sqsscirc1 - Mathbox for Thierry Arnoux

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Theorem sqsscirc1 32883

Description: The complex square of side 𝐷 is a subset of the complex circle of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.)

**Assertion**
Ref	Expression
sqsscirc1	⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷))

**Proof of Theorem sqsscirc1**
Step	Hyp	Ref	Expression
1		simp-4l 781	. . . . . 6 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝑋 ∈ ℝ)
2	1	resqcld 14089	. . . . 5 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑋↑2) ∈ ℝ)
3		simpllr 774	. . . . . . 7 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌))
4	3	simpld 495	. . . . . 6 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝑌 ∈ ℝ)
5	4	resqcld 14089	. . . . 5 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑌↑2) ∈ ℝ)
6	2, 5	readdcld 11242	. . . 4 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → ((𝑋↑2) + (𝑌↑2)) ∈ ℝ)
7	1	sqge0d 14101	. . . . 5 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ (𝑋↑2))
8	4	sqge0d 14101	. . . . 5 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ (𝑌↑2))
9	2, 5, 7, 8	addge0d 11789	. . . 4 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ ((𝑋↑2) + (𝑌↑2)))
10	6, 9	resqrtcld 15363	. . 3 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (√‘((𝑋↑2) + (𝑌↑2))) ∈ ℝ)
11		simplr 767	. . . . . . . 8 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝐷 ∈ ℝ⁺)
12	11	rpred 13015	. . . . . . 7 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝐷 ∈ ℝ)
13	12	rehalfcld 12458	. . . . . 6 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝐷 / 2) ∈ ℝ)
14	13	resqcld 14089	. . . . 5 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → ((𝐷 / 2)↑2) ∈ ℝ)
15	14, 14	readdcld 11242	. . . 4 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (((𝐷 / 2)↑2) + ((𝐷 / 2)↑2)) ∈ ℝ)
16	13	sqge0d 14101	. . . . 5 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ ((𝐷 / 2)↑2))
17	14, 14, 16, 16	addge0d 11789	. . . 4 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ (((𝐷 / 2)↑2) + ((𝐷 / 2)↑2)))
18	15, 17	resqrtcld 15363	. . 3 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (√‘(((𝐷 / 2)↑2) + ((𝐷 / 2)↑2))) ∈ ℝ)
19		simprl 769	. . . . . 6 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝑋 < (𝐷 / 2))
20		simp-4r 782	. . . . . . 7 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ 𝑋)
21		2rp 12978	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
22	21	a1i 11	. . . . . . . 8 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 2 ∈ ℝ⁺)
23	11	rpge0d 13019	. . . . . . . 8 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ 𝐷)
24	12, 22, 23	divge0d 13055	. . . . . . 7 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ (𝐷 / 2))
25	1, 13, 20, 24	lt2sqd 14218	. . . . . 6 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑋 < (𝐷 / 2) ↔ (𝑋↑2) < ((𝐷 / 2)↑2)))
