Step | Hyp | Ref
| Expression |
1 | | ply1mulgsum.a |
. . . 4
⊢ 𝐴 = (coe1‘𝐾) |
2 | | ply1mulgsum.b |
. . . 4
⊢ 𝐵 = (Base‘𝑃) |
3 | | ply1mulgsum.p |
. . . 4
⊢ 𝑃 = (Poly1‘𝑅) |
4 | | eqid 2738 |
. . . 4
⊢
(0g‘𝑅) = (0g‘𝑅) |
5 | 1, 2, 3, 4 | coe1ae0 21387 |
. . 3
⊢ (𝐾 ∈ 𝐵 → ∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) |
6 | 5 | 3ad2ant2 1133 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) |
7 | | ply1mulgsum.c |
. . . . 5
⊢ 𝐶 = (coe1‘𝐿) |
8 | 7, 2, 3, 4 | coe1ae0 21387 |
. . . 4
⊢ (𝐿 ∈ 𝐵 → ∃𝑎 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) |
9 | 8 | 3ad2ant3 1134 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑎 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) |
10 | | nn0addcl 12268 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) → (𝑎 + 𝑏) ∈
ℕ0) |
11 | 10 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑎 + 𝑏) ∈
ℕ0) |
12 | 11 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → (𝑎 + 𝑏) ∈
ℕ0) |
13 | | breq1 5077 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 = (𝑎 + 𝑏) → (𝑠 < 𝑛 ↔ (𝑎 + 𝑏) < 𝑛)) |
14 | 13 | imbi1d 342 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 = (𝑎 + 𝑏) → ((𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))) ↔ ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))) |
15 | 14 | ralbidv 3112 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = (𝑎 + 𝑏) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))) |
16 | 15 | adantl 482 |
. . . . . . . . . . . . . 14
⊢
(((((𝑎 ∈
ℕ0 ∧ 𝑏
∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) ∧ 𝑠 = (𝑎 + 𝑏)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))) |
17 | | r19.26 3095 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑛 ∈
ℕ0 ((𝑎
< 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) ↔ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) |
18 | | nn0cn 12243 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑎 ∈ ℕ0
→ 𝑎 ∈
ℂ) |
19 | 18 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑏 ∈ ℕ0
∧ 𝑎 ∈
ℕ0) → 𝑎 ∈ ℂ) |
20 | | nn0cn 12243 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑏 ∈ ℕ0
→ 𝑏 ∈
ℂ) |
21 | 20 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑏 ∈ ℕ0
∧ 𝑎 ∈
ℕ0) → 𝑏 ∈ ℂ) |
22 | 19, 21 | addcomd 11177 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑏 ∈ ℕ0
∧ 𝑎 ∈
ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎)) |
23 | 22 | 3adant3 1131 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑏 ∈ ℕ0
∧ 𝑎 ∈
ℕ0 ∧ 𝑛
∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎)) |
24 | 23 | breq1d 5084 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑏 ∈ ℕ0
∧ 𝑎 ∈
ℕ0 ∧ 𝑛
∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛)) |
25 | | nn0sumltlt 45686 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑏 ∈ ℕ0
∧ 𝑎 ∈
ℕ0 ∧ 𝑛
∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛 → 𝑎 < 𝑛)) |
26 | 24, 25 | sylbid 239 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑏 ∈ ℕ0
∧ 𝑎 ∈
ℕ0 ∧ 𝑛
∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛)) |
27 | 26 | 3expia 1120 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑏 ∈ ℕ0
∧ 𝑎 ∈
ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛))) |
28 | 27 | ancoms 459 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛))) |
29 | 28 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛))) |
30 | 29 | imp 407 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑎 < 𝑛)) |
31 | 30 | imim1d 82 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → ((𝑎 + 𝑏) < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)))) |
32 | 31 | com23 86 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → (𝐶‘𝑛) = (0g‘𝑅)))) |
33 | 32 | imp 407 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑎 ∈
ℕ0 ∧ 𝑏
∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → (𝐶‘𝑛) = (0g‘𝑅))) |
34 | | nn0sumltlt 45686 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0 ∧ 𝑛
∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛)) |
35 | 34 | 3expia 1120 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛))) |
