Proof of Theorem sqsscirc1
Step | Hyp | Ref
| Expression |
1 | | simp-4l 783 |
. . . . . 6
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝑋 ∈ ℝ) |
2 | 1 | resqcld 13817 |
. . . . 5
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑋↑2) ∈ ℝ) |
3 | | simpllr 776 |
. . . . . . 7
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) |
4 | 3 | simpld 498 |
. . . . . 6
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝑌 ∈ ℝ) |
5 | 4 | resqcld 13817 |
. . . . 5
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑌↑2) ∈ ℝ) |
6 | 2, 5 | readdcld 10862 |
. . . 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 13818 |
. . . . 5
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ (𝑋↑2)) |
8 | 4 | sqge0d 13818 |
. . . . 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 11408 |
. . . 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 14981 |
. . 3
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (√‘((𝑋↑2) + (𝑌↑2))) ∈ ℝ) |
11 | | simplr 769 |
. . . . . . . 8
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝐷 ∈
ℝ+) |
12 | 11 | rpred 12628 |
. . . . . . 7
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝐷 ∈ ℝ) |
13 | 12 | rehalfcld 12077 |
. . . . . 6
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝐷 / 2) ∈ ℝ) |
14 | 13 | resqcld 13817 |
. . . . 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 10862 |
. . . 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 13818 |
. . . . 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 11408 |
. . . 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 14981 |
. . 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 771 |
. . . . . 6
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝑋 < (𝐷 / 2)) |
20 | | simp-4r 784 |
. . . . . . 7
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ 𝑋) |
21 | | 2rp 12591 |
. . . . . . . . 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 12632 |
. . . . . . . 8
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 0 ≤ 𝐷) |
24 | 12, 22, 23 | divge0d 12668 |
. . . . . . 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 13825 |
. . . . . 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 235 |
. . . . 5
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → (𝑋↑2) < ((𝐷 / 2)↑2)) |
27 | | simprr 773 |
. . . . . 6
⊢
(((((𝑋 ∈
ℝ ∧ 0 ≤ 𝑋)
∧ (𝑌 ∈ ℝ
∧ 0 ≤ 𝑌)) ∧
𝐷 ∈
ℝ+) ∧ (𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2))) → 𝑌 < (𝐷 / 2)) |
28 | 3 | simprd 499 |
. . . . . . 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 13825 |
. . . . . 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 235 |
. . . . 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 11455 |
. . . 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 14991 |
. . . 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 235 |
. . 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 12594 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ ℝ+
→ 𝐷 ∈
ℝ) |
35 | 34 | rehalfcld 12077 |
. . . . . . . . . 10
⊢ (𝐷 ∈ ℝ+
→ (𝐷 / 2) ∈
ℝ) |
36 | 35 | resqcld 13817 |
. . . . . . . . 9
⊢ (𝐷 ∈ ℝ+
→ ((𝐷 / 2)↑2)
∈ ℝ) |
37 | 36 | recnd 10861 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ ((𝐷 / 2)↑2)
∈ ℂ) |
38 | 37 | 2timesd 12073 |
. . . . . . 7
⊢ (𝐷 ∈ ℝ+
→ (2 · ((𝐷 /
2)↑2)) = (((𝐷 /
2)↑2) + ((𝐷 /
2)↑2))) |
39 | 38 | fveq2d 6721 |
. . . . . 6
⊢ (𝐷 ∈ ℝ+
→ (√‘(2 · ((𝐷 / 2)↑2))) = (√‘(((𝐷 / 2)↑2) + ((𝐷 /
2)↑2)))) |
40 | 21 | a1i 11 |
. . . . . . . . . 10
⊢ (𝐷 ∈ ℝ+
→ 2 ∈ ℝ+) |
41 | | rpge0 12599 |
. . . . . . . . . 10
⊢ (𝐷 ∈ ℝ+
→ 0 ≤ 𝐷) |
42 | 34, 40, 41 | divge0d 12668 |
. . . . . . . . 9
⊢ (𝐷 ∈ ℝ+
→ 0 ≤ (𝐷 /
2)) |
43 | 35, 42 | sqrtsqd 14983 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ (√‘((𝐷 /
2)↑2)) = (𝐷 /
2)) |
44 | 43 | oveq2d 7229 |
