Step |
Hyp |
Ref |
Expression |
1 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
2 |
|
binom2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐴 + 1 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 1 ) ) ) + ( 1 ↑ 2 ) ) ) |
3 |
1 2
|
mpan2 |
⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 1 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 1 ) ) ) + ( 1 ↑ 2 ) ) ) |
4 |
|
mulid1 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · 1 ) = 𝐴 ) |
5 |
4
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( 𝐴 · 1 ) ) = ( 2 · 𝐴 ) ) |
6 |
5
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 1 ) ) ) = ( ( 𝐴 ↑ 2 ) + ( 2 · 𝐴 ) ) ) |
7 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
8 |
7
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( 1 ↑ 2 ) = 1 ) |
9 |
6 8
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 1 ) ) ) + ( 1 ↑ 2 ) ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · 𝐴 ) ) + 1 ) ) |
10 |
3 9
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 1 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · 𝐴 ) ) + 1 ) ) |