areacirclem2 - Mathbox for Brendan Leahy

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Theorem areacirclem2 33833

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 13138	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
2	1	adantr 466	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑅↑2) ∈ ℝ)
3	2	adantr 466	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑅↑2) ∈ ℝ)
4		renegcl 10546	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → -𝑅 ∈ ℝ)
5		iccssre 12460	. . . . . . . . . 10 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ)
6	4, 5	mpancom 660	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ)
7	6	sselda 3752	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 𝑡 ∈ ℝ)
8	7	resqcld 13242	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑡↑2) ∈ ℝ)
9	8	adantlr 686	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑡↑2) ∈ ℝ)
10	3, 9	resubcld 10660	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
11		elicc2 12443	. . . . . . . . 9 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
12	4, 11	mpancom 660	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
13	12	adantr 466	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
14	1	3ad2ant1 1127	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
15		resqcl 13138	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
16	15	3ad2ant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
17	14, 16	subge0d 10819	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (𝑡↑2) ≤ (𝑅↑2)))
18		absresq 14250	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → ((abs‘𝑡)↑2) = (𝑡↑2))
19	18	3ad2ant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡)↑2) = (𝑡↑2))
20	19	breq1d 4796	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (((abs‘𝑡)↑2) ≤ (𝑅↑2) ↔ (𝑡↑2) ≤ (𝑅↑2)))
21	17, 20	bitr4d 271	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
22		recn 10228	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
23	22	abscld 14383	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → (abs‘𝑡) ∈ ℝ)
24	23	3ad2ant3 1129	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (abs‘𝑡) ∈ ℝ)
25		simp1 1130	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑅 ∈ ℝ)
26	22	absge0d 14391	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → 0 ≤ (abs‘𝑡))
27	26	3ad2ant3 1129	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘𝑡))
28		simp2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ 𝑅)
29	24, 25, 27, 28	le2sqd 13251	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
30		simp3 1132	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
31	30, 25	absled 14377	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ (-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
32	21, 29, 31	3bitr2d 296	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
33	32	biimprd 238	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
34	33	3expa 1111	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ ℝ) → ((-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
35	34	exp4b 417	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ ℝ → (-𝑅 ≤ 𝑡 → (𝑡 ≤ 𝑅 → 0 ≤ ((𝑅↑2) − (𝑡↑2))))))
36	35	3impd 1441	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ((𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
37	13, 36	sylbid 230	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
38	37	imp 393	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))
39		elrege0 12485	. . . . 5 ⊢ (((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞) ↔ (((𝑅↑2) − (𝑡↑2)) ∈ ℝ ∧ 0 ≤ ((𝑅↑2) − (𝑡↑2))))
40	10, 38, 39	sylanbrc 564	. . . 4 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞))
41		fvres 6348	. . . 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 4878	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))))
44		eqid 2771	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 22806	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
46		ax-resscn 10195	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	6, 46	syl6ss 3764	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℂ)
48		resttopon 21186	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (-𝑅[,]𝑅) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
49	45, 47, 48	sylancr 567	. . . . 5 ⊢ (𝑅 ∈ ℝ → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
50	49	adantr 466	. . . 4 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
51	47	resmptd 5593	. . . . . . 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 10228	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℝ → 𝑅 ∈ ℂ)
54	53	sqcld 13213	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℂ)
55	52, 52, 54	cnmptc 21686	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ (𝑅↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
56	44	sqcn 22897	. . . . . . . . . 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 22889	. . . . . . . . . 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 21690	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
61	45	toponunii 20941	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
62	61	cnrest 21310	. . . . . . . 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 565	. . . . . . 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 2851	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂ_fld)))
65	64	adantr 466	. . . . 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 2771	. . . . . . . 8 ⊢ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2)))
68	67	rnmpt 5509	. . . . . . 7 ⊢ ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))}
69		simp3 1132	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 = ((𝑅↑2) − (𝑡↑2)))
70	40	3adant3 1126	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → ((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞))
71	69, 70	eqeltrd 2850	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 ∈ (0[,)+∞))
72	71	rexlimdv3a 3181	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2)) → 𝑢 ∈ (0[,)+∞)))
73	72	abssdv 3825	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))} ⊆ (0[,)+∞))
74	68, 73	syl5eqss 3798	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ⊆ (0[,)+∞))
75		rge0ssre 12487	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
76	75, 46	sstri 3761	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
77	76	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (0[,)+∞) ⊆ ℂ)
78		cnrest2 21311	. . . . . 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 1476	. . . . 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 222	. . . 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 3773	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
82		cncfss 22922	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0[,)+∞)–cn→ℝ) ⊆ ((0[,)+∞)–cn→ℂ))
83	46, 81, 82	mp2an 664	. . . . . . 7 ⊢ ((0[,)+∞)–cn→ℝ) ⊆ ((0[,)+∞)–cn→ℂ)
84		resqrtcn 24711	. . . . . . 7 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
85	83, 84	sselii 3749	. . . . . 6 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℂ)
86		eqid 2771	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) = ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))
87		eqid 2771	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
88	44, 86, 87	cncfcn 22932	. . . . . . 7 ⊢ (((0[,)+∞) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0[,)+∞)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
89	76, 81, 88	mp2an 664	. . . . . 6 ⊢ ((0[,)+∞)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ))
90	85, 89	eleqtri 2848	. . . . 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 21688	. . 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 2771	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) = ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅))
94	44, 93, 87	cncfcn 22932	. . . . 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 566	. . . 4 ⊢ (𝑅 ∈ ℝ → ((-𝑅[,]𝑅)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
96	95	adantr 466	. . 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 2853	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))
98	43, 97	eqeltrrd 2851	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 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 {cab 2757 ∃wrex 3062 ⊆ wss 3723 class class class wbr 4786 ↦ cmpt 4863 ran crn 5250 ↾ cres 5251 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 ℝcr 10137 0cc0 10138 +∞cpnf 10273 ≤ cle 10277 − cmin 10468 -cneg 10469 2c2 11272 [,)cico 12382 [,]cicc 12383 ↑cexp 13067 √csqrt 14181 abscabs 14182 ↾_t crest 16289 TopOpenctopn 16290 ℂ_fldccnfld 19961 TopOnctopon 20935 Cn ccn 21249 ×_t ctx 21584 –cn→ccncf 22899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ioc 12385 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 df-sin 15006 df-cos 15007 df-tan 15008 df-pi 15009 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-lp 21161 df-perf 21162 df-cn 21252 df-cnp 21253 df-haus 21340 df-cmp 21411 df-tx 21586 df-hmeo 21779 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-xms 22345 df-ms 22346 df-tms 22347 df-cncf 22901 df-limc 23850 df-dv 23851 df-log 24524 df-cxp 24525

This theorem is referenced by: areacirclem3 33834 areacirclem4 33835 areacirc 33837

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