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Theorem areacirclem2 30314
Description: Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
Assertion
Ref Expression
areacirclem2  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  |->  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )  e.  ( ( -u R [,] R ) -cn-> CC ) )
Distinct variable group:    t, R

Proof of Theorem areacirclem2
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 resqcl 12161 . . . . . . . 8  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  RR )
21adantr 463 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( R ^ 2 )  e.  RR )
32adantr 463 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( R ^ 2 )  e.  RR )
4 renegcl 9817 . . . . . . . . . 10  |-  ( R  e.  RR  ->  -u R  e.  RR )
5 iccssre 11549 . . . . . . . . . 10  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( -u R [,] R )  C_  RR )
64, 5mpancom 667 . . . . . . . . 9  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  RR )
76sselda 3434 . . . . . . . 8  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
t  e.  RR )
87resqcld 12261 . . . . . . 7  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
( t ^ 2 )  e.  RR )
98adantlr 712 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( t ^ 2 )  e.  RR )
103, 9resubcld 9927 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  RR )
11 elicc2 11532 . . . . . . . . 9  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
124, 11mpancom 667 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1312adantr 463 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1413ad2ant1 1015 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  RR )
15 resqcl 12161 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  RR )
16153ad2ant3 1017 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  RR )
1714, 16subge0d 10081 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( t ^ 2 )  <_ 
( R ^ 2 ) ) )
18 absresq 13160 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
19183ad2ant3 1017 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
2019breq1d 4394 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( ( abs `  t
) ^ 2 )  <_  ( R ^
2 )  <->  ( t ^ 2 )  <_ 
( R ^ 2 ) ) )
2117, 20bitr4d 256 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( ( abs `  t ) ^
2 )  <_  ( R ^ 2 ) ) )
22 recn 9515 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  t  e.  CC )
2322abscld 13292 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  ( abs `  t )  e.  RR )
24233ad2ant3 1017 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( abs `  t )  e.  RR )
25 simp1 994 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  R  e.  RR )
2622absge0d 13300 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  0  <_  ( abs `  t
) )
27263ad2ant3 1017 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  ( abs `  t
) )
28 simp2 995 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  R )
2924, 25, 27, 28le2sqd 12270 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
)  <_  R  <->  ( ( abs `  t ) ^
2 )  <_  ( R ^ 2 ) ) )
30 simp3 996 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  t  e.  RR )
3130, 25absled 13287 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
)  <_  R  <->  ( -u R  <_  t  /\  t  <_  R ) ) )
3221, 29, 313bitr2d 281 . . . . . . . . . . 11  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( -u R  <_  t  /\  t  <_  R ) ) )
3332biimprd 223 . . . . . . . . . 10  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( -u R  <_  t  /\  t  <_  R )  ->  0  <_  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
34333expa 1194 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  RR )  ->  ( ( -u R  <_  t  /\  t  <_  R )  ->  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
3534exp4b 605 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  RR  ->  ( -u R  <_ 
t  ->  ( t  <_  R  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) ) ) )
36353impd 1208 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R )  ->  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
3713, 36sylbid 215 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
3837imp 427 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )
39 elrege0 11570 . . . . 5  |-  ( ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )  <->  ( ( ( R ^ 2 )  -  ( t ^
2 ) )  e.  RR  /\  0  <_ 
( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
4010, 38, 39sylanbrc 662 . . . 4  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )
)
41 fvres 5805 . . . 4  |-  ( ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )  ->  ( ( sqr  |`  ( 0 [,) +oo ) ) `
 ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  =  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
4240, 41syl 16 . . 3  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( sqr  |`  ( 0 [,) +oo ) ) `  (
( R ^ 2 )  -  ( t ^ 2 ) ) )  =  ( sqr `  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
4342mpteq2dva 4470 . 2  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  |->  ( ( sqr  |`  ( 0 [,) +oo ) ) `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )  =  ( t  e.  ( -u R [,] R )  |->  ( sqr `  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) ) )
