areacirclem5 - Mathbox for Brendan Leahy

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Theorem areacirclem5 37672

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 6086	. . 3 ⊢ (𝑆 “ {𝑡}) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))} “ {𝑡})
3		vex 3492	. . . 4 ⊢ 𝑡 ∈ V
4		imasng 6113	. . . 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 5167	. . . . 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 3492	. . . . . 6 ⊢ 𝑢 ∈ V
8		eleq1w 2827	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ ℝ ↔ 𝑡 ∈ ℝ))
9	8	anbi1d 630	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (𝑡 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
10		oveq1 7455	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝑥↑2) = (𝑡↑2))
11	10	oveq1d 7463	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → ((𝑥↑2) + (𝑦↑2)) = ((𝑡↑2) + (𝑦↑2)))
12	11	breq1d 5176	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2) ↔ ((𝑡↑2) + (𝑦↑2)) ≤ (𝑅↑2)))
13	9, 12	anbi12d 631	. . . . . 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 2827	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ ℝ ↔ 𝑢 ∈ ℝ))
15	14	anbi2d 629	. . . . . . 7 ⊢ (𝑦 = 𝑢 → ((𝑡 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)))
16		oveq1 7455	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (𝑦↑2) = (𝑢↑2))
17	16	oveq2d 7464	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → ((𝑡↑2) + (𝑦↑2)) = ((𝑡↑2) + (𝑢↑2)))
18	17	breq1d 5176	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (((𝑡↑2) + (𝑦↑2)) ≤ (𝑅↑2) ↔ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))
19	15, 18	anbi12d 631	. . . . . 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 5561	. . . . 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 468	. . . . 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 297	. . . 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 2812	. . 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 2772	. 2 ⊢ (𝑆 “ {𝑡}) = {𝑢 ∣ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))}
25		simp3 1138	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
26	25	biantrurd 532	. . . 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 2811	. . 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 14174	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
29	28	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
30		resqcl 14174	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
31	30	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
32	29, 31	resubcld 11718	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
33	32	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
34		absresq 15351	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → ((abs‘𝑡)↑2) = (𝑡↑2))
35	34	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡)↑2) = (𝑡↑2))
36	35	breq1d 5176	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (((abs‘𝑡)↑2) ≤ (𝑅↑2) ↔ (𝑡↑2) ≤ (𝑅↑2)))
37		recn 11274	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
38	37	abscld 15485	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → (abs‘𝑡) ∈ ℝ)
39	38	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (abs‘𝑡) ∈ ℝ)
40		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑅 ∈ ℝ)
41	37	absge0d 15493	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → 0 ≤ (abs‘𝑡))
42	41	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘𝑡))
43		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ 𝑅)
44	39, 40, 42, 43	le2sqd 14306	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
45	29, 31	subge0d 11880	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (𝑡↑2) ≤ (𝑅↑2)))
46	36, 44, 45	3bitr4d 311	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ 0 ≤ ((𝑅↑2) − (𝑡↑2))))
47	46	biimpa 476	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))
48	33, 47	resqrtcld 15466	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
49	48	renegcld 11717	. . . . . . 7 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
50	49	rexrd 11340	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^*)
51	48	rexrd 11340	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^*)
52		iccval 13446	. . . . . 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 583	. . . . 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 4554	. . . . . 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 481	. . . . 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 15351	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℝ → ((abs‘𝑢)↑2) = (𝑢↑2))
57	32	recnd 11318	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) − (𝑡↑2)) ∈ ℂ)
58	57	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑅↑2) − (𝑡↑2)) ∈ ℂ)
59	58	sqsqrtd 15488	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((√‘((𝑅↑2) − (𝑡↑2)))↑2) = ((𝑅↑2) − (𝑡↑2)))
60	56, 59	breqan12rd 5183	. . . . . . . . . . 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 11274	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
62	61	abscld 15485	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → (abs‘𝑢) ∈ ℝ)
63	62	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (abs‘𝑢) ∈ ℝ)
64	48	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
65	61	absge0d 15493	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → 0 ≤ (abs‘𝑢))
66	65	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 0 ≤ (abs‘𝑢))
67	33, 47	sqrtge0d 15469	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → 0 ≤ (√‘((𝑅↑2) − (𝑡↑2))))
68	67	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 0 ≤ (√‘((𝑅↑2) − (𝑡↑2))))
69	63, 64, 66, 68	le2sqd 14306	. . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
71		resqcl 14174	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → (𝑢↑2) ∈ ℝ)
72	71	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑢↑2) ∈ ℝ)
