Step |
Hyp |
Ref |
Expression |
1 |
|
eluznn |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑛 ∈ ℕ ) |
2 |
|
eqidd |
⊢ ( 𝑛 ∈ ℕ → ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) = ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ) |
3 |
|
oveq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝐴 / 𝑚 ) = ( 𝐴 / 𝑛 ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 = 𝑛 ) → ( 𝐴 / 𝑚 ) = ( 𝐴 / 𝑛 ) ) |
5 |
|
id |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ ) |
6 |
|
ovexd |
⊢ ( 𝑛 ∈ ℕ → ( 𝐴 / 𝑛 ) ∈ V ) |
7 |
2 4 5 6
|
fvmptd |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) = ( 𝐴 / 𝑛 ) ) |
8 |
7
|
eqcomd |
⊢ ( 𝑛 ∈ ℕ → ( 𝐴 / 𝑛 ) = ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) |
9 |
1 8
|
syl |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐴 / 𝑛 ) = ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) |
10 |
9
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐴 / 𝑛 ) = ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) |
11 |
10
|
mpteq2dva |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ ) → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↦ ( 𝐴 / 𝑛 ) ) = ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↦ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) ) |
12 |
|
divcnv |
⊢ ( 𝐴 ∈ ℂ → ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ⇝ 0 ) |
13 |
12
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ ) → ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ⇝ 0 ) |
14 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℕ ) |
15 |
14
|
nnzd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ ) → 𝑀 ∈ ℤ ) |
16 |
|
nnex |
⊢ ℕ ∈ V |
17 |
16
|
mptex |
⊢ ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ∈ V |
18 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ 𝑀 ) |
19 |
|
eqid |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↦ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) = ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↦ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) |
20 |
18 19
|
climmpt |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ∈ V ) → ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ⇝ 0 ↔ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↦ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) ⇝ 0 ) ) |
21 |
15 17 20
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ ) → ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ⇝ 0 ↔ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↦ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) ⇝ 0 ) ) |
22 |
13 21
|
mpbid |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ ) → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↦ ( ( 𝑚 ∈ ℕ ↦ ( 𝐴 / 𝑚 ) ) ‘ 𝑛 ) ) ⇝ 0 ) |
23 |
11 22
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ ) → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↦ ( 𝐴 / 𝑛 ) ) ⇝ 0 ) |