areacirclem5 - Mathbox for Brendan Leahy

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Theorem areacirclem5 38420

Description: Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)

**Hypothesis**
Ref	Expression
areacirc.1	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}

**Assertion**
Ref	Expression
areacirclem5	⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑆 “ {𝑡}) = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅))

Distinct variable groups: 𝑥,𝑦,𝑡,𝑅 𝑡,𝑆 Allowed substitution hints: 𝑆(𝑥, 𝑦)

Proof of Theorem areacirclem5 Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		areacirc.1	. . . 4 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}
2	1	imaeq1i 6061	. . 3 ⊢ (𝑆 “ {𝑡}) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))} “ {𝑡})
3		vex 3461	. . . 4 ⊢ 𝑡 ∈ V
4		imasng 6088	. . . 4 ⊢ (𝑡 ∈ V → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))} “ {𝑡}) = {𝑢 ∣ 𝑡{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}𝑢})
5	3, 4	ax-mp 5	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))} “ {𝑡}) = {𝑢 ∣ 𝑡{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}𝑢}
6		df-br 5112	. . . . 5 ⊢ (𝑡{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}𝑢 ↔ ⟨𝑡, 𝑢⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))})
7		vex 3461	. . . . . 6 ⊢ 𝑢 ∈ V
8		eleq1w 2848	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ ℝ ↔ 𝑡 ∈ ℝ))
9	8	anbi1d 643	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (𝑡 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
10		oveq1 7426	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝑥↑2) = (𝑡↑2))
11	10	oveq1d 7434	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → ((𝑥↑2) + (𝑦↑2)) = ((𝑡↑2) + (𝑦↑2)))
12	11	breq1d 5121	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2) ↔ ((𝑡↑2) + (𝑦↑2)) ≤ (𝑅↑2)))
13	9, 12	anbi12d 644	. . . . . 6 ⊢ (𝑥 = 𝑡 → (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2)) ↔ ((𝑡 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑡↑2) + (𝑦↑2)) ≤ (𝑅↑2))))
14		eleq1w 2848	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ ℝ ↔ 𝑢 ∈ ℝ))
15	14	anbi2d 642	. . . . . . 7 ⊢ (𝑦 = 𝑢 → ((𝑡 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)))
16		oveq1 7426	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (𝑦↑2) = (𝑢↑2))
17	16	oveq2d 7435	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → ((𝑡↑2) + (𝑦↑2)) = ((𝑡↑2) + (𝑢↑2)))
18	17	breq1d 5121	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (((𝑡↑2) + (𝑦↑2)) ≤ (𝑅↑2) ↔ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))
19	15, 18	anbi12d 644	. . . . . 6 ⊢ (𝑦 = 𝑢 → (((𝑡 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑡↑2) + (𝑦↑2)) ≤ (𝑅↑2)) ↔ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))))
20	3, 7, 13, 19	opelopab 5529	. . . . 5 ⊢ (⟨𝑡, 𝑢⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))} ↔ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))
21		anass 474	. . . . 5 ⊢ (((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)) ↔ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))))
22	6, 20, 21	3bitri 300	. . . 4 ⊢ (𝑡{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}𝑢 ↔ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))))
23	22	abbii 2832	. . 3 ⊢ {𝑢 ∣ 𝑡{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}𝑢} = {𝑢 ∣ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))}
24	2, 5, 23	3eqtri 2792	. 2 ⊢ (𝑆 “ {𝑡}) = {𝑢 ∣ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))}
25		simp3 1156	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
26	25	biantrurd 542	. . . 4 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)) ↔ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))))
27	26	abbidv 2831	. . 3 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = {𝑢 ∣ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))})
28		resqcl 14178	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
29	28	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
30		resqcl 14178	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
31	30	3ad2ant3 1153	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
32	29, 31	resubcld 11657	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
33	32	adantr 486	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
34		absresq 15377	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → ((abs‘𝑡)↑2) = (𝑡↑2))
35	34	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡)↑2) = (𝑡↑2))
36	35	breq1d 5121	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (((abs‘𝑡)↑2) ≤ (𝑅↑2) ↔ (𝑡↑2) ≤ (𝑅↑2)))
37		recn 11205	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
38	37	abscld 15514	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → (abs‘𝑡) ∈ ℝ)
39	38	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (abs‘𝑡) ∈ ℝ)
40		simp1 1154	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑅 ∈ ℝ)
41	37	absge0d 15522	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → 0 ≤ (abs‘𝑡))
42	41	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘𝑡))
43		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ 𝑅)
44	39, 40, 42, 43	le2sqd 14311	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
45	29, 31	subge0d 11819	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (𝑡↑2) ≤ (𝑅↑2)))
46	36, 44, 45	3bitr4d 314	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ 0 ≤ ((𝑅↑2) − (𝑡↑2))))
