areacirclem2 - Mathbox for Brendan Leahy

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Theorem areacirclem2 37679

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 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))

Distinct variable group: 𝑡,𝑅

Proof of Theorem areacirclem2 Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resqcl 14140	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑅↑2) ∈ ℝ)
3	2	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑅↑2) ∈ ℝ)
4		renegcl 11544	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → -𝑅 ∈ ℝ)
5		iccssre 13444	. . . . . . . . . 10 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ)
6	4, 5	mpancom 688	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ)
7	6	sselda 3958	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 𝑡 ∈ ℝ)
8	7	resqcld 14141	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑡↑2) ∈ ℝ)
9	8	adantlr 715	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑡↑2) ∈ ℝ)
10	3, 9	resubcld 11663	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
11		elicc2 13426	. . . . . . . . 9 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
12	4, 11	mpancom 688	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
13	12	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
14	1	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
15		resqcl 14140	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
16	15	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
17	14, 16	subge0d 11825	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (𝑡↑2) ≤ (𝑅↑2)))
18		absresq 15319	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → ((abs‘𝑡)↑2) = (𝑡↑2))
19	18	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡)↑2) = (𝑡↑2))
20	19	breq1d 5129	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (((abs‘𝑡)↑2) ≤ (𝑅↑2) ↔ (𝑡↑2) ≤ (𝑅↑2)))
21	17, 20	bitr4d 282	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
22		recn 11217	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
23	22	abscld 15453	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → (abs‘𝑡) ∈ ℝ)
24	23	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (abs‘𝑡) ∈ ℝ)
25		simp1 1136	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑅 ∈ ℝ)
26	22	absge0d 15461	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → 0 ≤ (abs‘𝑡))
27	26	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘𝑡))
28		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ 𝑅)
29	24, 25, 27, 28	le2sqd 14273	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
30		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
31	30, 25	absled 15447	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ (-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
32	21, 29, 31	3bitr2d 307	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
33	32	biimprd 248	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
34	33	3expa 1118	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ ℝ) → ((-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
35	34	exp4b 430	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ ℝ → (-𝑅 ≤ 𝑡 → (𝑡 ≤ 𝑅 → 0 ≤ ((𝑅↑2) − (𝑡↑2))))))
36	35	3impd 1349	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ((𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
37	13, 36	sylbid 240	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
38	37	imp 406	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))
39		elrege0 13469	. . . . 5 ⊢ (((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞) ↔ (((𝑅↑2) − (𝑡↑2)) ∈ ℝ ∧ 0 ≤ ((𝑅↑2) − (𝑡↑2))))
40	10, 38, 39	sylanbrc 583	. . . 4 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞))
41		fvres 6894	. . . 4 ⊢ (((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞) → ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2))) = (√‘((𝑅↑2) − (𝑡↑2))))
42	40, 41	syl 17	. . 3 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2))) = (√‘((𝑅↑2) − (𝑡↑2))))
43	42	mpteq2dva 5214	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))))
44		eqid 2735	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 24719	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
46		ax-resscn 11184	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	6, 46	sstrdi 3971	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℂ)
48		resttopon 23097	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (-𝑅[,]𝑅) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
49	45, 47, 48	sylancr 587	. . . . 5 ⊢ (𝑅 ∈ ℝ → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
50	49	adantr 480	. . . 4 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
51	47	resmptd 6027	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ↾ (-𝑅[,]𝑅)) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))))
52	45	a1i 11	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
53		recn 11217	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℝ → 𝑅 ∈ ℂ)
54	53	sqcld 14160	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℂ)
55	52, 52, 54	cnmptc 23598	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ (𝑅↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
56	44	sqcn 24816	. . . . . . . . . 10 ⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
57	56	a1i 11	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
