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Theorem areacirclem2 29685
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 12199 . . . . . . . 8  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  RR )
21adantr 465 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( R ^ 2 )  e.  RR )
32adantr 465 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( R ^ 2 )  e.  RR )
4 renegcl 9878 . . . . . . . . . 10  |-  ( R  e.  RR  ->  -u R  e.  RR )
5 iccssre 11602 . . . . . . . . . 10  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( -u R [,] R )  C_  RR )
64, 5mpancom 669 . . . . . . . . 9  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  RR )
76sselda 3504 . . . . . . . 8  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
t  e.  RR )
87resqcld 12300 . . . . . . 7  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
( t ^ 2 )  e.  RR )
98adantlr 714 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( t ^ 2 )  e.  RR )
103, 9resubcld 9983 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  RR )
11 elicc2 11585 . . . . . . . . 9  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
124, 11mpancom 669 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1312adantr 465 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1413ad2ant1 1017 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  RR )
15 resqcl 12199 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  RR )
16153ad2ant3 1019 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  RR )
1714, 16subge0d 10138 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( t ^ 2 )  <_ 
( R ^ 2 ) ) )
18 absresq 13094 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
19183ad2ant3 1019 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
2019breq1d 4457 . . . . . . . . . . . . 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 9578 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  t  e.  CC )
2322abscld 13226 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  ( abs `  t )  e.  RR )
24233ad2ant3 1019 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( abs `  t )  e.  RR )
25 simp1 996 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  R  e.  RR )
2622absge0d 13234 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  0  <_  ( abs `  t
) )
27263ad2ant3 1019 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  ( abs `  t
) )
28 simp2 997 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  R )
2924, 25, 27, 28le2sqd 12309 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
)  <_  R  <->  ( ( abs `  t ) ^
2 )  <_  ( R ^ 2 ) ) )
30 simp3 998 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  t  e.  RR )
3130, 25absled 13221 . . . . . . . . . . . 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 1196 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  RR )  ->  ( ( -u R  <_  t  /\  t  <_  R )  ->  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
3534exp4b 607 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  RR  ->  ( -u R  <_ 
t  ->  ( t  <_  R  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) ) ) )
36353impd 1210 . . . . . . 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 429 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )
39 elrege0 11623 . . . . 5  |-  ( ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )  <->  ( ( ( R ^ 2 )  -  ( t ^
2 ) )  e.  RR  /\  0  <_ 
( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
4010, 38, 39sylanbrc 664 . . . 4  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )
)
41 fvres 5878 . . . 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 4533 . 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 2467 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 21025 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
46 ax-resscn 9545 . . . . . . 7  |-  RR  C_  CC
476, 46syl6ss 3516 . . . . . 6  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  CC )
48 resttopon 19428 . . . . . 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 663 . . . . 5  |-  ( R  e.  RR  ->  (
( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
5049adantr 465 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
51 resmpt 5321 . . . . . . . 8  |-  ( (
-u R [,] R
)  C_  CC  ->  ( ( t  e.  CC  |->  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  |`  ( -u R [,] R ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )
5247, 51syl 16 . . . . . . 7  |-  ( R  e.  RR  ->  (
( t  e.  CC  |->  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  |`  ( -u R [,] R ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )
5345a1i 11 . . . . . . . . 9  |-  ( R  e.  RR  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
54 recn 9578 . . . . . . . . . . 11  |-  ( R  e.  RR  ->  R  e.  CC )
5554sqcld 12272 . . . . . . . . . 10  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  CC )
5653, 53, 55cnmptc 19898 . . . . . . . . 9  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( R ^ 2 ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
5744sqcn 21113 . . . . . . . . . 10  |-  ( t  e.  CC  |->  ( t ^ 2 ) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen ` fld ) )
5857a1i 11 . . . . . . . . 9  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( t ^ 2 ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
5944subcn 21105 . . . . . . . . . 10  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
