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Theorem areacirclem3 25518
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 2358 . 2  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21mulcn 18474 . . 3  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
32a1i 10 . 2  |-  ( R  e.  RR+  ->  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
4 2cn 9906 . . . 4  |-  2  e.  CC
5 ioossre 10804 . . . . 5  |-  ( -u R (,) R )  C_  RR
6 ax-resscn 8884 . . . . 5  |-  RR  C_  CC
75, 6sstri 3264 . . . 4  |-  ( -u R (,) R )  C_  CC
8 ssid 3273 . . . 4  |-  CC  C_  CC
9 cncfmptc 18518 . . . 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 1277 . . 3  |-  ( t  e.  ( -u R (,) R )  |->  2 )  e.  ( ( -u R (,) R ) -cn-> CC )
1110a1i 10 . 2  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  |->  2 )  e.  ( ( -u R (,) R ) -cn-> CC ) )
126a1i 10 . . 3  |-  ( R  e.  RR+  ->  RR  C_  CC )
13 rpcn 10454 . . . . . . . 8  |-  ( R  e.  RR+  ->  R  e.  CC )
1413sqcld 11336 . . . . . . 7  |-  ( R  e.  RR+  ->  ( R ^ 2 )  e.  CC )
1514adantr 451 . . . . . 6  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( R ^ 2 )  e.  CC )
16 elioore 10778 . . . . . . . . 9  |-  ( t  e.  ( -u R (,) R )  ->  t  e.  RR )
1716recnd 8951 . . . . . . . 8  |-  ( t  e.  ( -u R (,) R )  ->  t  e.  CC )
1817sqcld 11336 . . . . . . 7  |-  ( t  e.  ( -u R (,) R )  ->  (
t ^ 2 )  e.  CC )
1918adantl 452 . . . . . 6  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( t ^ 2 )  e.  CC )
2015, 19subcld 9247 . . . . 5  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( ( R ^
2 )  -  (
t ^ 2 ) )  e.  CC )
2120sqrcld 12015 . . . 4  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  CC )
22 eqid 2358 . . . 4  |-  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )  =  ( t  e.  (
-u R (,) R
)  |->  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
2321, 22fmptd 5767 . . 3  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) ) : ( -u R (,) R ) --> CC )
245a1i 10 . . 3  |-  ( R  e.  RR+  ->  ( -u R (,) R )  C_  RR )
25 reex 8918 . . . . . . . 8  |-  RR  e.  _V
2625prid1 3810 . . . . . . 7  |-  RR  e.  { RR ,  CC }
2726a1i 10 . . . . . 6  |-  ( R  e.  RR+  ->  RR  e.  { RR ,  CC }
)
28 rpre 10452 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  e.  RR )
2928resqcld 11364 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( R ^ 2 )  e.  RR )
3029adantr 451 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( R ^ 2 )  e.  RR )
3116resqcld 11364 . . . . . . . . 9  |-  ( t  e.  ( -u R (,) R )  ->  (
t ^ 2 )  e.  RR )
3231adantl 452 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( t ^ 2 )  e.  RR )
3330, 32resubcld 9301 . . . . . . 7  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( ( R ^
2 )  -  (
t ^ 2 ) )  e.  RR )
3428renegcld 9300 . . . . . . . . . . 11  |-  ( R  e.  RR+  ->  -u R  e.  RR )
3534rexrd 8971 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  -u R  e.  RR* )
36 rpxr 10453 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  R  e. 