36 | 35 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛))) |
37 | 36 | imp 407 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → 𝑏 < 𝑛)) |
38 | 37 | imim1d 82 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ((𝑎 + 𝑏) < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) |
39 | 38 | com23 86 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → ((𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → (𝐴‘𝑛) = (0g‘𝑅)))) |
40 | 39 | imp 407 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑎 ∈
ℕ0 ∧ 𝑏
∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → (𝐴‘𝑛) = (0g‘𝑅))) |
41 | 33, 40 | anim12d 609 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑎 ∈
ℕ0 ∧ 𝑏
∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → (((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝐶‘𝑛) = (0g‘𝑅) ∧ (𝐴‘𝑛) = (0g‘𝑅)))) |
42 | 41 | imp 407 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑎 ∈
ℕ0 ∧ 𝑏
∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) ∧ ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → ((𝐶‘𝑛) = (0g‘𝑅) ∧ (𝐴‘𝑛) = (0g‘𝑅))) |
43 | 42 | ancomd 462 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝑎 ∈
ℕ0 ∧ 𝑏
∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) ∧ ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))) |
44 | 43 | exp31 420 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → (((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))) |
45 | 44 | com23 86 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))) |
46 | 45 | ralimdva 3108 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → (∀𝑛 ∈ ℕ0 ((𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))) |
47 | 17, 46 | syl5bir 242 |
. . . . . . . . . . . . . . 15
⊢ (((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))) |
48 | 47 | imp 407 |
. . . . . . . . . . . . . 14
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))) |
49 | 12, 16, 48 | rspcedvd 3563 |
. . . . . . . . . . . . 13
⊢ ((((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))) |
50 | 49 | exp31 420 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))) |
51 | 50 | com23 86 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))) |
52 | 51 | expd 416 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) → (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))) |
53 | 52 | com34 91 |
. . . . . . . . 9
⊢ ((𝑎 ∈ ℕ0
∧ 𝑏 ∈
ℕ0) → (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))) |
54 | 53 | impancom 452 |
. . . . . . . 8
⊢ ((𝑎 ∈ ℕ0
∧ ∀𝑛 ∈
ℕ0 (𝑎 <
𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → (𝑏 ∈ ℕ0 → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))) |
55 | 54 | com14 96 |
. . . . . . 7
⊢
(∀𝑛 ∈
ℕ0 (𝑏 <
𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → (𝑏 ∈ ℕ0 → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ((𝑎 ∈ ℕ0 ∧
∀𝑛 ∈
ℕ0 (𝑎 <
𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))))) |
56 | 55 | impcom 408 |
. . . . . 6
⊢ ((𝑏 ∈ ℕ0
∧ ∀𝑛 ∈
ℕ0 (𝑏 <
𝑛 → (𝐴‘𝑛) = (0g‘𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ((𝑎 ∈ ℕ0 ∧
∀𝑛 ∈
ℕ0 (𝑎 <
𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))) |
57 | 56 | rexlimiva 3210 |
. . . . 5
⊢
(∃𝑏 ∈
ℕ0 ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ((𝑎 ∈ ℕ0 ∧
∀𝑛 ∈
ℕ0 (𝑎 <
𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))) |
58 | 57 | com13 88 |
. . . 4
⊢ ((𝑎 ∈ ℕ0
∧ ∀𝑛 ∈
ℕ0 (𝑎 <
𝑛 → (𝐶‘𝑛) = (0g‘𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))) |
59 | 58 | rexlimiva 3210 |
. . 3
⊢
(∃𝑎 ∈
ℕ0 ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶‘𝑛) = (0g‘𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))))) |
60 | 9, 59 | mpcom 38 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∃𝑏 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑏 < 𝑛 → (𝐴‘𝑛) = (0g‘𝑅)) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅))))) |
61 | 6, 60 | mpd 15 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0
(𝑠 < 𝑛 → ((𝐴‘𝑛) = (0g‘𝑅) ∧ (𝐶‘𝑛) = (0g‘𝑅)))) |