Step |
Hyp |
Ref |
Expression |
1 |
|
nmoubi.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
nmoubi.y |
⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) |
3 |
|
nmoubi.l |
⊢ 𝐿 = ( normCV ‘ 𝑈 ) |
4 |
|
nmoubi.m |
⊢ 𝑀 = ( normCV ‘ 𝑊 ) |
5 |
|
nmoubi.3 |
⊢ 𝑁 = ( 𝑈 normOpOLD 𝑊 ) |
6 |
|
nmoubi.u |
⊢ 𝑈 ∈ NrmCVec |
7 |
|
nmoubi.w |
⊢ 𝑊 ∈ NrmCVec |
8 |
1 2 3 4 5 6 7
|
nmounbi |
⊢ ( 𝑇 : 𝑋 ⟶ 𝑌 → ( ( 𝑁 ‘ 𝑇 ) = +∞ ↔ ∀ 𝑘 ∈ ℝ ∃ 𝑦 ∈ 𝑋 ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ) ) ) |
9 |
8
|
biimpa |
⊢ ( ( 𝑇 : 𝑋 ⟶ 𝑌 ∧ ( 𝑁 ‘ 𝑇 ) = +∞ ) → ∀ 𝑘 ∈ ℝ ∃ 𝑦 ∈ 𝑋 ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ) ) |
10 |
|
nnre |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ ) |
11 |
10
|
imim1i |
⊢ ( ( 𝑘 ∈ ℝ → ∃ 𝑦 ∈ 𝑋 ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ) ) → ( 𝑘 ∈ ℕ → ∃ 𝑦 ∈ 𝑋 ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ) ) ) |
12 |
11
|
ralimi2 |
⊢ ( ∀ 𝑘 ∈ ℝ ∃ 𝑦 ∈ 𝑋 ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ) → ∀ 𝑘 ∈ ℕ ∃ 𝑦 ∈ 𝑋 ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ) ) |
13 |
1
|
fvexi |
⊢ 𝑋 ∈ V |
14 |
|
nnenom |
⊢ ℕ ≈ ω |
15 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑘 ) → ( 𝐿 ‘ 𝑦 ) = ( 𝐿 ‘ ( 𝑓 ‘ 𝑘 ) ) ) |
16 |
15
|
breq1d |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑘 ) → ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ↔ ( 𝐿 ‘ ( 𝑓 ‘ 𝑘 ) ) ≤ 1 ) ) |
17 |
|
2fveq3 |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑘 ) → ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) = ( 𝑀 ‘ ( 𝑇 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) |
18 |
17
|
breq2d |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑘 ) → ( 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ↔ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) |
19 |
16 18
|
anbi12d |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑘 ) → ( ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ) ↔ ( ( 𝐿 ‘ ( 𝑓 ‘ 𝑘 ) ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) ) |
20 |
13 14 19
|
axcc4 |
⊢ ( ∀ 𝑘 ∈ ℕ ∃ 𝑦 ∈ 𝑋 ( ( 𝐿 ‘ 𝑦 ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ 𝑦 ) ) ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ 𝑋 ∧ ∀ 𝑘 ∈ ℕ ( ( 𝐿 ‘ ( 𝑓 ‘ 𝑘 ) ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) ) |
21 |
9 12 20
|
3syl |
⊢ ( ( 𝑇 : 𝑋 ⟶ 𝑌 ∧ ( 𝑁 ‘ 𝑇 ) = +∞ ) → ∃ 𝑓 ( 𝑓 : ℕ ⟶ 𝑋 ∧ ∀ 𝑘 ∈ ℕ ( ( 𝐿 ‘ ( 𝑓 ‘ 𝑘 ) ) ≤ 1 ∧ 𝑘 < ( 𝑀 ‘ ( 𝑇 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) ) |