| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cnheibor.2 |
⊢ 𝐽 = ( TopOpen ‘ ℂfld ) |
| 2 |
|
cnheibor.3 |
⊢ 𝑇 = ( 𝐽 ↾t 𝑋 ) |
| 3 |
|
cnheibor.4 |
⊢ 𝐹 = ( 𝑥 ∈ ℝ , 𝑦 ∈ ℝ ↦ ( 𝑥 + ( i · 𝑦 ) ) ) |
| 4 |
|
cnheibor.5 |
⊢ 𝑌 = ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) |
| 5 |
1
|
cnfldtop |
⊢ 𝐽 ∈ Top |
| 6 |
3
|
cnref1o |
⊢ 𝐹 : ( ℝ × ℝ ) –1-1-onto→ ℂ |
| 7 |
|
f1ofn |
⊢ ( 𝐹 : ( ℝ × ℝ ) –1-1-onto→ ℂ → 𝐹 Fn ( ℝ × ℝ ) ) |
| 8 |
|
elpreima |
⊢ ( 𝐹 Fn ( ℝ × ℝ ) → ( 𝑢 ∈ ( ◡ 𝐹 “ 𝑋 ) ↔ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) ) |
| 9 |
6 7 8
|
mp2b |
⊢ ( 𝑢 ∈ ( ◡ 𝐹 “ 𝑋 ) ↔ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) |
| 10 |
|
1st2nd2 |
⊢ ( 𝑢 ∈ ( ℝ × ℝ ) → 𝑢 = ⟨ ( 1st ‘ 𝑢 ) , ( 2nd ‘ 𝑢 ) ⟩ ) |
| 11 |
10
|
ad2antrl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → 𝑢 = ⟨ ( 1st ‘ 𝑢 ) , ( 2nd ‘ 𝑢 ) ⟩ ) |
| 12 |
|
xp1st |
⊢ ( 𝑢 ∈ ( ℝ × ℝ ) → ( 1st ‘ 𝑢 ) ∈ ℝ ) |
| 13 |
12
|
ad2antrl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 1st ‘ 𝑢 ) ∈ ℝ ) |
| 14 |
13
|
recnd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 1st ‘ 𝑢 ) ∈ ℂ ) |
| 15 |
14
|
abscld |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 1st ‘ 𝑢 ) ) ∈ ℝ ) |
| 16 |
1
|
cnfldtopon |
⊢ 𝐽 ∈ ( TopOn ‘ ℂ ) |
| 17 |
16
|
toponunii |
⊢ ℂ = ∪ 𝐽 |
| 18 |
17
|
cldss |
⊢ ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) → 𝑋 ⊆ ℂ ) |
| 19 |
18
|
adantr |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → 𝑋 ⊆ ℂ ) |
| 20 |
19
|
adantr |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → 𝑋 ⊆ ℂ ) |
| 21 |
|
simprr |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) |
| 22 |
20 21
|
sseldd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝑢 ) ∈ ℂ ) |
| 23 |
22
|
abscld |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ∈ ℝ ) |
| 24 |
|
simplrl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → 𝑅 ∈ ℝ ) |
| 25 |
|
simprl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → 𝑢 ∈ ( ℝ × ℝ ) ) |
| 26 |
|
f1ocnvfv1 |
⊢ ( ( 𝐹 : ( ℝ × ℝ ) –1-1-onto→ ℂ ∧ 𝑢 ∈ ( ℝ × ℝ ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑢 ) ) = 𝑢 ) |
| 27 |
6 25 26
|
sylancr |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑢 ) ) = 𝑢 ) |
| 28 |
|
fveq2 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ( ℜ ‘ 𝑧 ) = ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 29 |
|
fveq2 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ( ℑ ‘ 𝑧 ) = ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 30 |
28 29
|
opeq12d |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ⟨ ( ℜ ‘ 𝑧 ) , ( ℑ ‘ 𝑧 ) ⟩ = ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ) |
| 31 |
3
|
cnrecnv |
⊢ ◡ 𝐹 = ( 𝑧 ∈ ℂ ↦ ⟨ ( ℜ ‘ 𝑧 ) , ( ℑ ‘ 𝑧 ) ⟩ ) |
| 32 |
|
opex |
⊢ ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ∈ V |
| 33 |
30 31 32
|
fvmpt |
⊢ ( ( 𝐹 ‘ 𝑢 ) ∈ ℂ → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑢 ) ) = ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ) |
| 34 |
22 33
|
syl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑢 ) ) = ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ) |
| 35 |
27 34
|
eqtr3d |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → 𝑢 = ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ) |
