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Theorem areacirclem2 28410
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 11929 . . . . . . . 8  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  RR )
21adantr 462 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( R ^ 2 )  e.  RR )
32adantr 462 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( R ^ 2 )  e.  RR )
4 renegcl 9668 . . . . . . . . . 10  |-  ( R  e.  RR  ->  -u R  e.  RR )
5 iccssre 11373 . . . . . . . . . 10  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( -u R [,] R )  C_  RR )
64, 5mpancom 664 . . . . . . . . 9  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  RR )
76sselda 3353 . . . . . . . 8  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
t  e.  RR )
87resqcld 12030 . . . . . . 7  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
( t ^ 2 )  e.  RR )
98adantlr 709 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( t ^ 2 )  e.  RR )
103, 9resubcld 9772 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  RR )
11 elicc2 11356 . . . . . . . . 9  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
124, 11mpancom 664 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1312adantr 462 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1413ad2ant1 1004 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  RR )
15 resqcl 11929 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  RR )
16153ad2ant3 1006 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  RR )
1714, 16subge0d 9925 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( t ^ 2 )  <_ 
( R ^ 2 ) ) )
18 absresq 12787 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
19183ad2ant3 1006 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
2019breq1d 4299 . . . . . . . . . . . . 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 9368 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  t  e.  CC )
2322abscld 12918 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  ( abs `  t )  e.  RR )
24233ad2ant3 1006 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( abs `  t )  e.  RR )
25 simp1 983 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  R  e.  RR )
2622absge0d 12926 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  0  <_  ( abs `  t
) )
27263ad2ant3 1006 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  ( abs `  t
) )
28 simp2 984 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  R )
2924, 25, 27, 28le2sqd 12039 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
)  <_  R  <->  ( ( abs `  t ) ^
2 )  <_  ( R ^ 2 ) ) )
30 simp3 985 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  t  e.  RR )
3130, 25absled 12913 . . . . . . . . . . . 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 1182 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  RR )  ->  ( ( -u R  <_  t  /\  t  <_  R )  ->  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
3534exp4b 604 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  RR  ->  ( -u R  <_ 
t  ->  ( t  <_  R  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) ) ) )
36353impd 1196 . . . . . . 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 11388 . . . . 5  |-  ( ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )  <->  ( ( ( R ^ 2 )  -  ( t ^
2 ) )  e.  RR  /\  0  <_ 
( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
4010, 38, 39sylanbrc 659 . . . 4  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) +oo )
)
41 fvres 5701 . . . 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 4375 . 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 2441 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 20321 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
46 ax-resscn 9335 . . . . . . 7  |-  RR  C_  CC
476, 46syl6ss 3365 . . . . . 6  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  CC )
48 resttopon 18724 . . . . . 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 658 . . . . 5  |-  ( R  e.  RR  ->  (
( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
5049adantr 462 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
51 resmpt 5153 . . . . . . . 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 9368 . . . . . . . . . . 11  |-  ( R  e.  RR  ->  R  e.  CC )
5554sqcld 12002 . . . . . . . . . 10  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  CC )
5653, 53, 55cnmptc 19194 . . . . . . . . 9  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( R ^ 2 ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
5744sqcn 20409 . . . . . . . . . 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 20401 . . . . . . . . . 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 19198 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
6245toponunii 18496 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
6362cnrest 18848 . . . . . . . 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 656 . . . . . . 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 2516 . . . . . 6  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) ) )
6665adantr 462 . . . . 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 2441 . . . . . . . 8  |-  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )
6968rnmpt 5081 . . . . . . 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 985 . . . . . . . . . 10  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) )
71403adant3 1003 . . . . . . . . . 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 2515 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  e.  ( 0 [,) +oo ) )
7372rexlimdv3a 2841 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) )  ->  u  e.  ( 0 [,) +oo ) ) )
7473abssdv 3423 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  { u  |  E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) }  C_  (
0 [,) +oo )
)
7569, 74syl5eqss 3397 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  ran  ( t  e.  (
-u R [,] R
)  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  C_  ( 0 [,) +oo ) )
76 0re 9382 . . . . . . . . 9  |-  0  e.  RR
77 pnfxr 11088 . . . . . . . . 9  |- +oo  e.  RR*
78 icossre 11372 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
7976, 77, 78mp2an 667 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
8079, 46sstri 3362 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
8180a1i 11 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( 0 [,) +oo )  C_  CC )
82 cnrest2 18849 . . . . . 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 1213 . . . . 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 3372 . . . . . . . 8  |-  CC  C_  CC
86 cncfss 20434 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( 0 [,) +oo ) -cn-> RR )  C_  (
( 0 [,) +oo ) -cn-> CC ) )
8746, 85, 86mp2an 667 . . . . . . 7  |-  ( ( 0 [,) +oo ) -cn->
RR )  C_  (
( 0 [,) +oo ) -cn-> CC )
88 resqrcn 22146 . . . . . . 7  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> RR )
8987, 88sselii 3350 . . . . . 6  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> CC )
90 eqid 2441 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  ( 0 [,) +oo ) )  =  ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )
91 eqid 2441 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  CC )  =  ( ( TopOpen ` fld )t  CC )
9244, 90, 91cncfcn 20444 . . . . . . 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 667 . . . . . 6  |-  ( ( 0 [,) +oo ) -cn->
CC )  =  ( ( ( TopOpen ` fld )t  ( 0 [,) +oo ) )  Cn  (
( TopOpen ` fld )t  CC ) )
9489, 93eleqtri 2513 . . . . 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 19196 . . 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 2441 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( -u R [,] R ) )  =  ( ( TopOpen ` fld )t  ( -u R [,] R ) )
9844, 97, 91cncfcn 20444 . . . . 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 657 . . . 4  |-  ( R  e.  RR  ->  (
( -u R [,] R
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
10099adantr 462 . . 3  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( -u R [,] R ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
10196, 100eleqtrrd 2518 . 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 2516 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 960    = wceq 1364    e. wcel 1761   {cab 2427   E.wrex 2714    C_ wss 3325   class class class wbr 4289    e. cmpt 4347   ran crn 4837    |` cres 4838   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   +oocpnf 9411   RR*cxr 9413    <_ cle 9415    - cmin 9591   -ucneg 9592   2c2 10367   [,)cico 11298   [,]cicc 11299   ^cexp 11861   sqrcsqr 12718   abscabs 12719   ↾t crest 14355   TopOpenctopn 14356  ℂfldccnfld 17777  TopOnctopon 18458    Cn ccn 18787    tX ctx 19092   -cn->ccncf 20411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-tan 13353  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-cmp 18949  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968
This theorem is referenced by:  areacirclem3  28411  areacirclem4  28412  areacirc  28414
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