Theorem int-sqgeq0d

Description: SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-sqgeq0d.1	$$⊢\phi \to A\in ℝ$$
	int-sqgeq0d.2	$$⊢\phi \to B\in ℝ$$
	int-sqgeq0d.3	$$⊢\phi \to A=B$$
Assertion	int-sqgeq0d	$$⊢\phi \to 0\le AB$$

Proof

Step	Hyp	Ref	Expression
1		int-sqgeq0d.1	$$⊢\phi \to A\in ℝ$$
2		int-sqgeq0d.2	$$⊢\phi \to B\in ℝ$$
3		int-sqgeq0d.3	$$⊢\phi \to A=B$$
4	1	sqge0d	$$⊢\phi \to 0\le A^2$$
5	3	oveq1d	$$⊢\phi \to A^2=B^2$$
6	2	recnd	$$⊢\phi \to B\in ℂ$$
7	6	sqvald	$$⊢\phi \to B^2=BB$$
8		eqcom	$$⊢A=B\leftrightarrow B=A$$
9	8	imbi2i	$$⊢\left(\phi \to A=B\right)\leftrightarrow \left(\phi \to B=A\right)$$
10	3 9	mpbi	$$⊢\phi \to B=A$$
11	10	oveq1d	$$⊢\phi \to BB=AB$$
12	7 11	eqtrd	$$⊢\phi \to B^2=AB$$
13	5 12	eqtrd	$$⊢\phi \to A^2=AB$$
14	4 13	breqtrd	$$⊢\phi \to 0\le AB$$