dom  ( RR  _D  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) ) )  =  dom  ( t  e.  ( -u R (,) R )  |->  ( ( 1  /  ( 2  x.  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )  x.  ( 0  -  (
2  x.  ( t ^ ( 2  -  1 ) ) ) ) ) ) )
107 ovex 6065 . . . . 5  |-  ( ( 1  /  ( 2  x.  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )  x.  ( 0  -  (
2  x.  ( t ^ ( 2  -  1 ) ) ) ) )  e.  _V
108 eqid 2404 . . . . 5  |-  ( t  e.  ( -u R (,) R )  |->  ( ( 1  /  ( 2  x.  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )  x.  ( 0  -  (
2  x.  ( t ^ ( 2  -  1 ) ) ) ) ) )  =  ( t  e.  (
-u R (,) R
)  |->  ( ( 1  /  ( 2  x.  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )  x.  ( 0  -  (
2  x.  ( t ^ ( 2  -  1 ) ) ) ) ) )
109107, 108dmmpti 5533 . . . 4  |-  dom  (
t  e.  ( -u R (,) R )  |->  ( ( 1  /  (
2  x.  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) ) )  x.  ( 0  -  ( 2  x.  (
t ^ ( 2  -  1 ) ) ) ) ) )  =  ( -u R (,) R )
110106, 109syl6eq 2452 . . 3  |-  ( R  e.  RR+  ->  dom  ( RR  _D  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) ) )  =  ( -u R (,) R ) )
111 dvcn 19760 . . 3  |-  ( ( ( RR  C_  CC  /\  ( t  e.  (
-u R (,) R
)  |->  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) : (
-u R (,) R
) --> CC  /\  ( -u R (,) R ) 
C_  RR )  /\  dom  ( RR  _D  (
t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) ) )  =  (
-u R (,) R
) )  ->  (
t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )  e.  ( (
-u R (,) R
) -cn-> CC ) )
11212, 23, 24, 110, 111syl31anc 1187 . 2  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )  e.  ( ( -u R (,) R ) -cn-> CC ) )
1131, 3, 11, 112cncfmpt2f 18897 1  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )  e.  ( ( -u R (,) R ) -cn-> CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279    C_ wss 3280   {cpr 3775   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   RR+crp 10568   (,)cioo 10872   ^cexp 11337   sqrcsqr 11993   abscabs 11994   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  areacirc  26187
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-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  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  df-log 20407  df-cxp 20408
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