Theorem leneg3d

Description: Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	leneg3d.1	$$⊢\phi \to A\in ℝ$$
	leneg3d.2	$$⊢\phi \to B\in ℝ$$
Assertion	leneg3d	$$⊢\phi \to \left(-A\le B\leftrightarrow -B\le A\right)$$

Proof

Step	Hyp	Ref	Expression
1		leneg3d.1	$$⊢\phi \to A\in ℝ$$
2		leneg3d.2	$$⊢\phi \to B\in ℝ$$
3	1	renegcld	$$⊢\phi \to -A\in ℝ$$
4	3 2	lenegd	$$⊢\phi \to \left(-A\le B\leftrightarrow -B\le -\left(-A\right)\right)$$
5	1	recnd	$$⊢\phi \to A\in ℂ$$
6	5	negnegd	$$⊢\phi \to -\left(-A\right)=A$$
7	6	breq2d	$$⊢\phi \to \left(-B\le -\left(-A\right)\leftrightarrow -B\le A\right)$$
8	4 7	bitrd	$$⊢\phi \to \left(-A\le B\leftrightarrow -B\le A\right)$$