26	19, 25	mpbid 231	. . . . 5 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑋↑2) < ((𝐷 / 2)↑2))
27		simprr 771	. . . . . 6 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝑌 < (𝐷 / 2))
28	3	simprd 496	. . . . . . 7 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ 𝑌)
29	4, 13, 28, 24	lt2sqd 14218	. . . . . 6 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑌 < (𝐷 / 2) ↔ (𝑌↑2) < ((𝐷 / 2)↑2)))
30	27, 29	mpbid 231	. . . . 5 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑌↑2) < ((𝐷 / 2)↑2))
31	2, 5, 14, 14, 26, 30	lt2addd 11836	. . . 4 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → ((𝑋↑2) + (𝑌↑2)) < (((𝐷 / 2)↑2) + ((𝐷 / 2)↑2)))
32	6, 9, 15, 17	sqrtltd 15373	. . . 4 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (((𝑋↑2) + (𝑌↑2)) < (((𝐷 / 2)↑2) + ((𝐷 / 2)↑2)) ↔ (√‘((𝑋↑2) + (𝑌↑2))) < (√‘(((𝐷 / 2)↑2) + ((𝐷 / 2)↑2)))))
33	31, 32	mpbid 231	. . 3 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (√‘((𝑋↑2) + (𝑌↑2))) < (√‘(((𝐷 / 2)↑2) + ((𝐷 / 2)↑2))))
34		rpre 12981	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ℝ⁺ → 𝐷 ∈ ℝ)
35	34	rehalfcld 12458	. . . . . . . . . 10 ⊢ (𝐷 ∈ ℝ⁺ → (𝐷 / 2) ∈ ℝ)
36	35	resqcld 14089	. . . . . . . . 9 ⊢ (𝐷 ∈ ℝ⁺ → ((𝐷 / 2)↑2) ∈ ℝ)
37	36	recnd 11241	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → ((𝐷 / 2)↑2) ∈ ℂ)
38	37	2timesd 12454	. . . . . . 7 ⊢ (𝐷 ∈ ℝ⁺ → (2 · ((𝐷 / 2)↑2)) = (((𝐷 / 2)↑2) + ((𝐷 / 2)↑2)))
39	38	fveq2d 6895	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → (√‘(2 · ((𝐷 / 2)↑2))) = (√‘(((𝐷 / 2)↑2) + ((𝐷 / 2)↑2))))
40	21	a1i 11	. . . . . . . . . 10 ⊢ (𝐷 ∈ ℝ⁺ → 2 ∈ ℝ⁺)
41		rpge0 12986	. . . . . . . . . 10 ⊢ (𝐷 ∈ ℝ⁺ → 0 ≤ 𝐷)
42	34, 40, 41	divge0d 13055	. . . . . . . . 9 ⊢ (𝐷 ∈ ℝ⁺ → 0 ≤ (𝐷 / 2))
43	35, 42	sqrtsqd 15365	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → (√‘((𝐷 / 2)↑2)) = (𝐷 / 2))
44	43	oveq2d 7424	. . . . . . 7 ⊢ (𝐷 ∈ ℝ⁺ → ((√‘2) · (√‘((𝐷 / 2)↑2))) = ((√‘2) · (𝐷 / 2)))
45		2re 12285	. . . . . . . . 9 ⊢ 2 ∈ ℝ
46	45	a1i 11	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → 2 ∈ ℝ)
47		0le2 12313	. . . . . . . . 9 ⊢ 0 ≤ 2
48	47	a1i 11	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → 0 ≤ 2)
49	35	sqge0d 14101	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → 0 ≤ ((𝐷 / 2)↑2))
50	46, 48, 36, 49	sqrtmuld 15370	. . . . . . 7 ⊢ (𝐷 ∈ ℝ⁺ → (√‘(2 · ((𝐷 / 2)↑2))) = ((√‘2) · (√‘((𝐷 / 2)↑2))))
51		2cnd 12289	. . . . . . . . 9 ⊢ (𝐷 ∈ ℝ⁺ → 2 ∈ ℂ)
52	51	sqrtcld 15383	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → (√‘2) ∈ ℂ)
53		rpcn 12983	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → 𝐷 ∈ ℂ)
54		2ne0 12315	. . . . . . . . 9 ⊢ 2 ≠ 0
55	54	a1i 11	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → 2 ≠ 0)
56	52, 51, 53, 55	div32d 12012	. . . . . . 7 ⊢ (𝐷 ∈ ℝ⁺ → (((√‘2) / 2) · 𝐷) = ((√‘2) · (𝐷 / 2)))
57	44, 50, 56	3eqtr4d 2782	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → (√‘(2 · ((𝐷 / 2)↑2))) = (((√‘2) / 2) · 𝐷))
