Step |
Hyp |
Ref |
Expression |
1 |
|
climfsum.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
climfsum.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
climfsum.3 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
4 |
|
climfsum.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐹 ⇝ 𝐵 ) |
5 |
|
climfsum.6 |
⊢ ( 𝜑 → 𝐻 ∈ 𝑊 ) |
6 |
|
climfsum.7 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ 𝑛 ∈ 𝑍 ) ) → ( 𝐹 ‘ 𝑛 ) ∈ ℂ ) |
7 |
|
climfsum.8 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐻 ‘ 𝑛 ) = Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑛 ) ) |
8 |
7
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝐻 ‘ 𝑛 ) ) = ( 𝑛 ∈ 𝑍 ↦ Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑛 ) ) ) |
9 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
10 |
1 9
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
11 |
|
zssre |
⊢ ℤ ⊆ ℝ |
12 |
10 11
|
sstri |
⊢ 𝑍 ⊆ ℝ |
13 |
12
|
a1i |
⊢ ( 𝜑 → 𝑍 ⊆ ℝ ) |
14 |
|
fvexd |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ 𝑍 ∧ 𝑘 ∈ 𝐴 ) ) → ( 𝐹 ‘ 𝑛 ) ∈ V ) |
15 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑀 ∈ ℤ ) |
16 |
|
climrel |
⊢ Rel ⇝ |
17 |
16
|
brrelex1i |
⊢ ( 𝐹 ⇝ 𝐵 → 𝐹 ∈ V ) |
18 |
4 17
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐹 ∈ V ) |
19 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) |
20 |
1 19
|
climmpt |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐹 ∈ V ) → ( 𝐹 ⇝ 𝐵 ↔ ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) ⇝ 𝐵 ) ) |
21 |
15 18 20
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ⇝ 𝐵 ↔ ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) ⇝ 𝐵 ) ) |
22 |
4 21
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) ⇝ 𝐵 ) |
23 |
6
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ∈ ℂ ) |
24 |
23
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) : 𝑍 ⟶ ℂ ) |
25 |
1 15 24
|
rlimclim |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) ⇝𝑟 𝐵 ↔ ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) ⇝ 𝐵 ) ) |
26 |
22 25
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑛 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑛 ) ) ⇝𝑟 𝐵 ) |
27 |
13 3 14 26
|
fsumrlim |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑛 ) ) ⇝𝑟 Σ 𝑘 ∈ 𝐴 𝐵 ) |
28 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐴 ∈ Fin ) |
29 |
6
|
anass1rs |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑛 ) ∈ ℂ ) |
30 |
28 29
|
fsumcl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑛 ) ∈ ℂ ) |
31 |
30
|
fmpttd |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑛 ) ) : 𝑍 ⟶ ℂ ) |
32 |
1 2 31
|
rlimclim |
⊢ ( 𝜑 → ( ( 𝑛 ∈ 𝑍 ↦ Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑛 ) ) ⇝𝑟 Σ 𝑘 ∈ 𝐴 𝐵 ↔ ( 𝑛 ∈ 𝑍 ↦ Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑛 ) ) ⇝ Σ 𝑘 ∈ 𝐴 𝐵 ) ) |
33 |
27 32
|
mpbid |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ Σ 𝑘 ∈ 𝐴 ( 𝐹 ‘ 𝑛 ) ) ⇝ Σ 𝑘 ∈ 𝐴 𝐵 ) |
34 |
8 33
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 ↦ ( 𝐻 ‘ 𝑛 ) ) ⇝ Σ 𝑘 ∈ 𝐴 𝐵 ) |
35 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑍 ↦ ( 𝐻 ‘ 𝑛 ) ) = ( 𝑛 ∈ 𝑍 ↦ ( 𝐻 ‘ 𝑛 ) ) |
36 |
1 35
|
climmpt |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐻 ∈ 𝑊 ) → ( 𝐻 ⇝ Σ 𝑘 ∈ 𝐴 𝐵 ↔ ( 𝑛 ∈ 𝑍 ↦ ( 𝐻 ‘ 𝑛 ) ) ⇝ Σ 𝑘 ∈ 𝐴 𝐵 ) ) |
37 |
2 5 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝐻 ⇝ Σ 𝑘 ∈ 𝐴 𝐵 ↔ ( 𝑛 ∈ 𝑍 ↦ ( 𝐻 ‘ 𝑛 ) ) ⇝ Σ 𝑘 ∈ 𝐴 𝐵 ) ) |
38 |
34 37
|
mpbird |
⊢ ( 𝜑 → 𝐻 ⇝ Σ 𝑘 ∈ 𝐴 𝐵 ) |