| 36 |
35
|
fveq2d |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 1st ‘ 𝑢 ) = ( 1st ‘ ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ) ) |
| 37 |
|
fvex |
⊢ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) ∈ V |
| 38 |
|
fvex |
⊢ ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ∈ V |
| 39 |
37 38
|
op1st |
⊢ ( 1st ‘ ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ) = ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) |
| 40 |
36 39
|
eqtrdi |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 1st ‘ 𝑢 ) = ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 41 |
40
|
fveq2d |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 1st ‘ 𝑢 ) ) = ( abs ‘ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) ) ) |
| 42 |
|
absrele |
⊢ ( ( 𝐹 ‘ 𝑢 ) ∈ ℂ → ( abs ‘ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 43 |
22 42
|
syl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 44 |
41 43
|
eqbrtrd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 1st ‘ 𝑢 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 45 |
|
fveq2 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ( abs ‘ 𝑧 ) = ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 46 |
45
|
breq1d |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑢 ) → ( ( abs ‘ 𝑧 ) ≤ 𝑅 ↔ ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ≤ 𝑅 ) ) |
| 47 |
|
simplrr |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) |
| 48 |
46 47 21
|
rspcdva |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ≤ 𝑅 ) |
| 49 |
15 23 24 44 48
|
letrd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 1st ‘ 𝑢 ) ) ≤ 𝑅 ) |
| 50 |
13 24
|
absled |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( ( abs ‘ ( 1st ‘ 𝑢 ) ) ≤ 𝑅 ↔ ( - 𝑅 ≤ ( 1st ‘ 𝑢 ) ∧ ( 1st ‘ 𝑢 ) ≤ 𝑅 ) ) ) |
| 51 |
49 50
|
mpbid |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( - 𝑅 ≤ ( 1st ‘ 𝑢 ) ∧ ( 1st ‘ 𝑢 ) ≤ 𝑅 ) ) |
| 52 |
51
|
simpld |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → - 𝑅 ≤ ( 1st ‘ 𝑢 ) ) |
| 53 |
51
|
simprd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 1st ‘ 𝑢 ) ≤ 𝑅 ) |
| 54 |
|
renegcl |
⊢ ( 𝑅 ∈ ℝ → - 𝑅 ∈ ℝ ) |
| 55 |
24 54
|
syl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → - 𝑅 ∈ ℝ ) |
| 56 |
|
elicc2 |
⊢ ( ( - 𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ ) → ( ( 1st ‘ 𝑢 ) ∈ ( - 𝑅 [,] 𝑅 ) ↔ ( ( 1st ‘ 𝑢 ) ∈ ℝ ∧ - 𝑅 ≤ ( 1st ‘ 𝑢 ) ∧ ( 1st ‘ 𝑢 ) ≤ 𝑅 ) ) ) |
| 57 |
55 24 56
|
syl2anc |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( ( 1st ‘ 𝑢 ) ∈ ( - 𝑅 [,] 𝑅 ) ↔ ( ( 1st ‘ 𝑢 ) ∈ ℝ ∧ - 𝑅 ≤ ( 1st ‘ 𝑢 ) ∧ ( 1st ‘ 𝑢 ) ≤ 𝑅 ) ) ) |
| 58 |
13 52 53 57
|
mpbir3and |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 1st ‘ 𝑢 ) ∈ ( - 𝑅 [,] 𝑅 ) ) |
| 59 |
|
xp2nd |
⊢ ( 𝑢 ∈ ( ℝ × ℝ ) → ( 2nd ‘ 𝑢 ) ∈ ℝ ) |
| 60 |
59
|
ad2antrl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 2nd ‘ 𝑢 ) ∈ ℝ ) |
| 61 |
60
|
recnd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 2nd ‘ 𝑢 ) ∈ ℂ ) |
| 62 |
61
|
abscld |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 2nd ‘ 𝑢 ) ) ∈ ℝ ) |
| 63 |
35
|
fveq2d |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ) ) |
| 64 |
37 38
|
op2nd |
