Step |
Hyp |
Ref |
Expression |
1 |
|
ntrivcvgmullem.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
ntrivcvgmullem.2 |
⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) |
3 |
|
ntrivcvgmullem.3 |
⊢ ( 𝜑 → 𝑃 ∈ 𝑍 ) |
4 |
|
ntrivcvgmullem.4 |
⊢ ( 𝜑 → 𝑋 ≠ 0 ) |
5 |
|
ntrivcvgmullem.5 |
⊢ ( 𝜑 → 𝑌 ≠ 0 ) |
6 |
|
ntrivcvgmullem.6 |
⊢ ( 𝜑 → seq 𝑁 ( · , 𝐹 ) ⇝ 𝑋 ) |
7 |
|
ntrivcvgmullem.7 |
⊢ ( 𝜑 → seq 𝑃 ( · , 𝐺 ) ⇝ 𝑌 ) |
8 |
|
ntrivcvgmullem.8 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
9 |
|
ntrivcvgmullem.9 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
10 |
|
ntrivcvgmullem.a |
⊢ ( 𝜑 → 𝑁 ≤ 𝑃 ) |
11 |
|
ntrivcvgmullem.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ 𝑘 ) ) ) |
12 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝑁 ) |
13 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
14 |
1 13
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
15 |
14 2
|
sselid |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
16 |
14 3
|
sselid |
⊢ ( 𝜑 → 𝑃 ∈ ℤ ) |
17 |
|
eluz |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑃 ∈ ℤ ) → ( 𝑃 ∈ ( ℤ≥ ‘ 𝑁 ) ↔ 𝑁 ≤ 𝑃 ) ) |
18 |
15 16 17
|
syl2anc |
⊢ ( 𝜑 → ( 𝑃 ∈ ( ℤ≥ ‘ 𝑁 ) ↔ 𝑁 ≤ 𝑃 ) ) |
19 |
10 18
|
mpbird |
⊢ ( 𝜑 → 𝑃 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
20 |
1
|
uztrn2 |
⊢ ( ( 𝑁 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑘 ∈ 𝑍 ) |
21 |
2 20
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑘 ∈ 𝑍 ) |
22 |
21 8
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
23 |
12 19 6 4 22
|
ntrivcvgtail |
⊢ ( 𝜑 → ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) ≠ 0 ∧ seq 𝑃 ( · , 𝐹 ) ⇝ ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) ) ) |
24 |
23
|
simprd |
⊢ ( 𝜑 → seq 𝑃 ( · , 𝐹 ) ⇝ ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) ) |
25 |
|
climcl |
⊢ ( seq 𝑃 ( · , 𝐹 ) ⇝ ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) → ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) ∈ ℂ ) |
26 |
24 25
|
syl |
⊢ ( 𝜑 → ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) ∈ ℂ ) |
27 |
|
climcl |
⊢ ( seq 𝑃 ( · , 𝐺 ) ⇝ 𝑌 → 𝑌 ∈ ℂ ) |
28 |
7 27
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ ℂ ) |
29 |
23
|
simpld |
⊢ ( 𝜑 → ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) ≠ 0 ) |
30 |
26 28 29 5
|
mulne0d |
⊢ ( 𝜑 → ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ≠ 0 ) |
31 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑃 ) = ( ℤ≥ ‘ 𝑃 ) |
32 |
|
seqex |
⊢ seq 𝑃 ( · , 𝐻 ) ∈ V |
33 |
32
|
a1i |
⊢ ( 𝜑 → seq 𝑃 ( · , 𝐻 ) ∈ V ) |
34 |
1
|
uztrn2 |
⊢ ( ( 𝑃 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑃 ) ) → 𝑘 ∈ 𝑍 ) |
35 |
3 34
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑃 ) ) → 𝑘 ∈ 𝑍 ) |
36 |
35 8
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
37 |
31 16 36
|
prodf |
⊢ ( 𝜑 → seq 𝑃 ( · , 𝐹 ) : ( ℤ≥ ‘ 𝑃 ) ⟶ ℂ ) |
38 |
37
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) → ( seq 𝑃 ( · , 𝐹 ) ‘ 𝑗 ) ∈ ℂ ) |
39 |
35 9
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑃 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