RR* )
37 elioo2 10789 . . . . . . . . . 10  |-  ( (
-u R  e.  RR*  /\  R  e.  RR* )  ->  ( t  e.  (
-u R (,) R
)  <->  ( t  e.  RR  /\  -u R  <  t  /\  t  < 
R ) ) )
3835, 36, 37syl2anc 642 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  <->  ( t  e.  RR  /\  -u R  <  t  /\  t  < 
R ) ) )
39 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  t  e.  RR )
4028adantr 451 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  R  e.  RR )
4139, 40absltd 12008 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( abs `  t
)  <  R  <->  ( -u R  <  t  /\  t  < 
R ) ) )
42 recn 8917 . . . . . . . . . . . . . . . . 17  |-  ( t  e.  RR  ->  t  e.  CC )
4342abscld 12014 . . . . . . . . . . . . . . . 16  |-  ( t  e.  RR  ->  ( abs `  t )  e.  RR )
4443adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  ( abs `  t )  e.  RR )
4542absge0d 12022 . . . . . . . . . . . . . . . 16  |-  ( t  e.  RR  ->  0  <_  ( abs `  t
) )
4645adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  0  <_  ( abs `  t
) )
47 rpge0 10458 . . . . . . . . . . . . . . . 16  |-  ( R  e.  RR+  ->  0  <_  R )
4847adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  0  <_  R )
4944, 40, 46, 48lt2sqd 11372 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( abs `  t
)  <  R  <->  ( ( abs `  t ) ^
2 )  <  ( R ^ 2 ) ) )
50 absresq 11883 . . . . . . . . . . . . . . . 16  |-  ( t  e.  RR  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
5150adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
5251breq1d 4114 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( ( abs `  t
) ^ 2 )  <  ( R ^
2 )  <->  ( t ^ 2 )  < 
( R ^ 2 ) ) )
53 resqcl 11264 . . . . . . . . . . . . . . . 16  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  RR )
5453adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  RR )
5529adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  RR )
5654, 55posdifd 9449 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( t ^ 2 )  <  ( R ^ 2 )  <->  0  <  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
5749, 52, 563bitrd 270 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( abs `  t
)  <  R  <->  0  <  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
5841, 57bitr3d 246 . . . . . . . . . . . 12  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( -u R  <  t  /\  t  <  R )  <->  0  <  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )
5958biimpd 198 . . . . . . . . . . 11  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( -u R  <  t  /\  t  <  R )  ->  0  <  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
6059exp4b 590 . . . . . . . . . 10  |-  ( R  e.  RR+  ->  ( t  e.  RR  ->  ( -u R  <  t  -> 
( t  <  R  ->  0  <  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) ) ) )
61603impd 1165 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( ( t  e.  RR  /\  -u R  <  t  /\  t  <  R )  -> 
0  <  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
6238, 61sylbid 206 . . . . . . . 8  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  ->  0  <  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
6362imp 418 . . . . . . 7  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
0  <  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )
6433, 63elrpd 10480 . . . . . 6  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( ( R ^
2 )  -  (
t ^ 2 ) )  e.  RR+ )
65 ovex 5970 . . . . . . 7  |-  ( 0  -  ( 2  x.  ( t ^ (
2  -  1 ) ) ) )  e. 
_V
6665a1i 10 . . . . . 6  |-  ( ( R  e.  RR+  /\  t  e.  ( -u R (,) R ) )  -> 
( 0  -  (
2  x.  ( t ^ ( 2  -  1 ) ) ) )  e.  _V )
67 rpcn 10454 . . . . . . . 8  |-  ( u  e.  RR+  ->  u  e.  CC )
6867sqrcld 12015 . . . . . . 7  |-  ( u  e.  RR+  ->  ( sqr `  u )  e.  CC )
6968adantl 452 . . . . . 6  |-  ( ( R  e.  RR+  /\  u  e.  RR+ )  ->  ( sqr `  u )  e.  CC )
70 ovex 5970 . . . . . . 7  |-  ( 1  /  ( 2  x.  ( sqr `  u
) ) )  e. 
_V
7170a1i 10 . . . . . 6  |-  ( ( R  e.  RR+  /\  u  e.  RR+ )  ->  (
1  /  ( 2  x.  ( sqr `  u
) ) )  e. 
_V )
7214adantr 451 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  CC )
7342sqcld 11336 . . . . . . . . 9  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  CC )
7473adantl 452 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  CC )
7572, 74subcld 9247 . . . . . . 7  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
( R ^ 2 )  -  ( t ^ 2 ) )  e.  CC )
7665a1i 10 . . . . . . 7  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
0  -  ( 2  x.  ( t ^
( 2  -  1 ) ) ) )  e.  _V )
77 0re 8928 . . . . . . . . 9  |-  0  e.  RR
7877a1i 10 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  0  e.  RR )
7927, 14dvmptc 19411 . . . . . . . 8  |-  ( R  e.  RR+  ->  ( RR 
_D  ( t  e.  RR  |->  ( R ^
2 ) ) )  =  ( t  e.  RR  |->  0 ) )
80 ovex 5970 . . . . . . . . 9  |-  ( 2  x.  ( t ^
( 2  -  1 ) ) )  e. 