⊢ ( 2nd ‘ ⟨ ( ℜ ‘ ( 𝐹 ‘ 𝑢 ) ) , ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ⟩ ) = ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) |
| 65 |
63 64
|
eqtrdi |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 2nd ‘ 𝑢 ) = ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 66 |
65
|
fveq2d |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 2nd ‘ 𝑢 ) ) = ( abs ‘ ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ) ) |
| 67 |
|
absimle |
⊢ ( ( 𝐹 ‘ 𝑢 ) ∈ ℂ → ( abs ‘ ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 68 |
22 67
|
syl |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( ℑ ‘ ( 𝐹 ‘ 𝑢 ) ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 69 |
66 68
|
eqbrtrd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 2nd ‘ 𝑢 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑢 ) ) ) |
| 70 |
62 23 24 69 48
|
letrd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( abs ‘ ( 2nd ‘ 𝑢 ) ) ≤ 𝑅 ) |
| 71 |
60 24
|
absled |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( ( abs ‘ ( 2nd ‘ 𝑢 ) ) ≤ 𝑅 ↔ ( - 𝑅 ≤ ( 2nd ‘ 𝑢 ) ∧ ( 2nd ‘ 𝑢 ) ≤ 𝑅 ) ) ) |
| 72 |
70 71
|
mpbid |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( - 𝑅 ≤ ( 2nd ‘ 𝑢 ) ∧ ( 2nd ‘ 𝑢 ) ≤ 𝑅 ) ) |
| 73 |
72
|
simpld |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → - 𝑅 ≤ ( 2nd ‘ 𝑢 ) ) |
| 74 |
72
|
simprd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 2nd ‘ 𝑢 ) ≤ 𝑅 ) |
| 75 |
|
elicc2 |
⊢ ( ( - 𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ ) → ( ( 2nd ‘ 𝑢 ) ∈ ( - 𝑅 [,] 𝑅 ) ↔ ( ( 2nd ‘ 𝑢 ) ∈ ℝ ∧ - 𝑅 ≤ ( 2nd ‘ 𝑢 ) ∧ ( 2nd ‘ 𝑢 ) ≤ 𝑅 ) ) ) |
| 76 |
55 24 75
|
syl2anc |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( ( 2nd ‘ 𝑢 ) ∈ ( - 𝑅 [,] 𝑅 ) ↔ ( ( 2nd ‘ 𝑢 ) ∈ ℝ ∧ - 𝑅 ≤ ( 2nd ‘ 𝑢 ) ∧ ( 2nd ‘ 𝑢 ) ≤ 𝑅 ) ) ) |
| 77 |
60 73 74 76
|
mpbir3and |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ( 2nd ‘ 𝑢 ) ∈ ( - 𝑅 [,] 𝑅 ) ) |
| 78 |
58 77
|
opelxpd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → ⟨ ( 1st ‘ 𝑢 ) , ( 2nd ‘ 𝑢 ) ⟩ ∈ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) |
| 79 |
11 78
|
eqeltrd |
⊢ ( ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) ∧ ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) ) → 𝑢 ∈ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) |
| 80 |
79
|
ex |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → ( ( 𝑢 ∈ ( ℝ × ℝ ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝑋 ) → 𝑢 ∈ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) |
| 81 |
9 80
|
biimtrid |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → ( 𝑢 ∈ ( ◡ 𝐹 “ 𝑋 ) → 𝑢 ∈ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) |
| 82 |
81
|
ssrdv |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → ( ◡ 𝐹 “ 𝑋 ) ⊆ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) |
| 83 |
|
f1ofun |
⊢ ( 𝐹 : ( ℝ × ℝ ) –1-1-onto→ ℂ → Fun 𝐹 ) |
| 84 |
6 83
|
ax-mp |
⊢ Fun 𝐹 |
| 85 |
|
f1ofo |
⊢ ( 𝐹 : ( ℝ × ℝ ) –1-1-onto→ ℂ → 𝐹 : ( ℝ × ℝ ) –onto→ ℂ ) |
| 86 |
|
forn |
⊢ ( 𝐹 : ( ℝ × ℝ ) –onto→ ℂ → ran 𝐹 = ℂ ) |
| 87 |
6 85 86
|
mp2b |
⊢ ran 𝐹 = ℂ |
| 88 |
19 87
|
sseqtrrdi |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → 𝑋 ⊆ ran 𝐹 ) |
| 89 |
|
funimass1 |