40 |
31 16 39
|
prodf |
⊢ ( 𝜑 → seq 𝑃 ( · , 𝐺 ) : ( ℤ≥ ‘ 𝑃 ) ⟶ ℂ ) |
41 |
40
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) → ( seq 𝑃 ( · , 𝐺 ) ‘ 𝑗 ) ∈ ℂ ) |
42 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) |
43 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) ∧ 𝑘 ∈ ( 𝑃 ... 𝑗 ) ) → 𝜑 ) |
44 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) → 𝑃 ∈ 𝑍 ) |
45 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑃 ... 𝑗 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑃 ) ) |
46 |
44 45 34
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) ∧ 𝑘 ∈ ( 𝑃 ... 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
47 |
43 46 8
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) ∧ 𝑘 ∈ ( 𝑃 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
48 |
43 46 9
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) ∧ 𝑘 ∈ ( 𝑃 ... 𝑗 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ ) |
49 |
43 46 11
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) ∧ 𝑘 ∈ ( 𝑃 ... 𝑗 ) ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ 𝑘 ) ) ) |
50 |
42 47 48 49
|
prodfmul |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑃 ) ) → ( seq 𝑃 ( · , 𝐻 ) ‘ 𝑗 ) = ( ( seq 𝑃 ( · , 𝐹 ) ‘ 𝑗 ) · ( seq 𝑃 ( · , 𝐺 ) ‘ 𝑗 ) ) ) |
51 |
31 16 24 33 7 38 41 50
|
climmul |
⊢ ( 𝜑 → seq 𝑃 ( · , 𝐻 ) ⇝ ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ) |
52 |
|
ovex |
⊢ ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ∈ V |
53 |
|
neeq1 |
⊢ ( 𝑤 = ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) → ( 𝑤 ≠ 0 ↔ ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ≠ 0 ) ) |
54 |
|
breq2 |
⊢ ( 𝑤 = ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) → ( seq 𝑃 ( · , 𝐻 ) ⇝ 𝑤 ↔ seq 𝑃 ( · , 𝐻 ) ⇝ ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ) ) |
55 |
53 54
|
anbi12d |
⊢ ( 𝑤 = ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) → ( ( 𝑤 ≠ 0 ∧ seq 𝑃 ( · , 𝐻 ) ⇝ 𝑤 ) ↔ ( ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ≠ 0 ∧ seq 𝑃 ( · , 𝐻 ) ⇝ ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ) ) ) |
56 |
52 55
|
spcev |
⊢ ( ( ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ≠ 0 ∧ seq 𝑃 ( · , 𝐻 ) ⇝ ( ( ⇝ ‘ seq 𝑃 ( · , 𝐹 ) ) · 𝑌 ) ) → ∃ 𝑤 ( 𝑤 ≠ 0 ∧ seq 𝑃 ( · , 𝐻 ) ⇝ 𝑤 ) ) |
57 |
30 51 56
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑤 ( 𝑤 ≠ 0 ∧ seq 𝑃 ( · , 𝐻 ) ⇝ 𝑤 ) ) |
58 |
|
seqeq1 |
⊢ ( 𝑞 = 𝑃 → seq 𝑞 ( · , 𝐻 ) = seq 𝑃 ( · , 𝐻 ) ) |
59 |
58
|
breq1d |
⊢ ( 𝑞 = 𝑃 → ( seq 𝑞 ( · , 𝐻 ) ⇝ 𝑤 ↔ seq 𝑃 ( · , 𝐻 ) ⇝ 𝑤 ) ) |
60 |
59
|
anbi2d |
⊢ ( 𝑞 = 𝑃 → ( ( 𝑤 ≠ 0 ∧ seq 𝑞 ( · , 𝐻 ) ⇝ 𝑤 ) ↔ ( 𝑤 ≠ 0 ∧ seq 𝑃 ( · , 𝐻 ) ⇝ 𝑤 ) ) ) |
61 |
60
|
exbidv |
⊢ ( 𝑞 = 𝑃 → ( ∃ 𝑤 ( 𝑤 ≠ 0 ∧ seq 𝑞 ( · , 𝐻 ) ⇝ 𝑤 ) ↔ ∃ 𝑤 ( 𝑤 ≠ 0 ∧ seq 𝑃 ( · , 𝐻 ) ⇝ 𝑤 ) ) ) |
62 |
61
|
rspcev |
⊢ ( ( 𝑃 ∈ 𝑍 ∧ ∃ 𝑤 ( 𝑤 ≠ 0 ∧ seq 𝑃 ( · , 𝐻 ) ⇝ 𝑤 ) ) → ∃ 𝑞 ∈ 𝑍 ∃ 𝑤 ( 𝑤 ≠ 0 ∧ seq 𝑞 ( · , 𝐻 ) ⇝ 𝑤 ) ) |
63 |
3 57 62
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑞 ∈ 𝑍 ∃ 𝑤 ( 𝑤 ≠ 0 ∧ seq 𝑞 ( · , 𝐻 ) ⇝ 𝑤 ) ) |