. . . . . . 7
⊢ (𝐷 ∈ ℝ+
→ ((√‘2) · (√‘((𝐷 / 2)↑2))) = ((√‘2)
· (𝐷 /
2))) |
45 | | 2re 11904 |
. . . . . . . . 9
⊢ 2 ∈
ℝ |
46 | 45 | a1i 11 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ 2 ∈ ℝ) |
47 | | 0le2 11932 |
. . . . . . . . 9
⊢ 0 ≤
2 |
48 | 47 | a1i 11 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ 0 ≤ 2) |
49 | 35 | sqge0d 13818 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ 0 ≤ ((𝐷 /
2)↑2)) |
50 | 46, 48, 36, 49 | sqrtmuld 14988 |
. . . . . . 7
⊢ (𝐷 ∈ ℝ+
→ (√‘(2 · ((𝐷 / 2)↑2))) = ((√‘2)
· (√‘((𝐷
/ 2)↑2)))) |
51 | | 2cnd 11908 |
. . . . . . . . 9
⊢ (𝐷 ∈ ℝ+
→ 2 ∈ ℂ) |
52 | 51 | sqrtcld 15001 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ (√‘2) ∈ ℂ) |
53 | | rpcn 12596 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ 𝐷 ∈
ℂ) |
54 | | 2ne0 11934 |
. . . . . . . . 9
⊢ 2 ≠
0 |
55 | 54 | a1i 11 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ 2 ≠ 0) |
56 | 52, 51, 53, 55 | div32d 11631 |
. . . . . . 7
⊢ (𝐷 ∈ ℝ+
→ (((√‘2) / 2) · 𝐷) = ((√‘2) · (𝐷 / 2))) |
57 | 44, 50, 56 | 3eqtr4d 2787 |
. . . . . 6
⊢ (𝐷 ∈ ℝ+
→ (√‘(2 · ((𝐷 / 2)↑2))) = (((√‘2) / 2)
· 𝐷)) |
58 | 39, 57 | eqtr3d 2779 |
. . . . 5
⊢ (𝐷 ∈ ℝ+
→ (√‘(((𝐷
/ 2)↑2) + ((𝐷 /
2)↑2))) = (((√‘2) / 2) · 𝐷)) |
59 | | 2lt4 12005 |
. . . . . . . . . 10
⊢ 2 <
4 |
60 | | 4re 11914 |
. . . . . . . . . . 11
⊢ 4 ∈
ℝ |
61 | | 0re 10835 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
62 | | 4pos 11937 |
. . . . . . . . . . . 12
⊢ 0 <
4 |
63 | 61, 60, 62 | ltleii 10955 |
. . . . . . . . . . 11
⊢ 0 ≤
4 |
64 | | sqrtlt 14825 |
. . . . . . . . . . 11
⊢ (((2
∈ ℝ ∧ 0 ≤ 2) ∧ (4 ∈ ℝ ∧ 0 ≤ 4)) →
(2 < 4 ↔ (√‘2) < (√‘4))) |
65 | 45, 47, 60, 63, 64 | mp4an 693 |
. . . . . . . . . 10
⊢ (2 < 4
↔ (√‘2) < (√‘4)) |
66 | 59, 65 | mpbi 233 |
. . . . . . . . 9
⊢
(√‘2) < (√‘4) |
67 | | 2pos 11933 |
. . . . . . . . . . 11
⊢ 0 <
2 |
68 | 45, 67 | sqrtpclii 14946 |
. . . . . . . . . 10
⊢
(√‘2) ∈ ℝ |
69 | 60, 62 | sqrtpclii 14946 |
. . . . . . . . . 10
⊢
(√‘4) ∈ ℝ |
70 | 68, 69, 45, 67 | ltdiv1ii 11761 |
. . . . . . . . 9
⊢
((√‘2) < (√‘4) ↔ ((√‘2) / 2)
< ((√‘4) / 2)) |
71 | 66, 70 | mpbi 233 |
. . . . . . . 8
⊢
((√‘2) / 2) < ((√‘4) / 2) |
72 | | sqrtsq 14833 |
. . . . . . . . . . 11
⊢ ((2
∈ ℝ ∧ 0 ≤ 2) → (√‘(2↑2)) =
2) |
73 | 45, 47, 72 | mp2an 692 |
. . . . . . . . . 10
⊢
(√‘(2↑2)) = 2 |
74 | 73 | oveq1i 7223 |
. . . . . . . . 9
⊢
((√‘(2↑2)) / 2) = (2 / 2) |
75 | | sq2 13766 |
. . . . . . . . . . 11
⊢
(2↑2) = 4 |
76 | 75 | fveq2i 6720 |
. . . . . . . . . 10
⊢
(√‘(2↑2)) = (√‘4) |
77 | 76 | oveq1i 7223 |
. . . . . . . . 9
⊢
((√‘(2↑2)) / 2) = ((√‘4) /
2) |
78 | | 2div2e1 11971 |
. . . . . . . . 9
⊢ (2 / 2) =
1 |
79 | 74, 77, 78 | 3eqtr3i 2773 |
. . . . . . . 8
⊢
((√‘4) / 2) = 1 |
80 | 71, 79 | breqtri 5078 |
. . . . . . 7
⊢
((√‘2) / 2) < 1 |
81 | 46, 48 | resqrtcld 14981 |
. . . . . . . . 9
⊢ (𝐷 ∈ ℝ+
→ (√‘2) ∈ ℝ) |
82 | 81 | rehalfcld 12077 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ ((√‘2) / 2) ∈ ℝ) |
83 | | 1red 10834 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ 1 ∈ ℝ) |
84 | | id 22 |
. . . . . . . 8
⊢ (𝐷 ∈ ℝ+
→ 𝐷 ∈
ℝ+) |
85 | 82, 83, 84 | ltmul1d 12669 |
. . . . . . 7
⊢ (𝐷 ∈ ℝ+
→ (((√‘2) / 2) < 1 ↔ (((√‘2) / 2) ·
𝐷) < (1 · 𝐷))) |
86 | 80, 85 | mpbii 236 |
. . . . . 6
⊢ (𝐷 ∈ ℝ+
→ (((√‘2) / 2) · 𝐷) < (1 · 𝐷)) |
87 | 53 | mulid2d 10851 |
. . . . . 6
⊢ (𝐷 ∈ ℝ+
→ (1 · 𝐷) =
𝐷) |
88 | 86, 87 | breqtrd 5079 |
. . . . 5
⊢ (𝐷 ∈ ℝ+
→ (((√‘2) / 2) · 𝐷) < 𝐷) |
89 | 58, 88 | eqbrtrd 5075 |
. . . 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 10993 |
. 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 416 |
1
⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤
𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ+) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷)) |