6059a1i 11 . . . . . . . . 9  |-  ( R  e.  RR  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
6153, 56, 58, 60cnmpt12f 19902 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
6245toponunii 19200 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
6362cnrest 19552 . . . . . . . 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 ) ) )
6461, 47, 63syl2anc 661 . . . . . . 7  |-  ( R  e.  RR  ->  (
( t  e.  CC  |->  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  |`  ( -u R [,] R ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) ) )
6552, 64eqeltrrd 2556 . . . . . 6  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) ) )
6665adantr 465 . . . . 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 )
) )
6745a1i 11 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( TopOpen ` fld )  e.  (TopOn `  CC ) )
68 eqid 2467 . . . . . . . 8  |-  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )
6968rnmpt 5246 . . . . . . 7  |-  ran  (
t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  =  { u  |  E. t  e.  (
-u R [,] R
) u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) }
70 simp3 998 . . . . . . . . . 10  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) )
71403adant3 1016 . . . . . . . . . 10  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )
)
7270, 71eqeltrd 2555 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  e.  ( 0 [,) +oo ) )
7372rexlimdv3a 2957 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) )  ->  u  e.  ( 0 [,) +oo ) ) )
7473abssdv 3574 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  { u  |  E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) }  C_  (
0 [,) +oo )
)
7569, 74syl5eqss 3548 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  ran  ( t  e.  (
-u R [,] R
)  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  C_  ( 0 [,) +oo ) )
76 0re 9592 . . . . . . . . 9  |-  0  e.  RR
77 pnfxr 11317 . . . . . . . . 9  |- +oo  e.  RR*
78 icossre 11601 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
7976, 77, 78mp2an 672 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
8079, 46sstri 3513 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
8180a1i 11 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( 0 [,) +oo )  C_  CC )
82 cnrest2 19553 . . . . . 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 ) ) ) ) )
8367, 75, 81, 82syl3anc 1228 . . . . 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 ) ) ) ) )
8466, 83mpbid 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 ) ) ) )
85 ssid 3523 . . . . . . . 8  |-  CC  C_  CC
86 cncfss 21138 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( 0 [,) +oo ) -cn-> RR )  C_  (
( 0 [,) +oo ) -cn-> CC ) )
8746, 85, 86mp2an 672 . . . . . . 7  |-  ( ( 0 [,) +oo ) -cn->
RR )  C_  (
( 0 [,) +oo ) -cn-> CC )
88 resqrtcn 22851 . . . . . . 7  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> RR )
8987, 88sselii 3501 . . . . . 6  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> CC )
90 eqid 2467 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  ( 0 [,) +oo ) )  =  ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )
91 eqid 2467 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  CC )  =  ( ( TopOpen ` fld )t  CC )
9244, 90, 91cncfcn 21148 . . . . . . 7  |-  ( ( ( 0 [,) +oo )  C_  CC  /\  CC  C_  CC )  ->  (
( 0 [,) +oo ) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) ) )
9380, 85, 92mp2an 672 . . . . . 6  |-  ( ( 0 [,) +oo ) -cn->
CC )  =  ( ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) )
9489, 93eleqtri 2553 . . . . 5  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) )
9594a1i 11 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( (
TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) ) )
9650, 84, 95cnmpt11f 19900 . . 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 ) ) )
97 eqid 2467 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( -u R [,] R ) )  =  ( ( TopOpen ` fld )t  ( -u R [,] R ) )
9844, 97, 91cncfcn 21148 . . . . 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 ) ) )
9947, 85, 98sylancl 662 . . . 4  |-  ( R  e.  RR  ->  (
( -u R [,] R
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
10099adantr 465 . . 3  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( -u R [,] R ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
10196, 100eleqtrrd 2558 . 2  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  |->  ( ( sqr  |`  ( 0 [,) +oo ) ) `  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )  e.  ( ( -u R [,] R ) -cn-> CC ) )
10243, 101eqeltrrd 2556 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   ran crn 5000    |` cres 5001   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   +oocpnf 9621   RR*cxr 9623    <_ cle 9625    - cmin 9801   -ucneg 9802   2c2 10581   [,)cico 11527   [,]cicc 11528   ^cexp 12130   sqrcsqrt 13025   abscabs 13026   ↾t crest 14672   TopOpenctopn 14673  ℂfldccnfld 18191  TopOnctopon 19162    Cn ccn 19491    tX ctx 19796   -cn->ccncf 21115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-tan 13665  df-pi 13666  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-cmp 19653  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006  df-log 22672  df-cxp 22673
This theorem is referenced by:  areacirclem3  29686  areacirclem4  29687  areacirc  29689
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