58	44	subcn 24804	. . . . . . . . . 10 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
59	58	a1i 11	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
60	52, 55, 57, 59	cnmpt12f 23602	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
61	45	toponunii 22852	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
62	61	cnrest 23221	. . . . . . . 8 ⊢ (((𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)) ∧ (-𝑅[,]𝑅) ⊆ ℂ) → ((𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ↾ (-𝑅[,]𝑅)) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂ_fld)))
63	60, 47, 62	syl2anc 584	. . . . . . 7 ⊢ (𝑅 ∈ ℝ → ((𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ↾ (-𝑅[,]𝑅)) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂ_fld)))
64	51, 63	eqeltrrd 2835	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂ_fld)))
65	64	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂ_fld)))
66	45	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
67		eqid 2735	. . . . . . . 8 ⊢ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2)))
68	67	rnmpt 5937	. . . . . . 7 ⊢ ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))}
69		simp3 1138	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 = ((𝑅↑2) − (𝑡↑2)))
70	40	3adant3 1132	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → ((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞))
71	69, 70	eqeltrd 2834	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 ∈ (0[,)+∞))
72	71	rexlimdv3a 3145	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2)) → 𝑢 ∈ (0[,)+∞)))
73	72	abssdv 4043	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))} ⊆ (0[,)+∞))
74	68, 73	eqsstrid 3997	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ⊆ (0[,)+∞))
75		rge0ssre 13471	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
76	75, 46	sstri 3968	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
77	76	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (0[,)+∞) ⊆ ℂ)
78		cnrest2 23222	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ⊆ (0[,)+∞) ∧ (0[,)+∞) ⊆ ℂ) → ((𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂ_fld)) ↔ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)))))
79	66, 74, 77, 78	syl3anc 1373	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ((𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂ_fld)) ↔ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)))))
80	65, 79	mpbid 232	. . . 4 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))))
81		ssid 3981	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
82		cncfss 24841	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0[,)+∞)–cn→ℝ) ⊆ ((0[,)+∞)–cn→ℂ))
83	46, 81, 82	mp2an 692	. . . . . . 7 ⊢ ((0[,)+∞)–cn→ℝ) ⊆ ((0[,)+∞)–cn→ℂ)
84		resqrtcn 26709	. . . . . . 7 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
85	83, 84	sselii 3955	. . . . . 6 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℂ)
86		eqid 2735	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) = ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))
87		eqid 2735	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
88	44, 86, 87	cncfcn 24852	. . . . . . 7 ⊢ (((0[,)+∞) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0[,)+∞)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
89	76, 81, 88	mp2an 692	. . . . . 6 ⊢ ((0[,)+∞)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ))
90	85, 89	eleqtri 2832	. . . . 5 ⊢ (√ ↾ (0[,)+∞)) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ))
91	90	a1i 11	. . . 4 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (√ ↾ (0[,)+∞)) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
92	50, 80, 91	cnmpt11f 23600	. . 3 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
93		eqid 2735	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) = ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅))
94	44, 93, 87	cncfcn 24852	. . . . 5 ⊢ (((-𝑅[,]𝑅) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-𝑅[,]𝑅)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
95	47, 81, 94	sylancl 586	. . . 4 ⊢ (𝑅 ∈ ℝ → ((-𝑅[,]𝑅)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
96	95	adantr 480	. . 3 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ((-𝑅[,]𝑅)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
97	92, 96	eleqtrrd 2837	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))
98	43, 97	eqeltrrd 2835	1 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 {cab 2713 ∃wrex 3060 ⊆ wss 3926 class class class wbr 5119 ↦ cmpt 5201 ran crn 5655 ↾ cres 5656 ‘cfv 6530 (class class class)co 7403 ℂcc 11125 ℝcr 11126 0cc0 11127 +∞cpnf 11264 ≤ cle 11268 − cmin 11464 -cneg 11465 2c2 12293 [,)cico 13362 [,]cicc 13363 ↑cexp 14077 √csqrt 15250 abscabs 15251 ↾_t crest 17432 TopOpenctopn 17433 ℂ_fldccnfld 21313 TopOnctopon 22846 Cn ccn 23160 ×_t ctx 23496 –cn→ccncf 24818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-ef 16081 df-sin 16083 df-cos 16084 df-tan 16085 df-pi 16086 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-cmp 23323 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-limc 25817 df-dv 25818 df-log 26515 df-cxp 26516

This theorem is referenced by: areacirclem3 37680 areacirclem4 37681 areacirc 37683

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