areacirclem2 - Mathbox for Brendan Leahy

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Theorem areacirclem2 37235

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 14115	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
2	1	adantr 479	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑅↑2) ∈ ℝ)
3	2	adantr 479	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑅↑2) ∈ ℝ)
4		renegcl 11548	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → -𝑅 ∈ ℝ)
5		iccssre 13433	. . . . . . . . . 10 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ)
6	4, 5	mpancom 686	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ)
7	6	sselda 3973	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 𝑡 ∈ ℝ)
8	7	resqcld 14116	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑡↑2) ∈ ℝ)
9	8	adantlr 713	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑡↑2) ∈ ℝ)
10	3, 9	resubcld 11667	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
11		elicc2 13416	. . . . . . . . 9 ⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
12	4, 11	mpancom 686	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
13	12	adantr 479	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
14	1	3ad2ant1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
15		resqcl 14115	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
16	15	3ad2ant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
17	14, 16	subge0d 11829	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (𝑡↑2) ≤ (𝑅↑2)))
18		absresq 15276	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → ((abs‘𝑡)↑2) = (𝑡↑2))
19	18	3ad2ant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡)↑2) = (𝑡↑2))
20	19	breq1d 5154	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (((abs‘𝑡)↑2) ≤ (𝑅↑2) ↔ (𝑡↑2) ≤ (𝑅↑2)))
21	17, 20	bitr4d 281	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
22		recn 11223	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
23	22	abscld 15410	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → (abs‘𝑡) ∈ ℝ)
24	23	3ad2ant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (abs‘𝑡) ∈ ℝ)
25		simp1 1133	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑅 ∈ ℝ)
26	22	absge0d 15418	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → 0 ≤ (abs‘𝑡))
27	26	3ad2ant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘𝑡))
28		simp2 1134	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ 𝑅)
29	24, 25, 27, 28	le2sqd 14246	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
30		simp3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
31	30, 25	absled 15404	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ (-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
32	21, 29, 31	3bitr2d 306	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅)))
33	32	biimprd 247	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
34	33	3expa 1115	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ ℝ) → ((-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
35	34	exp4b 429	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ ℝ → (-𝑅 ≤ 𝑡 → (𝑡 ≤ 𝑅 → 0 ≤ ((𝑅↑2) − (𝑡↑2))))))
36	35	3impd 1345	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ((𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
37	13, 36	sylbid 239	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2))))
38	37	imp 405	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))
39		elrege0 13458	. . . . 5 ⊢ (((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞) ↔ (((𝑅↑2) − (𝑡↑2)) ∈ ℝ ∧ 0 ≤ ((𝑅↑2) − (𝑡↑2))))
40	10, 38, 39	sylanbrc 581	. . . 4 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞))
41		fvres 6909	. . . 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 5244	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))))
44		eqid 2725	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 24712	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
46		ax-resscn 11190	. . . . . . 7 ⊢ ℝ ⊆ ℂ
47	6, 46	sstrdi 3986	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℂ)
48		resttopon 23078	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (-𝑅[,]𝑅) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
49	45, 47, 48	sylancr 585	. . . . 5 ⊢ (𝑅 ∈ ℝ → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
50	49	adantr 479	. . . 4 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅)))
51	47	resmptd 6040	. . . . . . 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 11223	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℝ → 𝑅 ∈ ℂ)
54	53	sqcld 14135	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℂ)
55	52, 52, 54	cnmptc 23579	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ (𝑅↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
56	44	sqcn 24807	. . . . . . . . . 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 24795	. . . . . . . . . 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 23583	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
61	45	toponunii 22831	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
62	61	cnrest 23202	. . . . . . . 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 582	. . . . . . 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 2826	. . . . . 6 ⊢ (𝑅 ∈ ℝ → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈ (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂ_fld)))
65	64	adantr 479	. . . . 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 2725	. . . . . . . 8 ⊢ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2)))
68	67	rnmpt 5952	. . . . . . 7 ⊢ ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))}
69		simp3 1135	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 = ((𝑅↑2) − (𝑡↑2)))
70	40	3adant3 1129	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → ((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞))
71	69, 70	eqeltrd 2825	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 ∈ (0[,)+∞))
72	71	rexlimdv3a 3149	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2)) → 𝑢 ∈ (0[,)+∞)))
73	72	abssdv 4058	. . . . . . 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 13460	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
76	75, 46	sstri 3983	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
77	76	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (0[,)+∞) ⊆ ℂ)
78		cnrest2 23203	. . . . . 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 1368	. . . . 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 231	. . . 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 3996	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
82		cncfss 24832	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0[,)+∞)–cn→ℝ) ⊆ ((0[,)+∞)–cn→ℂ))
83	46, 81, 82	mp2an 690	. . . . . . 7 ⊢ ((0[,)+∞)–cn→ℝ) ⊆ ((0[,)+∞)–cn→ℂ)
84		resqrtcn 26697	. . . . . . 7 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ)
85	83, 84	sselii 3970	. . . . . 6 ⊢ (√ ↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℂ)
86		eqid 2725	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) = ((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))
87		eqid 2725	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
88	44, 86, 87	cncfcn 24843	. . . . . . 7 ⊢ (((0[,)+∞) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0[,)+∞)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
89	76, 81, 88	mp2an 690	. . . . . 6 ⊢ ((0[,)+∞)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (0[,)+∞)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ))
90	85, 89	eleqtri 2823	. . . . 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 23581	. . 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 2725	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) = ((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅))
94	44, 93, 87	cncfcn 24843	. . . . 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 584	. . . 4 ⊢ (𝑅 ∈ ℝ → ((-𝑅[,]𝑅)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂ_fld) ↾_t ℂ)))
96	95	adantr 479	. . 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 2828	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))
98	43, 97	eqeltrrd 2826	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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {cab 2702 ∃wrex 3060 ⊆ wss 3941 class class class wbr 5144 ↦ cmpt 5227 ran crn 5674 ↾ cres 5675 ‘cfv 6543 (class class class)co 7413 ℂcc 11131 ℝcr 11132 0cc0 11133 +∞cpnf 11270 ≤ cle 11274 − cmin 11469 -cneg 11470 2c2 12292 [,)cico 13353 [,]cicc 13354 ↑cexp 14053 √csqrt 15207 abscabs 15208 ↾_t crest 17396 TopOpenctopn 17397 ℂ_fldccnfld 21278 TopOnctopon 22825 Cn ccn 23141 ×_t ctx 23477 –cn→ccncf 24809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-fac 14260 df-bc 14289 df-hash 14317 df-shft 15041 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660 df-ef 16038 df-sin 16040 df-cos 16041 df-tan 16042 df-pi 16043 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-cmp 23304 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503 df-cxp 26504

This theorem is referenced by: areacirclem3 37236 areacirclem4 37237 areacirc 37239

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