areacirclem2 - Mathbox for Brendan Leahy

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Theorem areacirclem2 37716

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 14164	. . . . . . . 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 11572	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → -𝑅 ∈ ℝ)
5		iccssre 13469	. . . . . . . . . 10 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ)
6	4, 5	mpancom 688	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ)
7	6	sselda 3983	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 𝑡 ∈ ℝ)
8	7	resqcld 14165	. . . . . . 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 11691	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
11		elicc2 13452	. . . . . . . . 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 1134	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
15		resqcl 14164	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
16	15	3ad2ant3 1136	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
17	14, 16	subge0d 11853	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (𝑡↑2) ≤ (𝑅↑2)))
18		absresq 15341	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → ((abs‘𝑡)↑2) = (𝑡↑2))
19	18	3ad2ant3 1136	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡)↑2) = (𝑡↑2))
20	19	breq1d 5153	. . . . . . . . . . . . 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 11245	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
23	22	abscld 15475	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → (abs‘𝑡) ∈ ℝ)
24	23	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (abs‘𝑡) ∈ ℝ)
25		simp1 1137	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑅 ∈ ℝ)
26	22	absge0d 15483	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → 0 ≤ (abs‘𝑡))
27	26	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘𝑡))
28		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ 𝑅)
29	24, 25, 27, 28	le2sqd 14296	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
30		simp3 1139	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
31	30, 25	absled 15469	. . . . . . . . . . . 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 1119	. . . . . . . . 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 13494	. . . . 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 6925	. . . 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 5242	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))))
44		eqid 2737	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 24803	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
46		ax-resscn 11212	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	6, 46	sstrdi 3996	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℂ)
48		resttopon 23169	. . . . . 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 6058	. . . . . . 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 11245	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℝ → 𝑅 ∈ ℂ)
54	53	sqcld 14184	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℂ)
55	52, 52, 54	cnmptc 23670	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ (𝑅↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
56	44	sqcn 24900	. . . . . . . . . 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 24888	. . . . . . . . . 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 23674	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
61	45	toponunii 22922	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
62	61	cnrest 23293	. . . . . . . 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 2842	. . . . . 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 2737	. . . . . . . 8 ⊢ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2)))
68	67	rnmpt 5968	. . . . . . 7 ⊢ ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))}
69		simp3 1139	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 = ((𝑅↑2) − (𝑡↑2)))
70	40	3adant3 1133	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → ((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞))
71	69, 70	eqeltrd 2841	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 ∈ (0[,)+∞))
72	71	rexlimdv3a 3159	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2)) → 𝑢 ∈ (0[,)+∞)))
73	72	abssdv 4068	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))} ⊆ (0[,)+∞))
74	68, 73	eqsstrid 4022	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ⊆ (0[,)+∞))
75		rge0ssre 13496	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
76	75, 46	sstri 3993	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
77	76	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (0[,)+∞) ⊆ ℂ)
78		cnrest2 23294	. . . . . 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 4006	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
82		cncfss 24925	. . . . . . . 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 26792	. . . . . . 7 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
85	83, 84	sselii 3980	. . . . . 6 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℂ)
86		eqid 2737	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) = ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))
87		eqid 2737	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
88	44, 86, 87	cncfcn 24936	. . . . . . 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 2839	. . . . 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 23672	. . 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 2737	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) = ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅))
94	44, 93, 87	cncfcn 24936	. . . . 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 2844	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))
98	43, 97	eqeltrrd 2842	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 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ≤ cle 11296 − cmin 11492 -cneg 11493 2c2 12321 [,)cico 13389 [,]cicc 13390 ↑cexp 14102 √csqrt 15272 abscabs 15273 ↾_t crest 17465 TopOpenctopn 17466 ℂ_fldccnfld 21364 TopOnctopon 22916 Cn ccn 23232 ×_t ctx 23568 –cn→ccncf 24902

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599

This theorem is referenced by: areacirclem3 37717 areacirclem4 37718 areacirc 37720

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