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Metamath Proof Explorer

Theorem binom21

Description: Special case of binom2 where B = 1 . (Contributed by Scott Fenton, 11-May-2014)

Ref Expression
Assertion binom21 ( 𝐴 ∈ ℂ → ( ( 𝐴 + 1 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · 𝐴 ) ) + 1 ) )

Proof

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 mulrid ⊢ ( 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 ) )