Step |
Hyp |
Ref |
Expression |
1 |
|
clim2ser.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
isermulc2.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
isermulc2.4 |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
4 |
|
isermulc2.5 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ⇝ 𝐴 ) |
5 |
|
isermulc2.6 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
6 |
|
isermulc2.7 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐶 · ( 𝐹 ‘ 𝑘 ) ) ) |
7 |
|
seqex |
⊢ seq 𝑀 ( + , 𝐺 ) ∈ V |
8 |
7
|
a1i |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐺 ) ∈ V ) |
9 |
1 2 5
|
serf |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℂ ) |
10 |
9
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ∈ ℂ ) |
11 |
|
addcl |
⊢ ( ( 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( 𝑘 + 𝑥 ) ∈ ℂ ) |
12 |
11
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ) ) → ( 𝑘 + 𝑥 ) ∈ ℂ ) |
13 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝐶 ∈ ℂ ) |
14 |
|
adddi |
⊢ ( ( 𝐶 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( 𝐶 · ( 𝑘 + 𝑥 ) ) = ( ( 𝐶 · 𝑘 ) + ( 𝐶 · 𝑥 ) ) ) |
15 |
14
|
3expb |
⊢ ( ( 𝐶 ∈ ℂ ∧ ( 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ) ) → ( 𝐶 · ( 𝑘 + 𝑥 ) ) = ( ( 𝐶 · 𝑘 ) + ( 𝐶 · 𝑥 ) ) ) |
16 |
13 15
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ ( 𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ) ) → ( 𝐶 · ( 𝑘 + 𝑥 ) ) = ( ( 𝐶 · 𝑘 ) + ( 𝐶 · 𝑥 ) ) ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ 𝑍 ) |
18 |
17 1
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
19 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
20 |
19 1
|
eleqtrrdi |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → 𝑘 ∈ 𝑍 ) |
21 |
20 5
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
22 |
21
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
23 |
20 6
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐶 · ( 𝐹 ‘ 𝑘 ) ) ) |
24 |
23
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐶 · ( 𝐹 ‘ 𝑘 ) ) ) |
25 |
12 16 18 22 24
|
seqdistr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑗 ) = ( 𝐶 · ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ) ) |
26 |
1 2 4 3 8 10 25
|
climmulc2 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐺 ) ⇝ ( 𝐶 · 𝐴 ) ) |