| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fveq2 6906 | . . . . 5
⊢ (𝑗 = 1 → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (seq1( + , (ℕ ×
{𝐴}))‘1)) | 
| 2 |  | oveq1 7438 | . . . . 5
⊢ (𝑗 = 1 → (𝑗 · 𝐴) = (1 · 𝐴)) | 
| 3 | 1, 2 | eqeq12d 2753 | . . . 4
⊢ (𝑗 = 1 → ((seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘1) = (1 · 𝐴))) | 
| 4 | 3 | imbi2d 340 | . . 3
⊢ (𝑗 = 1 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘1) = (1
· 𝐴)))) | 
| 5 |  | fveq2 6906 | . . . . 5
⊢ (𝑗 = 𝑘 → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (seq1( + , (ℕ × {𝐴}))‘𝑘)) | 
| 6 |  | oveq1 7438 | . . . . 5
⊢ (𝑗 = 𝑘 → (𝑗 · 𝐴) = (𝑘 · 𝐴)) | 
| 7 | 5, 6 | eqeq12d 2753 | . . . 4
⊢ (𝑗 = 𝑘 → ((seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴))) | 
| 8 | 7 | imbi2d 340 | . . 3
⊢ (𝑗 = 𝑘 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑘) = (𝑘 · 𝐴)))) | 
| 9 |  | fveq2 6906 | . . . . 5
⊢ (𝑗 = (𝑘 + 1) → (seq1( + , (ℕ ×
{𝐴}))‘𝑗) = (seq1( + , (ℕ ×
{𝐴}))‘(𝑘 + 1))) | 
| 10 |  | oveq1 7438 | . . . . 5
⊢ (𝑗 = (𝑘 + 1) → (𝑗 · 𝐴) = ((𝑘 + 1) · 𝐴)) | 
| 11 | 9, 10 | eqeq12d 2753 | . . . 4
⊢ (𝑗 = (𝑘 + 1) → ((seq1( + , (ℕ ×
{𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴))) | 
| 12 | 11 | imbi2d 340 | . . 3
⊢ (𝑗 = (𝑘 + 1) → ((𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴)))) | 
| 13 |  | fveq2 6906 | . . . . 5
⊢ (𝑗 = 𝑁 → (seq1( + , (ℕ × {𝐴}))‘𝑗) = (seq1( + , (ℕ × {𝐴}))‘𝑁)) | 
| 14 |  | oveq1 7438 | . . . . 5
⊢ (𝑗 = 𝑁 → (𝑗 · 𝐴) = (𝑁 · 𝐴)) | 
| 15 | 13, 14 | eqeq12d 2753 | . . . 4
⊢ (𝑗 = 𝑁 → ((seq1( + , (ℕ × {𝐴}))‘𝑗) = (𝑗 · 𝐴) ↔ (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))) | 
| 16 | 15 | imbi2d 340 | . . 3
⊢ (𝑗 = 𝑁 → ((𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑗) = (𝑗 · 𝐴)) ↔ (𝐴 ∈ ℂ → (seq1( + , (ℕ
× {𝐴}))‘𝑁) = (𝑁 · 𝐴)))) | 
| 17 |  | 1z 12647 | . . . 4
⊢ 1 ∈
ℤ | 
| 18 |  | 1nn 12277 | . . . . . 6
⊢ 1 ∈
ℕ | 
| 19 |  | fvconst2g 7222 | . . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℕ) → ((ℕ × {𝐴})‘1) = 𝐴) | 
| 20 | 18, 19 | mpan2 691 | . . . . 5
⊢ (𝐴 ∈ ℂ → ((ℕ
× {𝐴})‘1) =
𝐴) | 
| 21 |  | mullid 11260 | . . . . 5
⊢ (𝐴 ∈ ℂ → (1
· 𝐴) = 𝐴) | 
| 22 | 20, 21 | eqtr4d 2780 | . . . 4
⊢ (𝐴 ∈ ℂ → ((ℕ
× {𝐴})‘1) = (1
· 𝐴)) | 
| 23 | 17, 22 | seq1i 14056 | . . 3
⊢ (𝐴 ∈ ℂ → (seq1( +
, (ℕ × {𝐴}))‘1) = (1 · 𝐴)) | 
| 24 |  | oveq1 7438 | . . . . . 6
⊢ ((seq1( +
, (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴) → ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴) = ((𝑘 · 𝐴) + 𝐴)) | 
| 25 |  | seqp1 14057 | . . . . . . . . . 10
⊢ (𝑘 ∈
(ℤ≥‘1) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1)))) | 
| 26 |  | nnuz 12921 | . . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) | 
| 27 | 25, 26 | eleq2s 2859 | . . . . . . . . 9
⊢ (𝑘 ∈ ℕ → (seq1( +
, (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1)))) | 
| 28 | 27 | adantl 481 | . . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (seq1( +
, (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1)))) | 
| 29 |  | peano2nn 12278 | . . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) | 
| 30 |  | fvconst2g 7222 | . . . . . . . . . 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 7447 | . . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1(
+ , (ℕ × {𝐴}))‘𝑘) + ((ℕ × {𝐴})‘(𝑘 + 1))) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴)) | 
| 33 | 28, 32 | eqtrd 2777 | . . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (seq1( +
, (ℕ × {𝐴}))‘(𝑘 + 1)) = ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴)) | 
| 34 |  | nncn 12274 | . . . . . . . . 9
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) | 
| 35 |  | id 22 | . . . . . . . . 9
⊢ (𝐴 ∈ ℂ → 𝐴 ∈
ℂ) | 
| 36 |  | ax-1cn 11213 | . . . . . . . . . 10
⊢ 1 ∈
ℂ | 
| 37 |  | adddir 11252 | . . . . . . . . . 10
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ ∧ 𝐴 ∈
ℂ) → ((𝑘 + 1)
· 𝐴) = ((𝑘 · 𝐴) + (1 · 𝐴))) | 
| 38 | 36, 37 | mp3an2 1451 | . . . . . . . . 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 480 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (1
· 𝐴) = 𝐴) | 
| 41 | 40 | oveq2d 7447 | . . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((𝑘 · 𝐴) + (1 · 𝐴)) = ((𝑘 · 𝐴) + 𝐴)) | 
| 42 | 39, 41 | eqtrd 2777 | . . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) · 𝐴) = ((𝑘 · 𝐴) + 𝐴)) | 
| 43 | 33, 42 | eqeq12d 2753 | . . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1(
+ , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴) ↔ ((seq1( + , (ℕ × {𝐴}))‘𝑘) + 𝐴) = ((𝑘 · 𝐴) + 𝐴))) | 
| 44 | 24, 43 | imbitrrid 246 | . . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((seq1(
+ , (ℕ × {𝐴}))‘𝑘) = (𝑘 · 𝐴) → (seq1( + , (ℕ × {𝐴}))‘(𝑘 + 1)) = ((𝑘 + 1) · 𝐴))) | 
| 45 | 44 | expcom 413 | . . . 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 12284 | . 2
⊢ (𝑁 ∈ ℕ → (𝐴 ∈ ℂ → (seq1( +
, (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))) | 
| 48 | 47 | impcom 407 | 1
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( +
, (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴)) |