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Theorem areacirclem4 24927
Description: Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.)
Assertion
Ref Expression
areacirclem4  |-  ( ( 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 areacirclem4
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 resqcl 11171 . . . . . . . 8  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  RR )
21adantr 451 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( R ^ 2 )  e.  RR )
32adantr 451 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( R ^ 2 )  e.  RR )
4 renegcl 9110 . . . . . . . . . 10  |-  ( R  e.  RR  ->  -u R  e.  RR )
5 iccssre 10731 . . . . . . . . . 10  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( -u R [,] R )  C_  RR )
64, 5mpancom 650 . . . . . . . . 9  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  RR )
76sselda 3180 . . . . . . . 8  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
t  e.  RR )
87resqcld 11271 . . . . . . 7  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
( t ^ 2 )  e.  RR )
98adantlr 695 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( t ^ 2 )  e.  RR )
103, 9resubcld 9211 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  RR )
11 elicc2 10715 . . . . . . . . 9  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
124, 11mpancom 650 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1312adantr 451 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1413ad2ant1 976 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  RR )
15 resqcl 11171 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  RR )
16153ad2ant3 978 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  RR )
1714, 16subge0d 9362 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( t ^ 2 )  <_ 
( R ^ 2 ) ) )
18 absresq 11787 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
19183ad2ant3 978 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
2019breq1d 4033 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( ( abs `  t
) ^ 2 )  <_  ( R ^
2 )  <->  ( t ^ 2 )  <_ 
( R ^ 2 ) ) )
2117, 20bitr4d 247 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( ( abs `  t ) ^
2 )  <_  ( R ^ 2 ) ) )
22 recn 8827 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  t  e.  CC )
2322abscld 11918 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  ( abs `  t )  e.  RR )
24233ad2ant3 978 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( abs `  t )  e.  RR )
25 simp1 955 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  R  e.  RR )
2622absge0d 11926 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  0  <_  ( abs `  t
) )
27263ad2ant3 978 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  ( abs `  t
) )
28 simp2 956 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  R )
2924, 25, 27, 28le2sqd 11280 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
)  <_  R  <->  ( ( abs `  t ) ^
2 )  <_  ( R ^ 2 ) ) )
30 simp3 957 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  t  e.  RR )
3130, 25absled 11913 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
)  <_  R  <->  ( -u R  <_  t  /\  t  <_  R ) ) )
3221, 29, 313bitr2d 272 . . . . . . . . . . 11  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( -u R  <_  t  /\  t  <_  R ) ) )
3332biimprd 214 . . . . . . . . . 10  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( -u R  <_  t  /\  t  <_  R )  ->  0  <_  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
34333expa 1151 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  RR )  ->  ( ( -u R  <_  t  /\  t  <_  R )  ->  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
3534exp4b 590 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  RR  ->  ( -u R  <_ 
t  ->  ( t  <_  R  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) ) ) )
36353impd 1165 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R )  ->  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
3713, 36sylbid 206 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
3837imp 418 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )
39 elrege0 10746 . . . . 5  |-  ( ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) 
+oo )  <->  ( (
( R ^ 2 )  -  ( t ^ 2 ) )  e.  RR  /\  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
4010, 38, 39sylanbrc 645 . . . 4  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,)  +oo )
)
41 fvres 5542 . . . 4  |-  ( ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) 
+oo )  ->  (
( sqr  |`  ( 0 [,)  +oo ) ) `  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  =  ( sqr `  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )
4240, 41syl 15 . . 3  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( sqr  |`  ( 0 [,) 
+oo ) ) `  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  =  ( sqr `  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )
4342mpteq2dva 4106 . 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 2283 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 18292 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
46 ax-resscn 8794 . . . . . . 7  |-  RR  C_  CC
476, 46syl6ss 3191 . . . . . 6  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  CC )
48 resttopon 16892 . . . . . 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 644 . . . . 5  |-  ( R  e.  RR  ->  (
( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
5049adantr 451 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
51 resmpt 5000 . . . . . . . 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 15 . . . . . . 7  |-  ( R  e.  RR  ->  (
