Step |
Hyp |
Ref |
Expression |
1 |
|
mbflim.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
mbflim.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
mbflim.4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ⇝ 𝐶 ) |
4 |
|
mbflim.5 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
5 |
|
mbflim.6 |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) ) → 𝐵 ∈ 𝑉 ) |
6 |
1
|
fvexi |
⊢ 𝑍 ∈ V |
7 |
6
|
mptex |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ∈ V |
8 |
7
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ∈ V ) |
9 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑀 ∈ ℤ ) |
10 |
5
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
11 |
4 10
|
mbfmptcl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
12 |
11
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → 𝐵 ∈ ℂ ) |
13 |
12
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) : 𝑍 ⟶ ℂ ) |
14 |
13
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ∈ ℂ ) |
15 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
16 |
12
|
recld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ℜ ‘ 𝐵 ) ∈ ℝ ) |
17 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) |
18 |
17
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ ( ℜ ‘ 𝐵 ) ∈ ℝ ) → ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℜ ‘ 𝐵 ) ) |
19 |
15 16 18
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℜ ‘ 𝐵 ) ) |
20 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) = ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) |
21 |
20
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℂ ) → ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = 𝐵 ) |
22 |
15 12 21
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) = 𝐵 ) |
23 |
22
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) = ( ℜ ‘ 𝐵 ) ) |
24 |
19 23
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
25 |
24
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑛 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
26 |
|
nffvmpt1 |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑘 ) |
27 |
|
nfcv |
⊢ Ⅎ 𝑛 ℜ |
28 |
|
nffvmpt1 |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) |
29 |
27 28
|
nffv |
⊢ Ⅎ 𝑛 ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) |
30 |
26 29
|
nfeq |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) |
31 |
|
nfv |
⊢ Ⅎ 𝑘 ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) |
32 |
|
fveq2 |
⊢ ( 𝑘 = 𝑛 → ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑛 ) ) |
33 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑛 → ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
34 |
32 33
|
eqeq12d |
⊢ ( 𝑘 = 𝑛 → ( ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) ↔ ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) ) |
35 |
30 31 34
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) ↔ ∀ 𝑛 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
36 |
25 35
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑘 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) ) |
37 |
36
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℜ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) ) |
38 |
1 3 8 9 14 37
|
climre |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ ( ℜ ‘ 𝐵 ) ) ⇝ ( ℜ ‘ 𝐶 ) ) |
39 |
11
|
ismbfcn2 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ↔ ( ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ MblFn ∧ ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ MblFn ) ) ) |
40 |
4 39
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ MblFn ∧ ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ MblFn ) ) |
41 |
40
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ MblFn ) |
42 |
11
|
anasss |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) ) → 𝐵 ∈ ℂ ) |
43 |
42
|
recld |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) ) → ( ℜ ‘ 𝐵 ) ∈ ℝ ) |
44 |
1 2 38 41 43
|
mbflimlem |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐶 ) ) ∈ MblFn ) |
45 |
6
|
mptex |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ∈ V |
46 |
45
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ∈ V ) |
47 |
12
|
imcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ℑ ‘ 𝐵 ) ∈ ℝ ) |
48 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) |
49 |
48
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ ( ℑ ‘ 𝐵 ) ∈ ℝ ) → ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℑ ‘ 𝐵 ) ) |
50 |
15 47 49
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℑ ‘ 𝐵 ) ) |
51 |
22
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) = ( ℑ ‘ 𝐵 ) ) |
52 |
50 51
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
53 |
52
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑛 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
54 |
|
nffvmpt1 |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑘 ) |
55 |
|
nfcv |
⊢ Ⅎ 𝑛 ℑ |
56 |
55 28
|
nffv |
⊢ Ⅎ 𝑛 ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) |
57 |
54 56
|
nfeq |
⊢ Ⅎ 𝑛 ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) |
58 |
|
nfv |
⊢ Ⅎ 𝑘 ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) |
59 |
|
fveq2 |
⊢ ( 𝑘 = 𝑛 → ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑛 ) ) |
60 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑛 → ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
61 |
59 60
|
eqeq12d |
⊢ ( 𝑘 = 𝑛 → ( ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) ↔ ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) ) |
62 |
57 58 61
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) ↔ ∀ 𝑛 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑛 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑛 ) ) ) |
63 |
53 62
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑘 ∈ 𝑍 ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) ) |
64 |
63
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ‘ 𝑘 ) = ( ℑ ‘ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ‘ 𝑘 ) ) ) |
65 |
1 3 46 9 14 64
|
climim |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ ( ℑ ‘ 𝐵 ) ) ⇝ ( ℑ ‘ 𝐶 ) ) |
66 |
40
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ MblFn ) |
67 |
42
|
imcld |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴 ) ) → ( ℑ ‘ 𝐵 ) ∈ ℝ ) |
68 |
1 2 65 66 67
|
mbflimlem |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐶 ) ) ∈ MblFn ) |
69 |
|
climcl |
⊢ ( ( 𝑛 ∈ 𝑍 ↦ 𝐵 ) ⇝ 𝐶 → 𝐶 ∈ ℂ ) |
70 |
3 69
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
71 |
70
|
ismbfcn2 |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ MblFn ↔ ( ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐶 ) ) ∈ MblFn ∧ ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐶 ) ) ∈ MblFn ) ) ) |
72 |
44 68 71
|
mpbir2and |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ MblFn ) |