_V
8180a1i 10 . . . . . . . 8  |-  ( ( R  e.  RR+  /\  t  e.  RR )  ->  (
2  x.  ( t ^ ( 2  -  1 ) ) )  e.  _V )
821cnfldtopon 18394 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
83 toponmax 16772 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
8482, 83mp1i 11 . . . . . . . . 9  |-  ( R  e.  RR+  ->  CC  e.  ( TopOpen ` fld ) )
85 df-ss 3242 . . . . . . . . . . 11  |-  ( RR  C_  CC  <->  ( RR  i^i  CC )  =  RR )
866, 85mpbi 199 . . . . . . . . . 10  |-  ( RR 
i^i  CC )  =  RR
8786a1i 10 . . . . . . . . 9  |-  ( R  e.  RR+  ->  ( RR 
i^i  CC )  =  RR )
88 sqcl 11259 . . . . . . . . . 10  |-  ( t  e.  CC  ->  (
t ^ 2 )  e.  CC )
8988adantl 452 . . . . . . . . 9  |-  ( ( R  e.  RR+  /\  t  e.  CC )  ->  (
t ^ 2 )  e.  CC )
9080a1i 10 . . . . . . . . 9  |-  ( ( R  e.  RR+  /\  t  e.  CC )  ->  (
2  x.  ( t ^ ( 2  -  1 ) ) )  e.  _V )
91 2nn 9969 . . . . . . . . . 10  |-  2  e.  NN
92 dvexp 19406 . . . . . . . . . 10  |-  ( 2  e.  NN  ->  ( CC  _D  ( t  e.  CC  |->  ( t ^
2 ) ) )  =  ( t  e.  CC  |->  ( 2  x.  ( t ^ (
2  -  1 ) ) ) ) )
9391, 92mp1i 11 . . . . . . . . 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 19409 . . . . . . . 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 19420 . . . . . . 7  |-  ( R  e.  RR+  ->  ( RR 
_D  ( t  e.  RR  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )  =  ( t  e.  RR  |->  ( 0  -  ( 2  x.  (
t ^ ( 2  -  1 ) ) ) ) ) )
961tgioo2 18411 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
97 iooretop 18377 . . . . . . . 8  |-  ( -u R (,) R )  e.  ( topGen `  ran  (,) )
9897a1i 10 . . . . . . 7  |-  ( R  e.  RR+  ->  ( -u R (,) R )  e.  ( topGen `  ran  (,) )
)
9927, 75, 76, 95, 24, 96, 1, 98dvmptres 19416 . . . . . 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 20195 . . . . . . 7  |-  ( RR 
_D  ( u  e.  RR+  |->  ( sqr `  u
) ) )  =  ( u  e.  RR+  |->  ( 1  /  (
2  x.  ( sqr `  u ) ) ) )
101100a1i 10 . . . . . 6  |-  ( R  e.  RR+  ->  ( RR 
_D  ( u  e.  RR+  |->  ( sqr `  u
) ) )  =  ( u  e.  RR+  |->  ( 1  /  (
2  x.  ( sqr `  u ) ) ) ) )
102 fveq2 5608 . . . . . 6  |-  ( u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) )  ->  ( sqr `  u )  =  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
103102oveq2d 5961 . . . . . . 7  |-  ( u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) )  ->  (
2  x.  ( sqr `  u ) )  =  ( 2  x.  ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )
104103oveq2d 5961 . . . . . 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 19425 . . . . 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 4963 . . . 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 5970 . . . . 5  |-  ( ( 1  /  ( 2  x.  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) ) )  x.  ( 0  -  (
2  x.  ( t ^ ( 2  -  1 ) ) ) ) )  e.  _V
108 eqid 2358 . . . . 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 5455 . . . 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 2406 . . 3  |-  ( R  e.  RR+  ->  dom  ( RR  _D  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) ) )  =  ( -u R (,) R ) )
111 dvcn 19374 . . 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 1185 . 2  |-  ( R  e.  RR+  ->  ( t  e.  ( -u R (,) R )  |->  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )  e.  ( ( -u R (,) R ) -cn-> CC ) )
1131, 3, 11, 112cncfmpt2f 18521 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   _Vcvv 2864    i^i cin 3227    C_ wss 3228   {cpr 3717   class class class wbr 4104    e. cmpt 4158   dom cdm 4771   ran crn 4772   -->wf 5333   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832   RR*cxr 8956    < clt 8957    <_ cle 8958    - cmin 9127   -ucneg 9128    / cdiv 9513   NNcn 9836   2c2 9885   RR+crp 10446   (,)cioo 10748   ^cexp 11197   sqrcsqr 11814   abscabs 11815   TopOpenctopn 13425   topGenctg 13441  ℂfldccnfld 16482  TopOnctopon 16738    Cn ccn 17060    tX ctx 17361   -cn->ccncf 18483    _D cdv 19317
This theorem is referenced by:  areacirc  25523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ioc 10753  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-sum 12256  df-ef 12446  df-sin 12448  df-cos 12449  df-pi 12451  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cn 17063  df-cnp 17064  df-haus 17149  df-cmp 17220  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cncf 18485  df-limc 19320  df-dv 19321  df-log 20021  df-cxp 20022
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