areacirclem5 - Mathbox for Brendan Leahy

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Theorem areacirclem5 38172

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 6042	. . 3 ⊢ (𝑆 “ {𝑡}) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))} “ {𝑡})
3		vex 3457	. . . 4 ⊢ 𝑡 ∈ V
4		imasng 6069	. . . 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 5098	. . . . 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 3457	. . . . . 6 ⊢ 𝑢 ∈ V
8		eleq1w 2844	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ ℝ ↔ 𝑡 ∈ ℝ))
9	8	anbi1d 640	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (𝑡 ∈ ℝ ∧ 𝑦 ∈ ℝ)))
10		oveq1 7398	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (𝑥↑2) = (𝑡↑2))
11	10	oveq1d 7406	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → ((𝑥↑2) + (𝑦↑2)) = ((𝑡↑2) + (𝑦↑2)))
12	11	breq1d 5107	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2) ↔ ((𝑡↑2) + (𝑦↑2)) ≤ (𝑅↑2)))
13	9, 12	anbi12d 641	. . . . . 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 2844	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ ℝ ↔ 𝑢 ∈ ℝ))
15	14	anbi2d 639	. . . . . . 7 ⊢ (𝑦 = 𝑢 → ((𝑡 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)))
16		oveq1 7398	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (𝑦↑2) = (𝑢↑2))
17	16	oveq2d 7407	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → ((𝑡↑2) + (𝑦↑2)) = ((𝑡↑2) + (𝑢↑2)))
18	17	breq1d 5107	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (((𝑡↑2) + (𝑦↑2)) ≤ (𝑅↑2) ↔ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))
19	15, 18	anbi12d 641	. . . . . 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 5509	. . . . 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 472	. . . . 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 299	. . . 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 2828	. . 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 2788	. 2 ⊢ (𝑆 “ {𝑡}) = {𝑢 ∣ (𝑡 ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)))}
25		simp3 1150	. . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
26	25	biantrurd 540	. . . 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 2827	. . 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 14131	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
29	28	3ad2ant1 1145	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
30		resqcl 14131	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
31	30	3ad2ant3 1147	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
32	29, 31	resubcld 11609	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
33	32	adantr 484	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ)
34		absresq 15320	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → ((abs‘𝑡)↑2) = (𝑡↑2))
35	34	3ad2ant3 1147	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡)↑2) = (𝑡↑2))
36	35	breq1d 5107	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (((abs‘𝑡)↑2) ≤ (𝑅↑2) ↔ (𝑡↑2) ≤ (𝑅↑2)))
37		recn 11157	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
38	37	abscld 15457	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → (abs‘𝑡) ∈ ℝ)
39	38	3ad2ant3 1147	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (abs‘𝑡) ∈ ℝ)
40		simp1 1148	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 𝑅 ∈ ℝ)
41	37	absge0d 15465	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ ℝ → 0 ≤ (abs‘𝑡))
42	41	3ad2ant3 1147	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘𝑡))
43		simp2 1149	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ 𝑅)
44	39, 40, 42, 43	le2sqd 14264	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2)))
45	29, 31	subge0d 11771	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (𝑡↑2) ≤ (𝑅↑2)))
46	36, 44, 45	3bitr4d 313	. . . . . . . . . 10 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ 0 ≤ ((𝑅↑2) − (𝑡↑2))))
47	46	biimpa 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))
48	33, 47	resqrtcld 15436	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
49	48	renegcld 11608	. . . . . . 7 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
50	49	rexrd 11226	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -(√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^*)
51	48	rexrd 11226	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ^*)
52		iccval 13382	. . . . . 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 593	. . . . 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 4483	. . . . . 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 485	. . . . 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 15320	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℝ → ((abs‘𝑢)↑2) = (𝑢↑2))
57	32	recnd 11204	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) − (𝑡↑2)) ∈ ℂ)
58	57	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((𝑅↑2) − (𝑡↑2)) ∈ ℂ)
59	58	sqsqrtd 15460	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → ((√‘((𝑅↑2) − (𝑡↑2)))↑2) = ((𝑅↑2) − (𝑡↑2)))
60	56, 59	breqan12rd 5114	. . . . . . . . . . 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 11157	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
62	61	abscld 15457	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → (abs‘𝑢) ∈ ℝ)
63	62	adantl 485	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (abs‘𝑢) ∈ ℝ)
64	48	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ)
65	61	absge0d 15465	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → 0 ≤ (abs‘𝑢))
66	65	adantl 485	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 0 ≤ (abs‘𝑢))
67	33, 47	sqrtge0d 15439	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → 0 ≤ (√‘((𝑅↑2) − (𝑡↑2))))
68	67	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 0 ≤ (√‘((𝑅↑2) − (𝑡↑2))))
69	63, 64, 66, 68	le2sqd 14264	. . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
71		resqcl 14131	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → (𝑢↑2) ∈ ℝ)
72	71	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑢↑2) ∈ ℝ)
73	29	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