⊢ ( ( Fun 𝐹 ∧ 𝑋 ⊆ ran 𝐹 ) → ( ( ◡ 𝐹 “ 𝑋 ) ⊆ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) → 𝑋 ⊆ ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) ) |
| 90 |
84 88 89
|
sylancr |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → ( ( ◡ 𝐹 “ 𝑋 ) ⊆ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) → 𝑋 ⊆ ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) ) |
| 91 |
82 90
|
mpd |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → 𝑋 ⊆ ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) |
| 92 |
91 4
|
sseqtrrdi |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → 𝑋 ⊆ 𝑌 ) |
| 93 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
| 94 |
3 93 1
|
cnrehmeo |
⊢ 𝐹 ∈ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) Homeo 𝐽 ) |
| 95 |
|
imaexg |
⊢ ( 𝐹 ∈ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) Homeo 𝐽 ) → ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ∈ V ) |
| 96 |
94 95
|
ax-mp |
⊢ ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ∈ V |
| 97 |
4 96
|
eqeltri |
⊢ 𝑌 ∈ V |
| 98 |
97
|
a1i |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → 𝑌 ∈ V ) |
| 99 |
|
restabs |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ V ) → ( ( 𝐽 ↾t 𝑌 ) ↾t 𝑋 ) = ( 𝐽 ↾t 𝑋 ) ) |
| 100 |
5 92 98 99
|
mp3an2i |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → ( ( 𝐽 ↾t 𝑌 ) ↾t 𝑋 ) = ( 𝐽 ↾t 𝑋 ) ) |
| 101 |
100 2
|
eqtr4di |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → ( ( 𝐽 ↾t 𝑌 ) ↾t 𝑋 ) = 𝑇 ) |
| 102 |
4
|
oveq2i |
⊢ ( 𝐽 ↾t 𝑌 ) = ( 𝐽 ↾t ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) |
| 103 |
|
ishmeo |
⊢ ( 𝐹 ∈ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) Homeo 𝐽 ) ↔ ( 𝐹 ∈ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) Cn 𝐽 ) ∧ ◡ 𝐹 ∈ ( 𝐽 Cn ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) ) ) ) |
| 104 |
94 103
|
mpbi |
⊢ ( 𝐹 ∈ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) Cn 𝐽 ) ∧ ◡ 𝐹 ∈ ( 𝐽 Cn ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) ) ) |
| 105 |
104
|
simpli |
⊢ 𝐹 ∈ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) Cn 𝐽 ) |
| 106 |
|
iccssre |
⊢ ( ( - 𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ ) → ( - 𝑅 [,] 𝑅 ) ⊆ ℝ ) |
| 107 |
54 106
|
mpancom |
⊢ ( 𝑅 ∈ ℝ → ( - 𝑅 [,] 𝑅 ) ⊆ ℝ ) |
| 108 |
1 93
|
rerest |
⊢ ( ( - 𝑅 [,] 𝑅 ) ⊆ ℝ → ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ) |
| 109 |
107 108
|
syl |
⊢ ( 𝑅 ∈ ℝ → ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ) |
| 110 |
109 109
|
oveq12d |
⊢ ( 𝑅 ∈ ℝ → ( ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ×t ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ) = ( ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ×t ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ) ) |
| 111 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
| 112 |
|
ovex |
⊢ ( - 𝑅 [,] 𝑅 ) ∈ V |
| 113 |
|
txrest |
⊢ ( ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( topGen ‘ ran (,) ) ∈ Top ) ∧ ( ( - 𝑅 [,] 𝑅 ) ∈ V ∧ ( - 𝑅 [,] 𝑅 ) ∈ V ) ) → ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) ↾t ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) = ( ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ×t ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ) ) |
