Step | Hyp | Ref
| Expression |
1 | | ntrivcvgfvn0.4 |
. 2
⊢ (𝜑 → 𝑋 ≠ 0) |
2 | | fclim 15114 |
. . . . . . . 8
⊢ ⇝
:dom ⇝ ⟶ℂ |
3 | | ffun 6548 |
. . . . . . . 8
⊢ ( ⇝
:dom ⇝ ⟶ℂ → Fun ⇝ ) |
4 | 2, 3 | ax-mp 5 |
. . . . . . 7
⊢ Fun
⇝ |
5 | | ntrivcvgfvn0.3 |
. . . . . . 7
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) |
6 | | funbrfv 6763 |
. . . . . . 7
⊢ (Fun
⇝ → (seq𝑀(
· , 𝐹) ⇝ 𝑋 → ( ⇝
‘seq𝑀( · ,
𝐹)) = 𝑋)) |
7 | 4, 5, 6 | mpsyl 68 |
. . . . . 6
⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋) |
8 | 7 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋) |
9 | | eqid 2737 |
. . . . . . 7
⊢
(ℤ≥‘𝑁) = (ℤ≥‘𝑁) |
10 | | ntrivcvgfvn0.1 |
. . . . . . . . . 10
⊢ 𝑍 =
(ℤ≥‘𝑀) |
11 | | uzssz 12459 |
. . . . . . . . . 10
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
12 | 10, 11 | eqsstri 3935 |
. . . . . . . . 9
⊢ 𝑍 ⊆
ℤ |
13 | | ntrivcvgfvn0.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
14 | 12, 13 | sselid 3898 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
15 | 14 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑁 ∈ ℤ) |
16 | | seqex 13576 |
. . . . . . . 8
⊢ seq𝑀( · , 𝐹) ∈ V |
17 | 16 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ∈ V) |
18 | | 0cnd 10826 |
. . . . . . 7
⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 0 ∈
ℂ) |
19 | | fveqeq2 6726 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) = 0)) |
20 | 19 | imbi2d 344 |
. . . . . . . . 9
⊢ (𝑚 = 𝑁 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0))) |
21 | | fveqeq2 6726 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) = 0)) |
22 | 21 | imbi2d 344 |
. . . . . . . . 9
⊢ (𝑚 = 𝑛 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0))) |
23 | | fveqeq2 6726 |
. . . . . . . . . 10
⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0)) |
24 | 23 | imbi2d 344 |
. . . . . . . . 9
⊢ (𝑚 = (𝑛 + 1) → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))) |
25 | | fveqeq2 6726 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑘 → ((seq𝑀( · , 𝐹)‘𝑚) = 0 ↔ (seq𝑀( · , 𝐹)‘𝑘) = 0)) |
26 | 25 | imbi2d 344 |
. . . . . . . . 9
⊢ (𝑚 = 𝑘 → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑚) = 0) ↔ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0))) |
27 | | simpr 488 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑁) = 0) |
28 | 13, 10 | eleqtrdi 2848 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
29 | | uztrn 12456 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
30 | 28, 29 | sylan2 596 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝜑) → 𝑛 ∈ (ℤ≥‘𝑀)) |
31 | 30 | 3adant3 1134 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → 𝑛 ∈ (ℤ≥‘𝑀)) |
32 | | seqp1 13589 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) |
33 | 31, 32 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) |
34 | | oveq1 7220 |
. . . . . . . . . . . . . 14
⊢
((seq𝑀( · ,
𝐹)‘𝑛) = 0 → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1)))) |
35 | 34 | 3ad2ant3 1137 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) = (0 · (𝐹‘(𝑛 + 1)))) |
36 | | peano2uz 12497 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈
(ℤ≥‘𝑁) → (𝑛 + 1) ∈
(ℤ≥‘𝑁)) |
37 | 10 | uztrn2 12457 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ 𝑍 ∧ (𝑛 + 1) ∈
(ℤ≥‘𝑁)) → (𝑛 + 1) ∈ 𝑍) |
38 | 13, 36, 37 | syl2an 599 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝑛 + 1) ∈ 𝑍) |
39 | | ntrivcvgfvn0.5 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
40 | 39 | ralrimiva 3105 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
41 | | fveq2 6717 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1))) |
42 | 41 | eleq1d 2822 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ)) |
43 | 42 | rspcv 3532 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝐹‘(𝑛 + 1)) ∈ ℂ)) |
44 | 40, 43 | mpan9 510 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ∈ ℂ) |
45 | 38, 44 | syldan 594 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ) |
46 | 45 | ancoms 462 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ) |
47 | 46 | mul02d 11030 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝜑) → (0 · (𝐹‘(𝑛 + 1))) = 0) |
48 | 47 | 3adant3 1134 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (0 · (𝐹‘(𝑛 + 1))) = 0) |
49 | 33, 35, 48 | 3eqtrd 2781 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈
(ℤ≥‘𝑁) ∧ 𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑛) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0) |
50 | 49 | 3exp 1121 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(ℤ≥‘𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))) |
51 | 50 | adantrd 495 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(ℤ≥‘𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ((seq𝑀( · , 𝐹)‘𝑛) = 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))) |
52 | 51 | a2d 29 |
. . . . . . . . 9
⊢ (𝑛 ∈
(ℤ≥‘𝑁) → (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑛) = 0) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = 0))) |
53 | 20, 22, 24, 26, 27, 52 | uzind4i 12506 |
. . . . . . . 8
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → (seq𝑀( · , 𝐹)‘𝑘) = 0)) |
54 | 53 | impcom 411 |
. . . . . . 7
⊢ (((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹)‘𝑘) = 0) |
55 | 9, 15, 17, 18, 54 | climconst 15104 |
. . . . . 6
⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → seq𝑀( · , 𝐹) ⇝ 0) |
56 | | funbrfv 6763 |
. . . . . 6
⊢ (Fun
⇝ → (seq𝑀(
· , 𝐹) ⇝ 0
→ ( ⇝ ‘seq𝑀( · , 𝐹)) = 0)) |
57 | 4, 55, 56 | mpsyl 68 |
. . . . 5
⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → ( ⇝ ‘seq𝑀( · , 𝐹)) = 0) |
58 | 8, 57 | eqtr3d 2779 |
. . . 4
⊢ ((𝜑 ∧ (seq𝑀( · , 𝐹)‘𝑁) = 0) → 𝑋 = 0) |
59 | 58 | ex 416 |
. . 3
⊢ (𝜑 → ((seq𝑀( · , 𝐹)‘𝑁) = 0 → 𝑋 = 0)) |
60 | 59 | necon3d 2961 |
. 2
⊢ (𝜑 → (𝑋 ≠ 0 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)) |
61 | 1, 60 | mpd 15 |
1
⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0) |