Step |
Hyp |
Ref |
Expression |
1 |
|
cnre2csqima.1 |
⊢ 𝐹 = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( 𝑥 + ( i · 𝑦 ) ) ) |
2 |
|
ioossre |
⊢ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ⊆ ℝ |
3 |
|
ioossre |
⊢ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ⊆ ℝ |
4 |
|
xpinpreima2 |
⊢ ( ( ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ⊆ ℝ ∧ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ⊆ ℝ ) → ( ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) × ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) = ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ) ) |
5 |
4
|
eleq2d |
⊢ ( ( ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ⊆ ℝ ∧ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ⊆ ℝ ) → ( 𝑌 ∈ ( ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) × ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ↔ 𝑌 ∈ ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ) ) ) |
6 |
2 3 5
|
mp2an |
⊢ ( 𝑌 ∈ ( ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) × ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ↔ 𝑌 ∈ ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ) ) |
7 |
|
elin |
⊢ ( 𝑌 ∈ ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ) ↔ ( 𝑌 ∈ ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ) ∧ 𝑌 ∈ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ) ) |
8 |
|
simpl |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → 𝑥 ∈ ℝ ) |
9 |
8
|
recnd |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → 𝑥 ∈ ℂ ) |
10 |
|
ax-icn |
⊢ i ∈ ℂ |
11 |
10
|
a1i |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → i ∈ ℂ ) |
12 |
|
simpr |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ ) |
13 |
12
|
recnd |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℂ ) |
14 |
11 13
|
mulcld |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( i · 𝑦 ) ∈ ℂ ) |
15 |
9 14
|
addcld |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 + ( i · 𝑦 ) ) ∈ ℂ ) |
16 |
|
reval |
⊢ ( ( 𝑥 + ( i · 𝑦 ) ) ∈ ℂ → ( ℜ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) = ( ( ( 𝑥 + ( i · 𝑦 ) ) + ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) / 2 ) ) |
17 |
15 16
|
syl |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ℜ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) = ( ( ( 𝑥 + ( i · 𝑦 ) ) + ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) / 2 ) ) |
18 |
|
crre |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ℜ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) = 𝑥 ) |
19 |
17 18
|
eqtr3d |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝑥 + ( i · 𝑦 ) ) + ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) / 2 ) = 𝑥 ) |
20 |
19
|
mpoeq3ia |
⊢ ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( ( ( 𝑥 + ( i · 𝑦 ) ) + ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) / 2 ) ) = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ 𝑥 ) |
21 |
15
|
adantl |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 + ( i · 𝑦 ) ) ∈ ℂ ) |
22 |
1
|
a1i |
⊢ ( ⊤ → 𝐹 = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( 𝑥 + ( i · 𝑦 ) ) ) ) |