( t  e.  CC  |->  ( ( R ^
2 )  -  (
t ^ 2 ) ) )  |`  ( -u R [,] R ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) ) )
5345a1i 10 . . . . . . . . 9  |-  ( R  e.  RR  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
54 recn 8827 . . . . . . . . . . 11  |-  ( R  e.  RR  ->  R  e.  CC )
5554sqcld 11243 . . . . . . . . . 10  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  CC )
5653, 53, 55cnmptc 17356 . . . . . . . . 9  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( R ^ 2 ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
5744sqcn 18378 . . . . . . . . . 10  |-  ( t  e.  CC  |->  ( t ^ 2 ) )  e.  ( ( TopOpen ` fld )  Cn  ( TopOpen ` fld ) )
5857a1i 10 . . . . . . . . 9  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( t ^ 2 ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
5944subcn 18370 . . . . . . . . . 10  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
6059a1i 10 . . . . . . . . 9  |-  ( R  e.  RR  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
6153, 56, 58, 60cnmpt12f 17360 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
6245toponunii 16670 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
6362cnrest 17013 . . . . . . . 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 642 . . . . . . 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 2358 . . . . . 6  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) ) )
6665adantr 451 . . . . 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 10 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( TopOpen ` fld )  e.  (TopOn `  CC ) )
68 eqid 2283 . . . . . . . 8  |-  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )
6968rnmpt 4925 . . . . . . 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 957 . . . . . . . . . 10  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) )
71403adant3 975 . . . . . . . . . 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 2357 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  e.  ( 0 [,)  +oo ) )
7372rexlimdv3a 2669 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) )  ->  u  e.  ( 0 [,)  +oo ) ) )
7473abssdv 3247 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  { u  |  E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) }  C_  (
0 [,)  +oo ) )
7569, 74syl5eqss 3222 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  ran  ( t  e.  (
-u R [,] R
)  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  C_  ( 0 [,)  +oo ) )
76 0re 8838 . . . . . . . . 9  |-  0  e.  RR
77 pnfxr 10455 . . . . . . . . 9  |-  +oo  e.  RR*
78 icossre 10730 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
7976, 77, 78mp2an 653 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  RR
8079, 46sstri 3188 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  CC
8180a1i 10 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( 0 [,)  +oo )  C_  CC )
82 cnrest2 17014 . . . . . 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 1182 . . . . 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 201 . . . 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 3197 . . . . . . . 8  |-  CC  C_  CC
86 cncfss 18403 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( 0 [,)  +oo ) -cn-> RR )  C_  (
( 0 [,)  +oo ) -cn-> CC ) )
8746, 85, 86mp2an 653 . . . . . . 7  |-  ( ( 0 [,)  +oo ) -cn->
RR )  C_  (
( 0 [,)  +oo ) -cn-> CC )
88 resqrcn 20089 . . . . . . 7  |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
8987, 88sselii 3177 . . . . . 6  |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> CC )
90 eqid 2283 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  ( 0 [,)  +oo ) )  =  ( ( TopOpen ` fld )t  ( 0 [,) 
+oo ) )
91 eqid 2283 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  CC )  =  ( ( TopOpen ` fld )t  CC )
9244, 90, 91cncfcn 18413 . . . . . . 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 653 . . . . . 6  |-  ( ( 0 [,)  +oo ) -cn->
CC )  =  ( ( ( TopOpen ` fld )t  ( 0 [,) 
+oo ) )  Cn  ( ( TopOpen ` fld )t  CC ) )
9489, 93eleqtri 2355 . . . . 5  |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( ( TopOpen ` fld )t  ( 0 [,) 
+oo ) )  Cn  ( ( TopOpen ` fld )t  CC ) )
9594a1i 10 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( ( TopOpen ` fld )t  (
0 [,)  +oo ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
9650, 84, 95cnmpt11f 17358 . . 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 2283 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( -u R [,] R ) )  =  ( ( TopOpen ` fld )t  ( -u R [,] R ) )
9844, 97, 91cncfcn 18413 . . . . 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 643 . . . 4  |-  ( R  e.  RR  ->  (
( -u R [,] R
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
10099adantr 451 . . 3  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( -u R [,] R ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
10196, 100eleqtrrd 2360 . 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 2358 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    +oocpnf 8864   RR*cxr 8866    <_ cle 8868    - cmin 9037   -ucneg 9038   2c2 9795   [,)cico 10658   [,]cicc 10659   ^cexp 11104   sqrcsqr 11718   abscabs 11719   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255   -cn->ccncf 18380
This theorem is referenced by:  areacirclem1  24928  areacirclem5  24929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-tan 12353  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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