44 eqid 2396 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 21398 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
46 ax-resscn 9482 . . . . . . 7  |-  RR  C_  CC
476, 46syl6ss 3446 . . . . . 6  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  CC )
48 resttopon 19771 . . . . . 6  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( -u R [,] R ) 
C_  CC )  -> 
( ( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
4945, 47, 48sylancr 661 . . . . 5  |-  ( R  e.  RR  ->  (
( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
5049adantr 463 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
5147resmptd 5254 . . . . . . 7  |-  ( R  e.  RR  ->  (
( t  e.  CC  |->  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  |`  ( -u R [,] R ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )
5245a1i 11 . . . . . . . . 9  |-  ( R  e.  RR  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
53 recn 9515 . . . . . . . . . . 11  |-  ( R  e.  RR  ->  R  e.  CC )
5453sqcld 12233 . . . . . . . . . 10  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  CC )
5552, 52, 54cnmptc 20271 . . . . . . . . 9  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( R ^ 2 ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
5644sqcn 21486 . . . . . . . . . 10  |-  ( t  e.  CC  |->  ( t ^ 2 ) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen ` fld ) )
5756a1i 11 . . . . . . . . 9  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( t ^ 2 ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
5844subcn 21478 . . . . . . . . . 10  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
5958a1i 11 . . . . . . . . 9  |-  ( R  e.  RR  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
6052, 55, 57, 59cnmpt12f 20275 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
6145toponunii 19541 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
6261cnrest 19895 . . . . . . . 8  |-  ( ( ( t  e.  CC  |->  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen
` fld
) )  /\  ( -u R [,] R ) 
C_  CC )  -> 
( ( t  e.  CC  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  |`  ( -u R [,] R
) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) ) )
6360, 47, 62syl2anc 659 . . . . . . 7  |-  ( R  e.  RR  ->  (
( t  e.  CC  |->  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  |`  ( -u R [,] R ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) ) )
6451, 63eqeltrrd 2485 . . . . . 6  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) ) )
6564adantr 463 . . . . 5  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld )
) )
6645a1i 11 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( TopOpen ` fld )  e.  (TopOn `  CC ) )
67 eqid 2396 . . . . . . . 8  |-  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )
6867rnmpt 5178 . . . . . . 7  |-  ran  (
t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  =  { u  |  E. t  e.  (
-u R [,] R
) u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) }
69 simp3 996 . . . . . . . . . 10  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) )
70403adant3 1014 . . . . . . . . . 10  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )
)
7169, 70eqeltrd 2484 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  e.  ( 0 [,) +oo ) )
7271rexlimdv3a 2890 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) )  ->  u  e.  ( 0 [,) +oo ) ) )
7372abssdv 3505 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  { u  |  E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) }  C_  (
0 [,) +oo )
)
7468, 73syl5eqss 3478 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  ran  ( t  e.  (
-u R [,] R
)  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  C_  ( 0 [,) +oo ) )
75 rge0ssre 11571 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
7675, 46sstri 3443 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
7776a1i 11 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( 0 [,) +oo )  C_  CC )
78 cnrest2 19896 . . . . . 6  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( t  e.  (
-u R [,] R
)  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  C_  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  ( ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  e.  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) )  <->  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  e.  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  ( 0 [,) +oo ) ) ) ) )
7966, 74, 77, 78syl3anc 1226 . . . . 5  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  e.  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) )  <->  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  e.  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  ( 0 [,) +oo ) ) ) ) )
8065, 79mpbid 210 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( (
TopOpen ` fld )t  ( 0 [,) +oo ) ) ) )
81 ssid 3453 . . . . . . . 8  |-  CC  C_  CC
82 cncfss 21511 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( 0 [,) +oo ) -cn-> RR )  C_  (
( 0 [,) +oo ) -cn-> CC ) )
8346, 81, 82mp2an 670 . . . . . . 7  |-  ( ( 0 [,) +oo ) -cn->