23 |
|
df-re |
⊢ ℜ = ( 𝑧 ∈ ℂ ↦ ( ( 𝑧 + ( ∗ ‘ 𝑧 ) ) / 2 ) ) |
24 |
23
|
a1i |
⊢ ( ⊤ → ℜ = ( 𝑧 ∈ ℂ ↦ ( ( 𝑧 + ( ∗ ‘ 𝑧 ) ) / 2 ) ) ) |
25 |
|
id |
⊢ ( 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) → 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) ) |
26 |
|
fveq2 |
⊢ ( 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) → ( ∗ ‘ 𝑧 ) = ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) |
27 |
25 26
|
oveq12d |
⊢ ( 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) → ( 𝑧 + ( ∗ ‘ 𝑧 ) ) = ( ( 𝑥 + ( i · 𝑦 ) ) + ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) ) |
28 |
27
|
oveq1d |
⊢ ( 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) → ( ( 𝑧 + ( ∗ ‘ 𝑧 ) ) / 2 ) = ( ( ( 𝑥 + ( i · 𝑦 ) ) + ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) / 2 ) ) |
29 |
21 22 24 28
|
fmpoco |
⊢ ( ⊤ → ( ℜ ∘ 𝐹 ) = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( ( ( 𝑥 + ( i · 𝑦 ) ) + ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) / 2 ) ) ) |
30 |
29
|
mptru |
⊢ ( ℜ ∘ 𝐹 ) = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( ( ( 𝑥 + ( i · 𝑦 ) ) + ( ∗ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) ) / 2 ) ) |
31 |
|
df1stres |
⊢ ( 1st ↾ ( ℝ × ℝ ) ) = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ 𝑥 ) |
32 |
20 30 31
|
3eqtr4ri |
⊢ ( 1st ↾ ( ℝ × ℝ ) ) = ( ℜ ∘ 𝐹 ) |
33 |
15
|
rgen2 |
⊢ ∀ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ℝ ( 𝑥 + ( i · 𝑦 ) ) ∈ ℂ |
34 |
1
|
fnmpo |
⊢ ( ∀ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ℝ ( 𝑥 + ( i · 𝑦 ) ) ∈ ℂ → 𝐹 Fn ( ℝ × ℝ ) ) |
35 |
33 34
|
ax-mp |
⊢ 𝐹 Fn ( ℝ × ℝ ) |
36 |
|
fo1st |
⊢ 1st : V –onto→ V |
37 |
|
fofn |
⊢ ( 1st : V –onto→ V → 1st Fn V ) |
38 |
36 37
|
ax-mp |
⊢ 1st Fn V |
39 |
|
xp1st |
⊢ ( 𝑧 ∈ ( ℝ × ℝ ) → ( 1st ‘ 𝑧 ) ∈ ℝ ) |
40 |
1
|
rnmpo |
⊢ ran 𝐹 = { 𝑧 ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) } |
41 |
|
simpr |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) ) → 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) ) |
42 |
15
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) ) → ( 𝑥 + ( i · 𝑦 ) ) ∈ ℂ ) |
43 |
41 42
|
eqeltrd |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) ) → 𝑧 ∈ ℂ ) |
44 |
43
|
ex |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) → 𝑧 ∈ ℂ ) ) |
45 |
44
|
rexlimivv |
⊢ ( ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) → 𝑧 ∈ ℂ ) |
46 |
45
|
abssi |
⊢ { 𝑧 ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) } ⊆ ℂ |
47 |
40 46
|
eqsstri |
⊢ ran 𝐹 ⊆ ℂ |
48 |
|
simpl |
⊢ ( ( 𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹 ) → 𝑧 ∈ ran 𝐹 ) |
49 |
47 48
|
sseldi |
⊢ ( ( 𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹 ) → 𝑧 ∈ ℂ ) |
50 |
|
simpr |
⊢ ( ( 𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹 ) → 𝑢 ∈ ran 𝐹 ) |
51 |
47 50
|
sseldi |
⊢ ( ( 𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹 ) → 𝑢 ∈ ℂ ) |
52 |
49 51
|
resubd |
⊢ ( ( 𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹 ) → ( ℜ ‘ ( 𝑧 − 𝑢 ) ) = ( ( ℜ ‘ 𝑧 ) − ( ℜ ‘ 𝑢 ) ) ) |