47	46	biimpa 482	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))
48	33, 47	resqrtcld 15493	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
49	48	renegcld 11656	. . . . . . 7 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
50	49	rexrd 11274	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^*)
51	48	rexrd 11274	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^*)
52		iccval 13427	. . . . . 6 ⊢ ((-(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^* ∧ (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^) → (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))) = {𝑢 ∈ ℝ^ ∣ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))})
53	50, 51, 52	syl2anc 596	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))) = {𝑢 ∈ ℝ^* ∣ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))})
54		iftrue 4495	. . . . . 6 ⊢ ((abs‘𝑡) ≤ 𝑅 → if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅) = (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))))
55	54	adantl 487	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅) = (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))))
56		absresq 15377	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℝ → ((abs‘𝑢)↑2) = (𝑢↑2))
57	32	recnd 11252	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) − (𝑡↑2)) ∈ ℂ)
58	57	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑅↑2) − (𝑡↑2)) ∈ ℂ)
59	58	sqsqrtd 15517	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((√‘((𝑅↑2) − (𝑡↑2)))↑2) = ((𝑅↑2) − (𝑡↑2)))
60	56, 59	breqan12rd 5128	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (((abs‘𝑢)↑2) ≤ ((√‘((𝑅↑2) − (𝑡↑2)))↑2) ↔ (𝑢↑2) ≤ ((𝑅↑2) − (𝑡↑2))))
61		recn 11205	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
62	61	abscld 15514	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → (abs‘𝑢) ∈ ℝ)
63	62	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (abs‘𝑢) ∈ ℝ)
64	48	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
65	61	absge0d 15522	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → 0 ≤ (abs‘𝑢))
66	65	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 0 ≤ (abs‘𝑢))
67	33, 47	sqrtge0d 15496	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → 0 ≤ (√‘((𝑅↑2) − (𝑡↑2))))
68	67	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 0 ≤ (√‘((𝑅↑2) − (𝑡↑2))))
69	63, 64, 66, 68	le2sqd 14311	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → ((abs‘𝑢) ≤ (√‘((𝑅↑2) − (𝑡↑2))) ↔ ((abs‘𝑢)↑2) ≤ ((√‘((𝑅↑2) − (𝑡↑2)))↑2)))
70	31	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
71		resqcl 14178	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → (𝑢↑2) ∈ ℝ)
72	71	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑢↑2) ∈ ℝ)
73	29	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
74	70, 72, 73	leaddsub2d 11831	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2) ↔ (𝑢↑2) ≤ ((𝑅↑2) − (𝑡↑2))))
75	74	adantlr 728	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2) ↔ (𝑢↑2) ≤ ((𝑅↑2) − (𝑡↑2))))
76	60, 69, 75	3bitr4rd 315	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2) ↔ (abs‘𝑢) ≤ (√‘((𝑅↑2) − (𝑡↑2)))))
77		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
78	77, 64	absled 15508	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → ((abs‘𝑢) ≤ (√‘((𝑅↑2) − (𝑡↑2))) ↔ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))))
79		rexr 11270	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℝ^*)
80	79	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ^*)
81	80	biantrurd 542	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → ((-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))) ↔ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))))
82	76, 78, 81	3bitrd 308	. . . . . . . . 9 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2) ↔ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))))
83	82	pm5.32da 590	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)) ↔ (𝑢 ∈ ℝ ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))))))
84		simprl 783	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → 𝑢 ∈ ℝ^*)
85	48	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
86		mnfxr 11281	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
87	86	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → -∞ ∈ ℝ^*)
88	49	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → -(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
89	88	rexrd 11274	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → -(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^*)
90	49	mnfltd 13165	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -∞ < -(√‘((𝑅↑2) − (𝑡↑2))))
91	90	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → -∞ < -(√‘((𝑅↑2) − (𝑡↑2))))
92		simprrl 793	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → -(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢)
93	87, 89, 84, 91, 92	xrltletrd 13202	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → -∞ < 𝑢)
94		simprrr 794	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))
95		xrre 13211	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ ℝ^* ∧ (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ) ∧ (-∞ < 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))) → 𝑢 ∈ ℝ)
96	84, 85, 93, 94, 95	syl22anc 852	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))) → 𝑢 ∈ ℝ)