73	29	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
74	70, 72, 73	leaddsub2d 11892	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2) ↔ (𝑢↑2) ≤ ((𝑅↑2) − (𝑡↑2))))
75	74	adantlr 714	. . . . . . . . . . 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 312	. . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
78	77, 64	absled 15479	. . . . . . . . . 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 11336	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℝ^*)
80	79	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ^*)
81	80	biantrurd 532	. . . . . . . . . 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 305	. . . . . . . . 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 578	. . . . . . . 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 770	. . . . . . . . . . 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 480	. . . . . . . . . . 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 11347	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 11340	. . . . . . . . . . . 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 13187	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -∞ < -(√‘((𝑅↑2) − (𝑡↑2))))
91	90	adantr 480	. . . . . . . . . . . 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 780	. . . . . . . . . . . 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 13223	. . . . . . . . . . 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 781	. . . . . . . . . . 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 13231	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ ℝ^* ∧ (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ) ∧ (-∞ < 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))) → 𝑢 ∈ ℝ)
96	84, 85, 93, 94, 95	syl22anc 838	. . . . . . . . . 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 412	. . . . . . . . 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 562	. . . . . . . 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 282	. . . . . . 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 2811	. . . . . 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 3444	. . . . . 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 2798	. . . . 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 2791	. . . 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 11437	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ ¬ (abs‘𝑡) ≤ 𝑅))
105	104	biimprd 248	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (¬ (abs‘𝑡) ≤ 𝑅 → 𝑅 < (abs‘𝑡)))
106	105	imdistani 568	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ ¬ (abs‘𝑡) ≤ 𝑅) → ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)))
107		df-rab 3444	. . . . . . 7 ⊢ {𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))}
108	29	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
109	31	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
110	71	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑢↑2) ∈ ℝ)
111	109, 110	readdcld 11319	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → ((𝑡↑2) + (𝑢↑2)) ∈ ℝ)
112	40, 39, 43, 42	lt2sqd 14305	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ (𝑅↑2) < ((abs‘𝑡)↑2)))
113	35	breq2d 5178	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) < ((abs‘𝑡)↑2) ↔ (𝑅↑2) < (𝑡↑2)))
114	112, 113	bitrd 279	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ (𝑅↑2) < (𝑡↑2)))
115	114	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → (𝑅↑2) < (𝑡↑2))
116	115	3adant3 1132	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) < (𝑡↑2))
117		sqge0 14186	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → 0 ≤ (𝑢↑2))
118	117	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → 0 ≤ (𝑢↑2))
119	109, 110	addge01d 11878	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 11450	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) < ((𝑡↑2) + (𝑢↑2)))
122	108, 111	ltnled 11437	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
124	123	3expa 1118	. . . . . . . . 9 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) ∧ 𝑢 ∈ ℝ) → ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
125	124	ralrimiva 3152	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → ∀𝑢 ∈ ℝ ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
126		rabeq0 4411	. . . . . . . 8 ⊢ ({𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = ∅ ↔ ∀𝑢 ∈ ℝ ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
127	125, 126	sylibr 234	. . . . . . 7 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → {𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = ∅)
128	107, 127	eqtr3id 2794	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = ∅)
129	106, 128	syl 17	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ ¬ (abs‘𝑡) ≤ 𝑅) → {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))} = ∅)
130		iffalse 4557	. . . . . 6 ⊢ (¬ (abs‘𝑡) ≤ 𝑅 → if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅) = ∅)
131	130	adantl 481	. . . . 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 2783	. . . 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 812	. . 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 2782	. 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 2792	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 {cab 2717 ∀wral 3067 {crab 3443 Vcvv 3488 ∅c0 4352 ifcif 4548 {csn 4648 ⟨cop 4654 class class class wbr 5166 {copab 5228 “ cima 5703 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 + caddc 11187 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 2c2 12348 [,]cicc 13410 ↑cexp 14112 √csqrt 15282 abscabs 15283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-icc 13414 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285

This theorem is referenced by: areacirc 37673

W3C validator