53 |
32 35 38 39 52
|
cnre2csqlem |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝑌 ∈ ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ) → ( abs ‘ ( ℜ ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ) ) |
54 |
|
imval |
⊢ ( ( 𝑥 + ( i · 𝑦 ) ) ∈ ℂ → ( ℑ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) = ( ℜ ‘ ( ( 𝑥 + ( i · 𝑦 ) ) / i ) ) ) |
55 |
15 54
|
syl |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ℑ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) = ( ℜ ‘ ( ( 𝑥 + ( i · 𝑦 ) ) / i ) ) ) |
56 |
|
crim |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ℑ ‘ ( 𝑥 + ( i · 𝑦 ) ) ) = 𝑦 ) |
57 |
55 56
|
eqtr3d |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ℜ ‘ ( ( 𝑥 + ( i · 𝑦 ) ) / i ) ) = 𝑦 ) |
58 |
57
|
mpoeq3ia |
⊢ ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( ℜ ‘ ( ( 𝑥 + ( i · 𝑦 ) ) / i ) ) ) = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ 𝑦 ) |
59 |
|
df-im |
⊢ ℑ = ( 𝑧 ∈ ℂ ↦ ( ℜ ‘ ( 𝑧 / i ) ) ) |
60 |
59
|
a1i |
⊢ ( ⊤ → ℑ = ( 𝑧 ∈ ℂ ↦ ( ℜ ‘ ( 𝑧 / i ) ) ) ) |
61 |
|
fvoveq1 |
⊢ ( 𝑧 = ( 𝑥 + ( i · 𝑦 ) ) → ( ℜ ‘ ( 𝑧 / i ) ) = ( ℜ ‘ ( ( 𝑥 + ( i · 𝑦 ) ) / i ) ) ) |
62 |
21 22 60 61
|
fmpoco |
⊢ ( ⊤ → ( ℑ ∘ 𝐹 ) = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( ℜ ‘ ( ( 𝑥 + ( i · 𝑦 ) ) / i ) ) ) ) |
63 |
62
|
mptru |
⊢ ( ℑ ∘ 𝐹 ) = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( ℜ ‘ ( ( 𝑥 + ( i · 𝑦 ) ) / i ) ) ) |
64 |
|
df2ndres |
⊢ ( 2nd ↾ ( ℝ × ℝ ) ) = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ 𝑦 ) |
65 |
58 63 64
|
3eqtr4ri |
⊢ ( 2nd ↾ ( ℝ × ℝ ) ) = ( ℑ ∘ 𝐹 ) |
66 |
|
fo2nd |
⊢ 2nd : V –onto→ V |
67 |
|
fofn |
⊢ ( 2nd : V –onto→ V → 2nd Fn V ) |
68 |
66 67
|
ax-mp |
⊢ 2nd Fn V |
69 |
|
xp2nd |
⊢ ( 𝑧 ∈ ( ℝ × ℝ ) → ( 2nd ‘ 𝑧 ) ∈ ℝ ) |
70 |
49 51
|
imsubd |
⊢ ( ( 𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹 ) → ( ℑ ‘ ( 𝑧 − 𝑢 ) ) = ( ( ℑ ‘ 𝑧 ) − ( ℑ ‘ 𝑢 ) ) ) |
71 |
65 35 68 69 70
|
cnre2csqlem |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝑌 ∈ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) → ( abs ‘ ( ℑ ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ) ) |
72 |
53 71
|
anim12d |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( ( 𝑌 ∈ ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ) ∧ 𝑌 ∈ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ) → ( ( abs ‘ ( ℜ ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ∧ ( abs ‘ ( ℑ ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ) ) ) |
73 |
7 72
|
syl5bi |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝑌 ∈ ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) ) → ( ( abs ‘ ( ℜ ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ∧ ( abs ‘ ( ℑ ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ) ) ) |
74 |
6 73
|
syl5bi |
⊢ ( ( 𝑋 ∈ ( ℝ × ℝ ) ∧ 𝑌 ∈ ( ℝ × ℝ ) ∧ 𝐷 ∈ ℝ+ ) → ( 𝑌 ∈ ( ( ( ( 1st ‘ 𝑋 ) − 𝐷 ) (,) ( ( 1st ‘ 𝑋 ) + 𝐷 ) ) × ( ( ( 2nd ‘ 𝑋 ) − 𝐷 ) (,) ( ( 2nd ‘ 𝑋 ) + 𝐷 ) ) ) → ( ( abs ‘ ( ℜ ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ∧ ( abs ‘ ( ℑ ‘ ( ( 𝐹 ‘ 𝑌 ) − ( 𝐹 ‘ 𝑋 ) ) ) ) < 𝐷 ) ) ) |