Theorem remulneg2d

Description: Product with negative is negative of product. (Contributed by SN, 25-Jan-2025)

	Ref	Expression
Hypotheses	remulneg2d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	remulneg2d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	remulneg2d	⊢ ( 𝜑 → ( 𝐴 · ( 0 −ℝ 𝐵 ) ) = ( 0 −ℝ ( 𝐴 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		remulneg2d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		remulneg2d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
4		resubdi	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · ( 0 −ℝ 𝐵 ) ) = ( ( 𝐴 · 0 ) −ℝ ( 𝐴 · 𝐵 ) ) )
5	1 3 2 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 · ( 0 −ℝ 𝐵 ) ) = ( ( 𝐴 · 0 ) −ℝ ( 𝐴 · 𝐵 ) ) )
6		remul01	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = 0 )
7	1 6	syl	⊢ ( 𝜑 → ( 𝐴 · 0 ) = 0 )
8	7	oveq1d	⊢ ( 𝜑 → ( ( 𝐴 · 0 ) −ℝ ( 𝐴 · 𝐵 ) ) = ( 0 −ℝ ( 𝐴 · 𝐵 ) ) )
9	5 8	eqtrd	⊢ ( 𝜑 → ( 𝐴 · ( 0 −ℝ 𝐵 ) ) = ( 0 −ℝ ( 𝐴 · 𝐵 ) ) )