RR )  C_  (
( 0 [,) +oo ) -cn-> CC )
84 resqrtcn 23233 . . . . . . 7  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> RR )
8583, 84sselii 3431 . . . . . 6  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> CC )
86 eqid 2396 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  ( 0 [,) +oo ) )  =  ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )
87 eqid 2396 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  CC )  =  ( ( TopOpen ` fld )t  CC )
8844, 86, 87cncfcn 21521 . . . . . . 7  |-  ( ( ( 0 [,) +oo )  C_  CC  /\  CC  C_  CC )  ->  (
( 0 [,) +oo ) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) ) )
8976, 81, 88mp2an 670 . . . . . 6  |-  ( ( 0 [,) +oo ) -cn->
CC )  =  ( ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) )
9085, 89eleqtri 2482 . . . . 5  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) )
9190a1i 11 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( (
TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) ) )
9250, 80, 91cnmpt11f 20273 . . 3  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  |->  ( ( sqr  |`  ( 0 [,) +oo ) ) `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
93 eqid 2396 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( -u R [,] R ) )  =  ( ( TopOpen ` fld )t  ( -u R [,] R ) )
9444, 93, 87cncfcn 21521 . . . . 5  |-  ( ( ( -u R [,] R )  C_  CC  /\  CC  C_  CC )  ->  ( ( -u R [,] R ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
9547, 81, 94sylancl 660 . . . 4  |-  ( R  e.  RR  ->  (
( -u R [,] R
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
9695adantr 463 . . 3  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( -u R [,] R ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
9792, 96eleqtrrd 2487 . 2  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  |->  ( ( sqr  |`  ( 0 [,) +oo ) ) `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )  e.  ( ( -u R [,] R ) -cn-> CC ) )
9843, 97eqeltrrd 2485 1  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  |->  ( sqr `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )  e.  ( ( -u R [,] R ) -cn-> CC ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   {cab 2381   E.wrex 2747    C_ wss 3406   class class class wbr 4384    |-> cmpt 4442   ran crn 4931    |` cres 4932   ` cfv 5513  (class class class)co 6218   CCcc 9423   RRcr 9424   0cc0 9425   +oocpnf 9558    <_ cle 9562    - cmin 9740   -ucneg 9741   2c2 10524   [,)cico 11474   [,]cicc 11475   ^cexp 12092   sqrcsqrt 13091   abscabs 13092   ↾t crest 14851   TopOpenctopn 14852  ℂfldccnfld 18556  TopOnctopon 19503    Cn ccn 19834    tX ctx 20169   -cn->ccncf 21488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-inf2 7994  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503  ax-addf 9504  ax-mulf 9505
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-se 4770  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-of 6461  df-om 6622  df-1st 6721  df-2nd 6722  df-supp 6840  df-recs 6982  df-rdg 7016  df-1o 7070  df-2o 7071  df-oadd 7074  df-er 7251  df-map 7362  df-pm 7363  df-ixp 7411  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-fsupp 7767  df-fi 7808  df-sup 7838  df-oi 7872  df-card 8255  df-cda 8483  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-2 10533  df-3 10534  df-4 10535  df-5 10536  df-6 10537  df-7 10538  df-8 10539  df-9 10540  df-10 10541  df-n0 10735  df-z 10804  df-dec 10918  df-uz 11024  df-q 11124  df-rp 11162  df-xneg 11261  df-xadd 11262  df-xmul 11263  df-ioo 11476  df-ioc 11477  df-ico 11478  df-icc 11479  df-fz 11616  df-fzo 11740  df-fl 11851  df-mod 11920  df-seq 12034  df-exp 12093  df-fac 12279  df-bc 12306  df-hash 12331  df-shft 12925  df-cj 12957  df-re 12958  df-im 12959  df-sqrt 13093  df-abs 13094  df-limsup 13319  df-clim 13336  df-rlim 13337  df-sum 13534  df-ef 13828  df-sin 13830  df-cos 13831  df-tan 13832  df-pi 13833  df-struct 14659  df-ndx 14660  df-slot 14661  df-base 14662  df-sets 14663  df-ress 14664  df-plusg 14738  df-mulr 14739  df-starv 14740  df-sca 14741  df-vsca 14742  df-ip 14743  df-tset 14744  df-ple 14745  df-ds 14747  df-unif 14748  df-hom 14749  df-cco 14750  df-rest 14853  df-topn 14854  df-0g 14872  df-gsum 14873  df-topgen 14874  df-pt 14875  df-prds 14878  df-xrs 14932  df-qtop 14937  df-imas 14938  df-xps 14940  df-mre 15016  df-mrc 15017  df-acs 15019  df-mgm 16012  df-sgrp 16051  df-mnd 16061  df-submnd 16107  df-mulg 16200  df-cntz 16495  df-cmn 16940  df-psmet 18547  df-xmet 18548  df-met 18549  df-bl 18550  df-mopn 18551  df-fbas 18552  df-fg 18553  df-cnfld 18557  df-top 19507  df-bases 19509  df-topon 19510  df-topsp 19511  df-cld 19628  df-ntr 19629  df-cls 19630  df-nei 19708  df-lp 19746  df-perf 19747  df-cn 19837  df-cnp 19838  df-haus 19925  df-cmp 19996  df-tx 20171  df-hmeo 20364  df-fil 20455  df-fm 20547  df-flim 20548  df-flf 20549  df-xms 20931  df-ms 20932  df-tms 20933  df-cncf 21490  df-limc 22378  df-dv 22379  df-log 23052  df-cxp 23053
This theorem is referenced by:  areacirclem3  30315  areacirclem4  30316  areacirc  30318
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