| 114 |
111 111 112 112 113
|
mp4an |
⊢ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) ↾t ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) = ( ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ×t ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ) |
| 115 |
110 114
|
eqtr4di |
⊢ ( 𝑅 ∈ ℝ → ( ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ×t ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ) = ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) ↾t ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) |
| 116 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) |
| 117 |
93 116
|
icccmp |
⊢ ( ( - 𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ∈ Comp ) |
| 118 |
54 117
|
mpancom |
⊢ ( 𝑅 ∈ ℝ → ( ( topGen ‘ ran (,) ) ↾t ( - 𝑅 [,] 𝑅 ) ) ∈ Comp ) |
| 119 |
109 118
|
eqeltrd |
⊢ ( 𝑅 ∈ ℝ → ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ∈ Comp ) |
| 120 |
|
txcmp |
⊢ ( ( ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ∈ Comp ∧ ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ∈ Comp ) → ( ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ×t ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ) ∈ Comp ) |
| 121 |
119 119 120
|
syl2anc |
⊢ ( 𝑅 ∈ ℝ → ( ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ×t ( 𝐽 ↾t ( - 𝑅 [,] 𝑅 ) ) ) ∈ Comp ) |
| 122 |
115 121
|
eqeltrrd |
⊢ ( 𝑅 ∈ ℝ → ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) ↾t ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ∈ Comp ) |
| 123 |
|
imacmp |
⊢ ( ( 𝐹 ∈ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) Cn 𝐽 ) ∧ ( ( ( topGen ‘ ran (,) ) ×t ( topGen ‘ ran (,) ) ) ↾t ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ∈ Comp ) → ( 𝐽 ↾t ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) ∈ Comp ) |
| 124 |
105 122 123
|
sylancr |
⊢ ( 𝑅 ∈ ℝ → ( 𝐽 ↾t ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ) ∈ Comp ) |
| 125 |
102 124
|
eqeltrid |
⊢ ( 𝑅 ∈ ℝ → ( 𝐽 ↾t 𝑌 ) ∈ Comp ) |
| 126 |
125
|
ad2antrl |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → ( 𝐽 ↾t 𝑌 ) ∈ Comp ) |
| 127 |
|
imassrn |
⊢ ( 𝐹 “ ( ( - 𝑅 [,] 𝑅 ) × ( - 𝑅 [,] 𝑅 ) ) ) ⊆ ran 𝐹 |
| 128 |
4 127
|
eqsstri |
⊢ 𝑌 ⊆ ran 𝐹 |
| 129 |
|
f1of |
⊢ ( 𝐹 : ( ℝ × ℝ ) –1-1-onto→ ℂ → 𝐹 : ( ℝ × ℝ ) ⟶ ℂ ) |
| 130 |
|
frn |
⊢ ( 𝐹 : ( ℝ × ℝ ) ⟶ ℂ → ran 𝐹 ⊆ ℂ ) |
| 131 |
6 129 130
|
mp2b |
⊢ ran 𝐹 ⊆ ℂ |
| 132 |
128 131
|
sstri |
⊢ 𝑌 ⊆ ℂ |
| 133 |
|
simpl |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → 𝑋 ∈ ( Clsd ‘ 𝐽 ) ) |
| 134 |
17
|
restcldi |
⊢ ( ( 𝑌 ⊆ ℂ ∧ 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ 𝑋 ⊆ 𝑌 ) → 𝑋 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑌 ) ) ) |
| 135 |
132 133 92 134
|
mp3an2i |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → 𝑋 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑌 ) ) ) |
| 136 |
|
cmpcld |
⊢ ( ( ( 𝐽 ↾t 𝑌 ) ∈ Comp ∧ 𝑋 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑌 ) ) ) → ( ( 𝐽 ↾t 𝑌 ) ↾t 𝑋 ) ∈ Comp ) |
| 137 |
126 135 136
|
syl2anc |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → ( ( 𝐽 ↾t 𝑌 ) ↾t 𝑋 ) ∈ Comp ) |
| 138 |
101 137
|
eqeltrrd |
⊢ ( ( 𝑋 ∈ ( Clsd ‘ 𝐽 ) ∧ ( 𝑅 ∈ ℝ ∧ ∀ 𝑧 ∈ 𝑋 ( abs ‘ 𝑧 ) ≤ 𝑅 ) ) → 𝑇 ∈ Comp ) |