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Metamath Proof Explorer

Theorem binom2subi

Description: Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013)

Ref Expression
Hypotheses binom2subi.1 ⊢ 𝐴 ∈ ℂ
binom2subi.2 ⊢ 𝐵 ∈ ℂ
Assertion binom2subi ( ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) − ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) )

Proof

Step Hyp Ref Expression
1 binom2subi.1 ⊢ 𝐴 ∈ ℂ
2 binom2subi.2 ⊢ 𝐵 ∈ ℂ
3 binom2sub ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) − ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) )
4 1 2 3 mp2an ⊢ ( ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) − ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) )