Theorem areacirclem2 33501

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.

Step	Hyp	Ref	Expression
1		resqcl 12931	. . . . . . . 8 |- ( R e. RR -> ( R ^ 2 ) e. RR )
2	1	adantr 481	. . . . . . 7 |- ( ( R e. RR /\ 0 <_ R ) -> ( R ^ 2 ) e. RR )
3	2	adantr 481	. . . . . 6 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) ) -> ( R ^ 2 ) e. RR )
4		renegcl 10344	. . . . . . . . . 10 |- ( R e. RR -> -u R e. RR )
5		iccssre 12255	. . . . . . . . . 10 |- ( ( -u R e. RR /\ R e. RR ) -> ( -u R [,] R ) C_ RR )
6	4, 5	mpancom 703	. . . . . . . . 9 |- ( R e. RR -> ( -u R [,] R ) C_ RR )
7	6	sselda 3603	. . . . . . . 8 |- ( ( R e. RR /\ t e. ( -u R [,] R ) ) -> t e. RR )
8	7	resqcld 13035	. . . . . . 7 |- ( ( R e. RR /\ t e. ( -u R [,] R ) ) -> ( t ^ 2 ) e. RR )
9	8	adantlr 751	. . . . . 6 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) ) -> ( t ^ 2 ) e. RR )
10	3, 9	resubcld 10458	. . . . 5 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) ) -> ( ( R ^ 2 ) - ( t ^ 2 ) ) e. RR )
11		elicc2 12238	. . . . . . . . 9 |- ( ( -u R e. RR /\ R e. RR ) -> ( t e. ( -u R [,] R ) <-> ( t e. RR /\ -u R <_ t /\ t <_ R ) ) )
12	4, 11	mpancom 703	. . . . . . . 8 |- ( R e. RR -> ( t e. ( -u R [,] R ) <-> ( t e. RR /\ -u R <_ t /\ t <_ R ) ) )
13	12	adantr 481	. . . . . . 7 |- ( ( R e. RR /\ 0 <_ R ) -> ( t e. ( -u R [,] R ) <-> ( t e. RR /\ -u R <_ t /\ t <_ R ) ) )
14	1	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( R ^ 2 ) e. RR )
15		resqcl 12931	. . . . . . . . . . . . . . 15 |- ( t e. RR -> ( t ^ 2 ) e. RR )
16	15	3ad2ant3 1084	. . . . . . . . . . . . . 14 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( t ^ 2 ) e. RR )
17	14, 16	subge0d 10617	. . . . . . . . . . . . 13 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) <-> ( t ^ 2 ) <_ ( R ^ 2 ) ) )
18		absresq 14042	. . . . . . . . . . . . . . 15 |- ( t e. RR -> ( ( abs ` t ) ^ 2 ) = ( t ^ 2 ) )
19	18	3ad2ant3 1084	. . . . . . . . . . . . . 14 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( ( abs ` t ) ^ 2 ) = ( t ^ 2 ) )
20	19	breq1d 4663	. . . . . . . . . . . . 13 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( ( ( abs ` t ) ^ 2 ) <_ ( R ^ 2 ) <-> ( t ^ 2 ) <_ ( R ^ 2 ) ) )
21	17, 20	bitr4d 271	. . . . . . . . . . . 12 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) <-> ( ( abs ` t ) ^ 2 ) <_ ( R ^ 2 ) ) )
22		recn 10026	. . . . . . . . . . . . . . 15 |- ( t e. RR -> t e. CC )
23	22	abscld 14175	. . . . . . . . . . . . . 14 |- ( t e. RR -> ( abs ` t ) e. RR )
24	23	3ad2ant3 1084	. . . . . . . . . . . . 13 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( abs ` t ) e. RR )
25		simp1 1061	. . . . . . . . . . . . 13 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> R e. RR )
26	22	absge0d 14183	. . . . . . . . . . . . . 14 |- ( t e. RR -> 0 <_ ( abs ` t ) )
27	26	3ad2ant3 1084	. . . . . . . . . . . . 13 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> 0 <_ ( abs ` t ) )
28		simp2 1062	. . . . . . . . . . . . 13 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> 0 <_ R )
29	24, 25, 27, 28	le2sqd 13044	. . . . . . . . . . . 12 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( ( abs ` t ) <_ R <-> ( ( abs ` t ) ^ 2 ) <_ ( R ^ 2 ) ) )