74	70, 72, 73	leaddsub2d 11783	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑢 ∈ ℝ) → (((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2) ↔ (𝑢↑2) ≤ ((𝑅↑2) − (𝑡↑2))))
75	74	adantlr 725	. . . . . . . . . . 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 314	. . . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
78	77, 64	absled 15451	. . . . . . . . . 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 11222	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℝ^*)
80	79	adantl 485	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ^*)
81	80	biantrurd 540	. . . . . . . . . 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 307	. . . . . . . . 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 587	. . . . . . . 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 780	. . . . . . . . . . 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 484	. . . . . . . . . . 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 11233	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 11226	. . . . . . . . . . . 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 13120	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ (abs‘𝑡) ≤ 𝑅) → -∞ < -(√‘((𝑅↑2) − (𝑡↑2))))
91	90	adantr 484	. . . . . . . . . . . 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 790	. . . . . . . . . . . 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 13157	. . . . . . . . . . 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 791	. . . . . . . . . . 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 13166	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ ℝ^* ∧ (√‘((𝑅↑2) − (𝑡↑2))) ∈ ℝ) ∧ (-∞ < 𝑢 ∧ 𝑢 ≤ (√‘((𝑅↑2) − (𝑡↑2))))) → 𝑢 ∈ ℝ)
96	84, 85, 93, 94, 95	syl22anc 849	. . . . . . . . . 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 416	. . . . . . . . 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 570	. . . . . . . 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 284	. . . . . . 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 2827	. . . . . 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 3414	. . . . . 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 2814	. . . . 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 2807	. . . 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 11324	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ ¬ (abs‘𝑡) ≤ 𝑅))
105	104	biimprd 250	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (¬ (abs‘𝑡) ≤ 𝑅 → 𝑅 < (abs‘𝑡)))
106	105	imdistani 576	. . . . . 6 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ ¬ (abs‘𝑡) ≤ 𝑅) → ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)))
107		df-rab 3414	. . . . . . 7 ⊢ {𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = {𝑢 ∣ (𝑢 ∈ ℝ ∧ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))}
108	29	3ad2ant1 1145	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) ∈ ℝ)
109	31	3ad2ant1 1145	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑡↑2) ∈ ℝ)
110	71	3ad2ant3 1147	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑢↑2) ∈ ℝ)
111	109, 110	readdcld 11205	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → ((𝑡↑2) + (𝑢↑2)) ∈ ℝ)
112	40, 39, 43, 42	lt2sqd 14263	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ (𝑅↑2) < ((abs‘𝑡)↑2)))
113	35	breq2d 5109	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → ((𝑅↑2) < ((abs‘𝑡)↑2) ↔ (𝑅↑2) < (𝑡↑2)))
114	112, 113	bitrd 281	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅 < (abs‘𝑡) ↔ (𝑅↑2) < (𝑡↑2)))
115	114	biimpa 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → (𝑅↑2) < (𝑡↑2))
116	115	3adant3 1144	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) < (𝑡↑2))
117		sqge0 14143	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ℝ → 0 ≤ (𝑢↑2))
118	117	3ad2ant3 1147	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → 0 ≤ (𝑢↑2))
119	109, 110	addge01d 11769	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 11337	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → (𝑅↑2) < ((𝑡↑2) + (𝑢↑2)))
122	108, 111	ltnled 11324	. . . . . . . . . . 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 234	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡) ∧ 𝑢 ∈ ℝ) → ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
124	123	3expa 1130	. . . . . . . . 9 ⊢ ((((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) ∧ 𝑢 ∈ ℝ) → ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
125	124	ralrimiva 3153	. . . . . . . 8 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → ∀𝑢 ∈ ℝ ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
126		rabeq0 4339	. . . . . . . 8 ⊢ ({𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = ∅ ↔ ∀𝑢 ∈ ℝ ¬ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2))
127	125, 126	sylibr 236	. . . . . . 7 ⊢ (((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) ∧ 𝑅 < (abs‘𝑡)) → {𝑢 ∈ ℝ ∣ ((𝑡↑2) + (𝑢↑2)) ≤ (𝑅↑2)} = ∅)
128	107, 127	eqtr3id 2810	. . . . . 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 4486	. . . . . 6 ⊢ (¬ (abs‘𝑡) ≤ 𝑅 → if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅) = ∅)
131	130	adantl 485	. . . . 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 2799	. . . 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 822	. . 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 2798	. 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 2808	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 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 {cab 2739 ∀wral 3075 {crab 3413 Vcvv 3453 ∅c0 4283 ifcif 4477 {csn 4579 ⟨cop 4585 class class class wbr 5097 {copab 5159 “ cima 5646 ‘cfv 6516 (class class class)co 7391 ℂcc 11065 ℝcr 11066 0cc0 11067 + caddc 11070 -∞cmnf 11208 ℝ^*cxr 11209 < clt 11210 ≤ cle 11211 − cmin 11408 -cneg 11409 2c2 12266 [,]cicc 13346 ↑cexp 14068 √csqrt 15251 abscabs 15252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9382 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-n0 12476 df-z 12563 df-uz 12834 df-rp 12988 df-icc 13350 df-seq 14009 df-exp 14069 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254

This theorem is referenced by: areacirc 38173

W3C validator