Step | Hyp | Ref
| Expression |
1 | | fveq2 6766 |
. . . . 5
⊢ (𝑗 = 1 → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (seq1( + , (ℕ ×
{𝐴}))‘1)) |
2 | | oveq1 7274 |
. . . . 5
⊢ (𝑗 = 1 → (𝑗 · 𝐴) = (1 · 𝐴)) |
3 | 1, 2 | eqeq12d 2754 |
. . . 4
⊢ (𝑗 = 1 → ((seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘1) = (1 · 𝐴))) |
4 | 3 | imbi2d 341 |
. . 3
⊢ (𝑗 = 1 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘1) = (1
· 𝐴)))) |
5 | | fveq2 6766 |
. . . . 5
⊢ (𝑗 = 𝑘 → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (seq1( + , (ℕ × {𝐴}))‘𝑘)) |
6 | | oveq1 7274 |
. . . . 5
⊢ (𝑗 = 𝑘 → (𝑗 · 𝐴) = (𝑘 · 𝐴)) |
7 | 5, 6 | eqeq12d 2754 |
. . . 4
⊢ (𝑗 = 𝑘 → ((seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴))) |
8 | 7 | imbi2d 341 |
. . 3
⊢ (𝑗 = 𝑘 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑘) = (𝑘 · 𝐴)))) |
9 | | fveq2 6766 |
. . . . 5
⊢ (𝑗 = (𝑘 + 1) → (seq1( + , (ℕ ×
{𝐴}))‘𝑗) = (seq1( + , (ℕ ×
{𝐴}))‘(𝑘 + 1))) |
10 | | oveq1 7274 |
. . . . 5
⊢ (𝑗 = (𝑘 + 1) → (𝑗 · 𝐴) = ((𝑘 + 1) · 𝐴)) |
11 | 9, 10 | eqeq12d 2754 |
. . . 4
⊢ (𝑗 = (𝑘 + 1) → ((seq1( + , (ℕ ×
{𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴))) |
12 | 11 | imbi2d 341 |
. . 3
⊢ (𝑗 = (𝑘 + 1) → ((𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴)))) |
13 | | fveq2 6766 |
. . . . 5
⊢ (𝑗 = 𝑁 → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (seq1( + , (ℕ × {𝐴}))‘𝑁)) |
14 | | oveq1 7274 |
. . . . 5
⊢ (𝑗 = 𝑁 → (𝑗 · 𝐴) = (𝑁 · 𝐴)) |
15 | 13, 14 | eqeq12d 2754 |
. . . 4
⊢ (𝑗 = 𝑁 → ((seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))) |
16 | 15 | imbi2d 341 |
. . 3
⊢ (𝑗 = 𝑁 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑁) = (𝑁 · 𝐴)))) |
17 | | 1z 12360 |
. . . 4
⊢ 1 ∈
ℤ |
18 | | 1nn 11994 |
. . . . . 6
⊢ 1 ∈
ℕ |
19 | | fvconst2g 7069 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℕ) → ((ℕ × {𝐴})‘1) = 𝐴) |
20 | 18, 19 | mpan2 688 |
. . . . 5
⊢ (𝐴 ∈ ℂ → ((ℕ
× {𝐴})‘1) =
𝐴) |
21 | | mulid2 10984 |
. . . . 5
⊢ (𝐴 ∈ ℂ → (1
· 𝐴) = 𝐴) |
22 | 20, 21 | eqtr4d 2781 |
. . . 4
⊢ (𝐴 ∈ ℂ → ((ℕ
× {𝐴})‘1) = (1
· 𝐴)) |
23 | 17, 22 | seq1i 13745 |
. . 3
⊢ (𝐴 ∈ ℂ → (seq1( +
, (ℕ × {𝐴}))‘1) = (1 · 𝐴)) |
24 | | oveq1 7274 |
. . . . . 6
⊢ ((seq1( +
, (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴) → ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴) = ((𝑘 · 𝐴) + 𝐴)) |
25 | | seqp1 13746 |
. . . . . . . . . 10
⊢ (𝑘 ∈
(ℤ≥‘1) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1)))) |
26 | | nnuz 12631 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
27 | 25, 26 | eleq2s 2857 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ → (seq1( +
, (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1)))) |
28 | 27 | adantl 482 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (seq1( +
, (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1)))) |
29 | | peano2nn 11995 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
30 | | fvconst2g 7069 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ (𝑘 + 1) ∈ ℕ) →
((ℕ × {𝐴})‘(𝑘 + 1)) = 𝐴) |
31 | 29, 30 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) →
((ℕ × {𝐴})‘(𝑘 + 1)) = 𝐴) |
32 | 31 | oveq2d 7283 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1(
+ , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1))) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴)) |
33 | 28, 32 | eqtrd 2778 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (seq1( +
, (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴)) |
34 | | nncn 11991 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
35 | | id 22 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℂ → 𝐴 ∈
ℂ) |
36 | | ax-1cn 10939 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
37 | | adddir 10976 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ ∧ 𝐴 ∈
ℂ) → ((𝑘 + 1)
· 𝐴) = ((𝑘 · 𝐴) + (1 · 𝐴))) |
38 | 36, 37 | mp3an2 1448 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑘 + 1) · 𝐴) = ((𝑘 · 𝐴) + (1 · 𝐴))) |
39 | 34, 35, 38 | syl2anr 597 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) · 𝐴) = ((𝑘 · 𝐴) + (1 · 𝐴))) |
40 | 21 | adantr 481 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (1
· 𝐴) = 𝐴) |
41 | 40 | oveq2d 7283 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((𝑘 · 𝐴) + (1 · 𝐴)) = ((𝑘 · 𝐴) + 𝐴)) |
42 | 39, 41 | eqtrd 2778 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) · 𝐴) = ((𝑘 · 𝐴) + 𝐴)) |
43 | 33, 42 | eqeq12d 2754 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1(
+ , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴) ↔ ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴) = ((𝑘 · 𝐴) + 𝐴))) |
44 | 24, 43 | syl5ibr 245 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1(
+ , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴))) |
45 | 44 | expcom 414 |
. . . 4
⊢ (𝑘 ∈ ℕ → (𝐴 ∈ ℂ → ((seq1( +
, (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴)))) |
46 | 45 | a2d 29 |
. . 3
⊢ (𝑘 ∈ ℕ → ((𝐴 ∈ ℂ → (seq1( +
, (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴)) → (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴)))) |
47 | 4, 8, 12, 16, 23, 46 | nnind 12001 |
. 2
⊢ (𝑁 ∈ ℕ → (𝐴 ∈ ℂ → (seq1( +
, (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))) |
48 | 47 | impcom 408 |
1
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( +
, (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴)) |