30		simp3 1063	. . . . . . . . . . . . 13 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> t e. RR )
31	30, 25	absled 14169	. . . . . . . . . . . 12 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( ( abs ` t ) <_ R <-> ( -u R <_ t /\ t <_ R ) ) )
32	21, 29, 31	3bitr2d 296	. . . . . . . . . . 11 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) <-> ( -u R <_ t /\ t <_ R ) ) )
33	32	biimprd 238	. . . . . . . . . 10 |- ( ( R e. RR /\ 0 <_ R /\ t e. RR ) -> ( ( -u R <_ t /\ t <_ R ) -> 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) ) )
34	33	3expa 1265	. . . . . . . . 9 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. RR ) -> ( ( -u R <_ t /\ t <_ R ) -> 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) ) )
35	34	exp4b 632	. . . . . . . 8 |- ( ( R e. RR /\ 0 <_ R ) -> ( t e. RR -> ( -u R <_ t -> ( t <_ R -> 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) ) ) ) )
36	35	3impd 1281	. . . . . . 7 |- ( ( R e. RR /\ 0 <_ R ) -> ( ( t e. RR /\ -u R <_ t /\ t <_ R ) -> 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) ) )
37	13, 36	sylbid 230	. . . . . 6 |- ( ( R e. RR /\ 0 <_ R ) -> ( t e. ( -u R [,] R ) -> 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) ) )
38	37	imp 445	. . . . 5 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) ) -> 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) )
39		elrege0 12278	. . . . 5 |- ( ( ( R ^ 2 ) - ( t ^ 2 ) ) e. ( 0 [,) +oo ) <-> ( ( ( R ^ 2 ) - ( t ^ 2 ) ) e. RR /\ 0 <_ ( ( R ^ 2 ) - ( t ^ 2 ) ) ) )
40	10, 38, 39	sylanbrc 698	. . . 4 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) ) -> ( ( R ^ 2 ) - ( t ^ 2 ) ) e. ( 0 [,) +oo ) )
41		fvres 6207	. . . 4 |- ( ( ( R ^ 2 ) - ( t ^ 2 ) ) e. ( 0 [,) +oo ) -> ( ( sqr |` ( 0 [,) +oo ) ) ` ( ( R ^ 2 ) - ( t ^ 2 ) ) ) = ( sqr ` ( ( R ^ 2 ) - ( t ^ 2 ) ) ) )
42	40, 41	syl 17	. . 3 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) ) -> ( ( sqr |` ( 0 [,) +oo ) ) ` ( ( R ^ 2 ) - ( t ^ 2 ) ) ) = ( sqr ` ( ( R ^ 2 ) - ( t ^ 2 ) ) ) )
43	42	mpteq2dva 4744	. 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 2622	. . . . . . 7 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
45	44	cnfldtopon 22586	. . . . . 6 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
46		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
47	6, 46	syl6ss 3615	. . . . . 6 |- ( R e. RR -> ( -u R [,] R ) C_ CC )
48		resttopon 20965	. . . . . 6 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( -u R [,] R ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) e. (TopOn ` ( -u R [,] R ) ) )
49	45, 47, 48	sylancr 695	. . . . 5 |- ( R e. RR -> ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) e. (TopOn ` ( -u R [,] R ) ) )
50	49	adantr 481	. . . 4 |- ( ( R e. RR /\ 0 <_ R ) -> ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) e. (TopOn ` ( -u R [,] R ) ) )
51	47	resmptd 5452	. . . . . . 7 |- ( R e. RR -> ( ( t e. CC |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) |` ( -u R [,] R ) ) = ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) )
52	45	a1i 11	. . . . . . . . 9 |- ( R e. RR -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
53		recn 10026	. . . . . . . . . . 11 |- ( R e. RR -> R e. CC )