97	96	ex 418	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))) → 𝑢 ∈ ℝ))
98	97	pm4.71rd 572	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))) ↔ (𝑢 ∈ ℝ ∧ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))))))
99	83, 98	bitr4d 285	. . . . . . 7 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)) ↔ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))))
100	99	abbidv 2831	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = {𝑢 ∣ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))})
101		df-rab 3419	. . . . . 6 ⊢ {𝑢 ∈ ℝ^* ∣ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))} = {𝑢 ∣ (𝑢 ∈ ℝ^* ∧ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2)))))}
102	100, 101	eqtr4di 2818	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = {𝑢 ∈ ℝ^* ∣ (-(√‘((𝑅↑2) − (𝑡↑2))) ≤ 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))})
103	53, 55, 102	3eqtr4rd 2811	. . . 4 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅))
104	40, 39	ltnled 11372	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ ¬ (abs‘𝑡) ≤ 𝑅))
105	104	biimprd 251	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (¬ (abs‘𝑡) ≤ 𝑅 → 𝑅 < (abs‘𝑡)))
106	105	imdistani 579	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ ¬ (abs‘𝑡) ≤ 𝑅) → ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)))
107		df-rab 3419	. . . . . . 7 ⊢ {𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))}
108	29	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
109	31	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
110	71	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑢↑2) ∈ ℝ)
111	109, 110	readdcld 11253	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → ((𝑡↑2) + (𝑢↑2)) ∈ ℝ)
112	40, 39, 43, 42	lt2sqd 14310	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ (𝑅↑2) < ((abs‘𝑡)↑2)))
113	35	breq2d 5123	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) < ((abs‘𝑡)↑2) ↔ (𝑅↑2) < (𝑡↑2)))
114	112, 113	bitrd 282	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ (𝑅↑2) < (𝑡↑2)))
115	114	biimpa 482	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → (𝑅↑2) < (𝑡↑2))
116	115	3adant3 1150	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) < (𝑡↑2))
117		sqge0 14190	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → 0 ≤ (𝑢↑2))
118	117	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → 0 ≤ (𝑢↑2))
119	109, 110	addge01d 11817	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (0 ≤ (𝑢↑2) ↔ (𝑡↑2) ≤ ((𝑡↑2) + (𝑢↑2))))
120	118, 119	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑡↑2) ≤ ((𝑡↑2) + (𝑢↑2)))
121	108, 109, 111, 116, 120	ltletrd 11385	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) < ((𝑡↑2) + (𝑢↑2)))
122	108, 111	ltnled 11372	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → ((𝑅↑2) < ((𝑡↑2) + (𝑢↑2)) ↔ ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))
123	121, 122	mpbid 235	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
124	123	3expa 1136	. . . . . . . . 9 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) ∧ 𝑢 ∈ ℝ) → ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
125	124	ralrimiva 3159	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → ∀𝑢 ∈ ℝ ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
126		rabeq0 4345	. . . . . . . 8 ⊢ ({𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = ∅ ↔ ∀𝑢 ∈ ℝ ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
127	125, 126	sylibr 237	. . . . . . 7 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → {𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = ∅)
128	107, 127	eqtr3id 2814	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = ∅)
129	106, 128	syl 18	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ ¬ (abs‘𝑡) ≤ 𝑅) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = ∅)
130		iffalse 4498	. . . . . 6 ⊢ (¬ (abs‘𝑡) ≤ 𝑅 → if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅) = ∅)
131	130	adantl 487	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ ¬ (abs‘𝑡) ≤ 𝑅) → if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅) = ∅)
132	129, 131	eqtr4d 2803	. . . 4 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ ¬ (abs‘𝑡) ≤ 𝑅) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅))
133	103, 132	pm2.61dan 825	. . 3 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅))
134	27, 133	eqtr3d 2802	. 2 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → {𝑢 ∣ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))} = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅))
135	24, 134	eqtrid 2812	1 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑆 “ {𝑡}) = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 {cab 2743 ∀wral 3081 {crab 3418 Vcvv 3457 ∅c0 4286 ifcif 4489 {csn 4591 ⟨cop 4597 class class class wbr 5111 {copab 5175 “ cima 5666 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 + caddc 11118 -∞cmnf 11256 ℝ^*cxr 11257 < clt 11258 ≤ cle 11259 − cmin 11456 -cneg 11457 2c2 12310 [,]cicc 13391 ↑cexp 14115 √csqrt 15308 abscabs 15309

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-sup 9409 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-n0 12520 df-z 12607 df-uz 12879 df-rp 13033 df-icc 13395 df-seq 14056 df-exp 14116 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311

This theorem is used by: areacirc 38421

W3C validator