Theorem remulneg2d

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

	Ref	Expression
Hypotheses	remulneg2d.a	$$⊢\phi \to A\in ℝ$$
	remulneg2d.b	$$⊢\phi \to B\in ℝ$$
Assertion	remulneg2d	$$⊢\phi \to A\left(0{-}_{ℝ}B\right)=0{-}_{ℝ}AB$$

Proof

Step	Hyp	Ref	Expression
1		remulneg2d.a	$$⊢\phi \to A\in ℝ$$
2		remulneg2d.b	$$⊢\phi \to B\in ℝ$$
3		0red	$$⊢\phi \to 0\in ℝ$$
4		resubdi	$$⊢\left(A\in ℝ\wedge 0\in ℝ\wedge B\in ℝ\right)\to A\left(0{-}_{ℝ}B\right)=A\cdot 0{-}_{ℝ}AB$$
5	1 3 2 4	syl3anc	$$⊢\phi \to A\left(0{-}_{ℝ}B\right)=A\cdot 0{-}_{ℝ}AB$$
6		remul01	$$⊢A\in ℝ\to A\cdot 0=0$$
7	1 6	syl	$$⊢\phi \to A\cdot 0=0$$
8	7	oveq1d	$$⊢\phi \to A\cdot 0{-}_{ℝ}AB=0{-}_{ℝ}AB$$
9	5 8	eqtrd	$$⊢\phi \to A\left(0{-}_{ℝ}B\right)=0{-}_{ℝ}AB$$