Theorem binom2d

Description: Deduction form of binom2 . (Contributed by Igor Ieskov, 14-Dec-2023)

	Ref	Expression
Hypotheses	binom2d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	binom2d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	binom2d	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		binom2d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		binom2d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		binom2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) )