Step |
Hyp |
Ref |
Expression |
1 |
|
pser.g |
⊢ 𝐺 = ( 𝑥 ∈ ℂ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑛 ) · ( 𝑥 ↑ 𝑛 ) ) ) ) |
2 |
|
radcnv.a |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
3 |
|
fveq2 |
⊢ ( 𝑟 = 0 → ( 𝐺 ‘ 𝑟 ) = ( 𝐺 ‘ 0 ) ) |
4 |
3
|
seqeq3d |
⊢ ( 𝑟 = 0 → seq 0 ( + , ( 𝐺 ‘ 𝑟 ) ) = seq 0 ( + , ( 𝐺 ‘ 0 ) ) ) |
5 |
4
|
eleq1d |
⊢ ( 𝑟 = 0 → ( seq 0 ( + , ( 𝐺 ‘ 𝑟 ) ) ∈ dom ⇝ ↔ seq 0 ( + , ( 𝐺 ‘ 0 ) ) ∈ dom ⇝ ) ) |
6 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
7 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
8 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
9 |
|
snfi |
⊢ { 0 } ∈ Fin |
10 |
9
|
a1i |
⊢ ( 𝜑 → { 0 } ∈ Fin ) |
11 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
12 |
11
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℕ0 ) |
13 |
12
|
snssd |
⊢ ( 𝜑 → { 0 } ⊆ ℕ0 ) |
14 |
|
ifid |
⊢ if ( 𝑘 ∈ { 0 } , ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) , ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) ) = ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) |
15 |
|
0cnd |
⊢ ( 𝜑 → 0 ∈ ℂ ) |
16 |
1
|
pserval2 |
⊢ ( ( 0 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) · ( 0 ↑ 𝑘 ) ) ) |
17 |
15 16
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) · ( 0 ↑ 𝑘 ) ) ) |
18 |
17
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 ∈ { 0 } ) → ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) · ( 0 ↑ 𝑘 ) ) ) |
19 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
20 |
|
elnn0 |
⊢ ( 𝑘 ∈ ℕ0 ↔ ( 𝑘 ∈ ℕ ∨ 𝑘 = 0 ) ) |
21 |
19 20
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ℕ ∨ 𝑘 = 0 ) ) |
22 |
21
|
ord |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ¬ 𝑘 ∈ ℕ → 𝑘 = 0 ) ) |
23 |
|
velsn |
⊢ ( 𝑘 ∈ { 0 } ↔ 𝑘 = 0 ) |
24 |
22 23
|
syl6ibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ¬ 𝑘 ∈ ℕ → 𝑘 ∈ { 0 } ) ) |
25 |
24
|
con1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ¬ 𝑘 ∈ { 0 } → 𝑘 ∈ ℕ ) ) |
26 |
25
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 ∈ { 0 } ) → 𝑘 ∈ ℕ ) |
27 |
26
|
0expd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 ∈ { 0 } ) → ( 0 ↑ 𝑘 ) = 0 ) |
28 |
27
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 ∈ { 0 } ) → ( ( 𝐴 ‘ 𝑘 ) · ( 0 ↑ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) · 0 ) ) |
29 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
30 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 ∈ { 0 } ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
31 |
30
|
mul01d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 ∈ { 0 } ) → ( ( 𝐴 ‘ 𝑘 ) · 0 ) = 0 ) |
32 |
18 28 31
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 ∈ { 0 } ) → ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) = 0 ) |
33 |
32
|
ifeq2da |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → if ( 𝑘 ∈ { 0 } , ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) , ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) ) = if ( 𝑘 ∈ { 0 } , ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) , 0 ) ) |
34 |
14 33
|
eqtr3id |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) = if ( 𝑘 ∈ { 0 } , ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) , 0 ) ) |
35 |
13
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 0 } ) → 𝑘 ∈ ℕ0 ) |
36 |
1 2 15
|
psergf |
⊢ ( 𝜑 → ( 𝐺 ‘ 0 ) : ℕ0 ⟶ ℂ ) |
37 |
36
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) ∈ ℂ ) |
38 |
35 37
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 0 } ) → ( ( 𝐺 ‘ 0 ) ‘ 𝑘 ) ∈ ℂ ) |
39 |
7 8 10 13 34 38
|
fsumcvg3 |
⊢ ( 𝜑 → seq 0 ( + , ( 𝐺 ‘ 0 ) ) ∈ dom ⇝ ) |
40 |
5 6 39
|
elrabd |
⊢ ( 𝜑 → 0 ∈ { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝐺 ‘ 𝑟 ) ) ∈ dom ⇝ } ) |