54	53	sqcld 13006	. . . . . . . . . 10 |- ( R e. RR -> ( R ^ 2 ) e. CC )
55	52, 52, 54	cnmptc 21465	. . . . . . . . 9 |- ( R e. RR -> ( t e. CC |-> ( R ^ 2 ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
56	44	sqcn 22677	. . . . . . . . . 10 |- ( t e. CC |-> ( t ^ 2 ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) )
57	56	a1i 11	. . . . . . . . 9 |- ( R e. RR -> ( t e. CC |-> ( t ^ 2 ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
58	44	subcn 22669	. . . . . . . . . 10 |- - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
59	58	a1i 11	. . . . . . . . 9 |- ( R e. RR -> - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) ) )
60	52, 55, 57, 59	cnmpt12f 21469	. . . . . . . 8 |- ( R e. RR -> ( t e. CC |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
61	45	toponunii 20721	. . . . . . . . 9 |- CC = U. ( TopOpen ` ℂfld )
62	61	cnrest 21089	. . . . . . . 8 |- ( ( ( t e. CC |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) /\ ( -u R [,] R ) C_ CC ) -> ( ( t e. CC |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) |` ( -u R [,] R ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( TopOpen ` ℂfld ) ) )
63	60, 47, 62	syl2anc 693	. . . . . . 7 |- ( R e. RR -> ( ( t e. CC |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) |` ( -u R [,] R ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( TopOpen ` ℂfld ) ) )
64	51, 63	eqeltrrd 2702	. . . . . 6 |- ( R e. RR -> ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( TopOpen ` ℂfld ) ) )
65	64	adantr 481	. . . . 5 |- ( ( R e. RR /\ 0 <_ R ) -> ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( TopOpen ` ℂfld ) ) )
66	45	a1i 11	. . . . . 6 |- ( ( R e. RR /\ 0 <_ R ) -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
67		eqid 2622	. . . . . . . 8 |- ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) = ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) )
68	67	rnmpt 5371	. . . . . . 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 1063	. . . . . . . . . 10 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) /\ u = ( ( R ^ 2 ) - ( t ^ 2 ) ) ) -> u = ( ( R ^ 2 ) - ( t ^ 2 ) ) )
70	40	3adant3 1081	. . . . . . . . . 10 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) /\ u = ( ( R ^ 2 ) - ( t ^ 2 ) ) ) -> ( ( R ^ 2 ) - ( t ^ 2 ) ) e. ( 0 [,) +oo ) )
71	69, 70	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( R e. RR /\ 0 <_ R ) /\ t e. ( -u R [,] R ) /\ u = ( ( R ^ 2 ) - ( t ^ 2 ) ) ) -> u e. ( 0 [,) +oo ) )
72	71	rexlimdv3a 3033	. . . . . . . 8 |- ( ( R e. RR /\ 0 <_ R ) -> ( E. t e. ( -u R [,] R ) u = ( ( R ^ 2 ) - ( t ^ 2 ) ) -> u e. ( 0 [,) +oo ) ) )
73	72	abssdv 3676	. . . . . . 7 |- ( ( R e. RR /\ 0 <_ R ) -> { u | E. t e. ( -u R [,] R ) u = ( ( R ^ 2 ) - ( t ^ 2 ) ) } C_ ( 0 [,) +oo ) )
74	68, 73	syl5eqss 3649	. . . . . 6 |- ( ( R e. RR /\ 0 <_ R ) -> ran ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) C_ ( 0 [,) +oo ) )
75		rge0ssre 12280	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
76	75, 46	sstri 3612	. . . . . . 7 |- ( 0 [,) +oo ) C_ CC
77	76	a1i 11	. . . . . 6 |- ( ( R e. RR /\ 0 <_ R ) -> ( 0 [,) +oo ) C_ CC )
