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Theorem areacirclem4 26183
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 11404 . . . . . . . 8  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  RR )
21adantr 452 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( R ^ 2 )  e.  RR )
32adantr 452 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( R ^ 2 )  e.  RR )
4 renegcl 9320 . . . . . . . . . 10  |-  ( R  e.  RR  ->  -u R  e.  RR )
5 iccssre 10948 . . . . . . . . . 10  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( -u R [,] R )  C_  RR )
64, 5mpancom 651 . . . . . . . . 9  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  RR )
76sselda 3308 . . . . . . . 8  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
t  e.  RR )
87resqcld 11504 . . . . . . 7  |-  ( ( R  e.  RR  /\  t  e.  ( -u R [,] R ) )  -> 
( t ^ 2 )  e.  RR )
98adantlr 696 . . . . . 6  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( t ^ 2 )  e.  RR )
103, 9resubcld 9421 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  RR )
11 elicc2 10931 . . . . . . . . 9  |-  ( (
-u R  e.  RR  /\  R  e.  RR )  ->  ( t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
124, 11mpancom 651 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1312adantr 452 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  <->  ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R ) ) )
1413ad2ant1 978 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( R ^ 2 )  e.  RR )
15 resqcl 11404 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
t ^ 2 )  e.  RR )
16153ad2ant3 980 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
t ^ 2 )  e.  RR )
1714, 16subge0d 9572 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( t ^ 2 )  <_ 
( R ^ 2 ) ) )
18 absresq 12062 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
19183ad2ant3 980 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
) ^ 2 )  =  ( t ^
2 ) )
2019breq1d 4182 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( ( abs `  t
) ^ 2 )  <_  ( R ^
2 )  <->  ( t ^ 2 )  <_ 
( R ^ 2 ) ) )
2117, 20bitr4d 248 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( ( abs `  t ) ^
2 )  <_  ( R ^ 2 ) ) )
22 recn 9036 . . . . . . . . . . . . . . 15  |-  ( t  e.  RR  ->  t  e.  CC )
2322abscld 12193 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  ( abs `  t )  e.  RR )
24233ad2ant3 980 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  ( abs `  t )  e.  RR )
25 simp1 957 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  R  e.  RR )
2622absge0d 12201 . . . . . . . . . . . . . 14  |-  ( t  e.  RR  ->  0  <_  ( abs `  t
) )
27263ad2ant3 980 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  ( abs `  t
) )
28 simp2 958 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  0  <_  R )
2924, 25, 27, 28le2sqd 11513 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
)  <_  R  <->  ( ( abs `  t ) ^
2 )  <_  ( R ^ 2 ) ) )
30 simp3 959 . . . . . . . . . . . . 13  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  t  e.  RR )
3130, 25absled 12188 . . . . . . . . . . . 12  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( abs `  t
)  <_  R  <->  ( -u R  <_  t  /\  t  <_  R ) ) )
3221, 29, 313bitr2d 273 . . . . . . . . . . 11  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) )  <->  ( -u R  <_  t  /\  t  <_  R ) ) )
3332biimprd 215 . . . . . . . . . 10  |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e.  RR )  ->  (
( -u R  <_  t  /\  t  <_  R )  ->  0  <_  (
( R ^ 2 )  -  ( t ^ 2 ) ) ) )
34333expa 1153 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  RR )  ->  ( ( -u R  <_  t  /\  t  <_  R )  ->  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
3534exp4b 591 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  RR  ->  ( -u R  <_ 
t  ->  ( t  <_  R  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) ) ) )
36353impd 1167 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( t  e.  RR  /\  -u R  <_  t  /\  t  <_  R )  ->  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
3713, 36sylbid 207 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( t  e.  (
-u R [,] R
)  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) ) )
3837imp 419 . . . . 5  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  0  <_  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )
39 elrege0 10963 . . . . 5  |-  ( ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,) 
+oo )  <->  ( (
( R ^ 2 )  -  ( t ^ 2 ) )  e.  RR  /\  0  <_  ( ( R ^
2 )  -  (
t ^ 2 ) ) ) )
4010, 38, 39sylanbrc 646 . . . 4  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R ) )  ->  ( ( R ^ 2 )  -  ( t ^ 2 ) )  e.  ( 0 [,)  +oo )
)
41 fvres 5704 . . . 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 4255 . 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 2404 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 18770 . . . . . 6  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
46 ax-resscn 9003 . . . . . . 7  |-  RR  C_  CC
476, 46syl6ss 3320 . . . . . 6  |-  ( R  e.  RR  ->  ( -u R [,] R ) 
C_  CC )
48 resttopon 17179 . . . . . 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 645 . . . . 5  |-  ( R  e.  RR  ->  (
( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