58	39, 57	eqtr3d 2774	. . . . 5 ⊢ (𝐷 ∈ ℝ⁺ → (√‘(((𝐷 / 2)↑2) + ((𝐷 / 2)↑2))) = (((√‘2) / 2) · 𝐷))
59		2lt4 12386	. . . . . . . . . 10 ⊢ 2 < 4
60		4re 12295	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
61		0re 11215	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
62		4pos 12318	. . . . . . . . . . . 12 ⊢ 0 < 4
63	61, 60, 62	ltleii 11336	. . . . . . . . . . 11 ⊢ 0 ≤ 4
64		sqrtlt 15207	. . . . . . . . . . 11 ⊢ (((2 ∈ ℝ ∧ 0 ≤ 2) ∧ (4 ∈ ℝ ∧ 0 ≤ 4)) → (2 < 4 ↔ (√‘2) < (√‘4)))
65	45, 47, 60, 63, 64	mp4an 691	. . . . . . . . . 10 ⊢ (2 < 4 ↔ (√‘2) < (√‘4))
66	59, 65	mpbi 229	. . . . . . . . 9 ⊢ (√‘2) < (√‘4)
67		2pos 12314	. . . . . . . . . . 11 ⊢ 0 < 2
68	45, 67	sqrtpclii 15328	. . . . . . . . . 10 ⊢ (√‘2) ∈ ℝ
69	60, 62	sqrtpclii 15328	. . . . . . . . . 10 ⊢ (√‘4) ∈ ℝ
70	68, 69, 45, 67	ltdiv1ii 12142	. . . . . . . . 9 ⊢ ((√‘2) < (√‘4) ↔ ((√‘2) / 2) < ((√‘4) / 2))
71	66, 70	mpbi 229	. . . . . . . 8 ⊢ ((√‘2) / 2) < ((√‘4) / 2)
72		sqrtsq 15215	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (√‘(2↑2)) = 2)
73	45, 47, 72	mp2an 690	. . . . . . . . . 10 ⊢ (√‘(2↑2)) = 2
74	73	oveq1i 7418	. . . . . . . . 9 ⊢ ((√‘(2↑2)) / 2) = (2 / 2)
75		sq2 14160	. . . . . . . . . . 11 ⊢ (2↑2) = 4
76	75	fveq2i 6894	. . . . . . . . . 10 ⊢ (√‘(2↑2)) = (√‘4)
77	76	oveq1i 7418	. . . . . . . . 9 ⊢ ((√‘(2↑2)) / 2) = ((√‘4) / 2)
78		2div2e1 12352	. . . . . . . . 9 ⊢ (2 / 2) = 1
79	74, 77, 78	3eqtr3i 2768	. . . . . . . 8 ⊢ ((√‘4) / 2) = 1
80	71, 79	breqtri 5173	. . . . . . 7 ⊢ ((√‘2) / 2) < 1
81	46, 48	resqrtcld 15363	. . . . . . . . 9 ⊢ (𝐷 ∈ ℝ⁺ → (√‘2) ∈ ℝ)
82	81	rehalfcld 12458	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → ((√‘2) / 2) ∈ ℝ)
83		1red 11214	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → 1 ∈ ℝ)
84		id 22	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ⁺ → 𝐷 ∈ ℝ⁺)
85	82, 83, 84	ltmul1d 13056	. . . . . . 7 ⊢ (𝐷 ∈ ℝ⁺ → (((√‘2) / 2) < 1 ↔ (((√‘2) / 2) · 𝐷) < (1 · 𝐷)))
86	80, 85	mpbii 232	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → (((√‘2) / 2) · 𝐷) < (1 · 𝐷))
87	53	mullidd 11231	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → (1 · 𝐷) = 𝐷)
88	86, 87	breqtrd 5174	. . . . 5 ⊢ (𝐷 ∈ ℝ⁺ → (((√‘2) / 2) · 𝐷) < 𝐷)
89	58, 88	eqbrtrd 5170	. . . 4 ⊢ (𝐷 ∈ ℝ⁺ → (√‘(((𝐷 / 2)↑2) + ((𝐷 / 2)↑2))) < 𝐷)
90	11, 89	syl 17	. . 3 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (√‘(((𝐷 / 2)↑2) + ((𝐷 / 2)↑2))) < 𝐷)
91	10, 18, 12, 33, 90	lttrd 11374	. 2 ⊢ (((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷)
92	91	ex 413	1 ⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ⁺) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 < clt 11247 ≤ cle 11248 / cdiv 11870 2c2 12266 4c4 12268 ℝ⁺crp 12973 ↑cexp 14026 √csqrt 15179

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182

This theorem is referenced by: sqsscirc2 32884

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