78		cnrest2 21090	. . . . . 6 |- ( ( ( TopOpen ` ℂfld ) 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 ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) ) ) )
79	66, 74, 77, 78	syl3anc 1326	. . . . 5 |- ( ( R e. RR /\ 0 <_ R ) -> ( ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) ) ) )
80	65, 79	mpbid 222	. . . 4 |- ( ( R e. RR /\ 0 <_ R ) -> ( t e. ( -u R [,] R ) |-> ( ( R ^ 2 ) - ( t ^ 2 ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) ) )
81		ssid 3624	. . . . . . . 8 |- CC C_ CC
82		cncfss 22702	. . . . . . . 8 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( 0 [,) +oo ) -cn-> RR ) C_ ( ( 0 [,) +oo ) -cn-> CC ) )
83	46, 81, 82	mp2an 708	. . . . . . 7 |- ( ( 0 [,) +oo ) -cn-> RR ) C_ ( ( 0 [,) +oo ) -cn-> CC )
84		resqrtcn 24490	. . . . . . 7 |- ( sqr |` ( 0 [,) +oo ) ) e. ( ( 0 [,) +oo ) -cn-> RR )
85	83, 84	sselii 3600	. . . . . 6 |- ( sqr |` ( 0 [,) +oo ) ) e. ( ( 0 [,) +oo ) -cn-> CC )
86		eqid 2622	. . . . . . . 8 |- ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) = ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) )
87		eqid 2622	. . . . . . . 8 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( ( TopOpen ` ℂfld ) ↾t CC )
88	44, 86, 87	cncfcn 22712	. . . . . . 7 |- ( ( ( 0 [,) +oo ) C_ CC /\ CC C_ CC ) -> ( ( 0 [,) +oo ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
89	76, 81, 88	mp2an 708	. . . . . 6 |- ( ( 0 [,) +oo ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) )
90	85, 89	eleqtri 2699	. . . . 5 |- ( sqr |` ( 0 [,) +oo ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) )
91	90	a1i 11	. . . 4 |- ( ( R e. RR /\ 0 <_ R ) -> ( sqr |` ( 0 [,) +oo ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( 0 [,) +oo ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
92	50, 80, 91	cnmpt11f 21467	. . 3 |- ( ( R e. RR /\ 0 <_ R ) -> ( t e. ( -u R [,] R ) |-> ( ( sqr |` ( 0 [,) +oo ) ) ` ( ( R ^ 2 ) - ( t ^ 2 ) ) ) ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
93		eqid 2622	. . . . . 6 |- ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) = ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) )
94	44, 93, 87	cncfcn 22712	. . . . 5 |- ( ( ( -u R [,] R ) C_ CC /\ CC C_ CC ) -> ( ( -u R [,] R ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
95	47, 81, 94	sylancl 694	. . . 4 |- ( R e. RR -> ( ( -u R [,] R ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
96	95	adantr 481	. . 3 |- ( ( R e. RR /\ 0 <_ R ) -> ( ( -u R [,] R ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( -u R [,] R ) ) Cn ( ( TopOpen ` ℂfld ) ↾t CC ) ) )
97	92, 96	eleqtrrd 2704	. 2 |- ( ( R e. RR /\ 0 <_ R ) -> ( t e. ( -u R [,] R ) |-> ( ( sqr |` ( 0 [,) +oo ) ) ` ( ( R ^ 2 ) - ( t ^ 2 ) ) ) ) e. ( ( -u R [,] R ) -cn-> CC ) )
98	43, 97	eqeltrrd 2702	1 |- ( ( R e. RR /\ 0 <_ R ) -> ( t e. ( -u R [,] R ) |-> ( sqr ` ( ( R ^ 2 ) - ( t ^ 2 ) ) ) ) e. ( ( -u R [,] R ) -cn-> CC ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   +oocpnf 10071   <_ cle 10075   - cmin 10266   -ucneg 10267   2c2 11070   [,)cico 12177   [,]cicc 12178   ^cexp 12860   sqrcsqrt 13973   abscabs 13974   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   -cn->ccncf 22679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  areacirclem3  33502  areacirclem4  33503  areacirc  33505