Step |
Hyp |
Ref |
Expression |
1 |
|
climreclf.k |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
climreclf.f |
⊢ Ⅎ 𝑘 𝐹 |
3 |
|
climreclf.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
4 |
|
climreclf.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
5 |
|
climreclf.a |
⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 ) |
6 |
|
climreclf.r |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
7 |
|
nfv |
⊢ Ⅎ 𝑘 𝑗 ∈ 𝑍 |
8 |
1 7
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) |
9 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑗 |
10 |
2 9
|
nffv |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 ) |
11 |
|
nfcv |
⊢ Ⅎ 𝑘 ℝ |
12 |
10 11
|
nfel |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 ) ∈ ℝ |
13 |
8 12
|
nfim |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ ) |
14 |
|
eleq1w |
⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍 ) ) |
15 |
14
|
anbi2d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) ) |
17 |
16
|
eleq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℝ ↔ ( 𝐹 ‘ 𝑗 ) ∈ ℝ ) ) |
18 |
15 17
|
imbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ ) ) ) |
19 |
13 18 6
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) ∈ ℝ ) |
20 |
3 4 5 19
|
climrecl |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |