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Metamath Proof Explorer

Theorem subsqi

Description: Factor the difference of two squares. (Contributed by NM, 7-Feb-2005)

Ref Expression
Hypotheses binom2.1 ⊢ 𝐴 ∈ ℂ
binom2.2 ⊢ 𝐵 ∈ ℂ
Assertion subsqi ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = ( ( 𝐴 + 𝐵 ) · ( 𝐴 − 𝐵 ) )

Proof

Step Hyp Ref Expression
1 binom2.1 ⊢ 𝐴 ∈ ℂ
2 binom2.2 ⊢ 𝐵 ∈ ℂ
3 subsq ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = ( ( 𝐴 + 𝐵 ) · ( 𝐴 − 𝐵 ) ) )
4 1 2 3 mp2an ⊢ ( ( 𝐴 ↑ 2 ) − ( 𝐵 ↑ 2 ) ) = ( ( 𝐴 + 𝐵 ) · ( 𝐴 − 𝐵 ) )