5049adantr 452 . . . 4  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( TopOpen ` fld )t  ( -u R [,] R ) )  e.  (TopOn `  ( -u R [,] R ) ) )
51 resmpt 5150 . . . . . . . 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 9036 . . . . . . . . . . 11  |-  ( R  e.  RR  ->  R  e.  CC )
5554sqcld 11476 . . . . . . . . . 10  |-  ( R  e.  RR  ->  ( R ^ 2 )  e.  CC )
5653, 53, 55cnmptc 17647 . . . . . . . . 9  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( R ^ 2 ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
5744sqcn 18857 . . . . . . . . . 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 18849 . . . . . . . . . 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 17651 . . . . . . . 8  |-  ( R  e.  RR  ->  (
t  e.  CC  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( (
TopOpen ` fld )  Cn  ( TopOpen ` fld )
) )
6245toponunii 16952 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
6362cnrest 17303 . . . . . . . 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 643 . . . . . . 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 2479 . . . . . 6  |-  ( R  e.  RR  ->  (
t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  e.  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( TopOpen ` fld ) ) )
6665adantr 452 . . . . 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 2404 . . . . . . . 8  |-  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )  =  ( t  e.  ( -u R [,] R )  |->  ( ( R ^ 2 )  -  ( t ^
2 ) ) )
6968rnmpt 5075 . . . . . . 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 959 . . . . . . . . . 10  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) )
71403adant3 977 . . . . . . . . . 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 2478 . . . . . . . . 9  |-  ( ( ( R  e.  RR  /\  0  <_  R )  /\  t  e.  ( -u R [,] R )  /\  u  =  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  ->  u  e.  ( 0 [,)  +oo ) )
7372rexlimdv3a 2792 . . . . . . . 8  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) )  ->  u  e.  ( 0 [,)  +oo ) ) )
7473abssdv 3377 . . . . . . 7  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  { u  |  E. t  e.  ( -u R [,] R ) u  =  ( ( R ^
2 )  -  (
t ^ 2 ) ) }  C_  (
0 [,)  +oo ) )
7569, 74syl5eqss 3352 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  ->  ran  ( t  e.  (
-u R [,] R
)  |->  ( ( R ^ 2 )  -  ( t ^ 2 ) ) )  C_  ( 0 [,)  +oo ) )
76 0re 9047 . . . . . . . . 9  |-  0  e.  RR
77 pnfxr 10669 . . . . . . . . 9  |-  +oo  e.  RR*
78 icossre 10947 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
7976, 77, 78mp2an 654 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  RR
8079, 46sstri 3317 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  CC
8180a1i 11 . . . . . 6  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( 0 [,)  +oo )  C_  CC )
82 cnrest2 17304 . . . . . 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 1184 . . . . 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 202 . . . 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 3327 . . . . . . . 8  |-  CC  C_  CC
86 cncfss 18882 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( 0 [,)  +oo ) -cn-> RR )  C_  (
( 0 [,)  +oo ) -cn-> CC ) )
8746, 85, 86mp2an 654 . . . . . . 7  |-  ( ( 0 [,)  +oo ) -cn->
RR )  C_  (
( 0 [,)  +oo ) -cn-> CC )
88 resqrcn 20586 . . . . . . 7  |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
8987, 88sselii 3305 . . . . . 6  |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> CC )
90 eqid 2404 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  ( 0 [,)  +oo ) )  =  ( ( TopOpen ` fld )t  ( 0 [,) 
+oo ) )
91 eqid 2404 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  CC )  =  ( ( TopOpen ` fld )t  CC )
9244, 90, 91cncfcn 18892 . . . . . . 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 654 . . . . . 6  |-  ( ( 0 [,)  +oo ) -cn->
CC )  =  ( ( ( TopOpen ` fld )t  ( 0 [,) 
+oo ) )  Cn  ( ( TopOpen ` fld )t  CC ) )
9489, 93eleqtri 2476 . . . . 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 17649 . . 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 2404 . . . . . 6  |-  ( (
TopOpen ` fld )t  ( -u R [,] R ) )  =  ( ( TopOpen ` fld )t  ( -u R [,] R ) )
9844, 97, 91cncfcn 18892 . . . . 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 644 . . . 4  |-  ( R  e.  RR  ->  (
( -u R [,] R
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
10099adantr 452 . . 3  |-  ( ( R  e.  RR  /\  0  <_  R )  -> 
( ( -u R [,] R ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( -u R [,] R ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
10196, 100eleqtrrd 2481 . 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 2479 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   ran crn 4838    |` cres 4839   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    +oocpnf 9073   RR*cxr 9075    <_ cle 9077    - cmin 9247   -ucneg 9248   2c2 10005   [,)cico 10874   [,]cicc 10875   ^cexp 11337   sqrcsqr 11993   abscabs 11994   ↾t crest 13603   TopOpenctopn 13604  ℂfldccnfld 16658  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545   -cn->ccncf 18859
This theorem is referenced by:  areacirclem1  26184  areacirclem5  26185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-tan 12629  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408
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