Theorem int-sqgeq0d

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

	Ref	Expression
Hypotheses	int-sqgeq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	int-sqgeq0d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	int-sqgeq0d.3	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	int-sqgeq0d	⊢ ( 𝜑 → 0 ≤ ( 𝐴 · 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		int-sqgeq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		int-sqgeq0d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		int-sqgeq0d.3	⊢ ( 𝜑 → 𝐴 = 𝐵 )
4	1	sqge0d	⊢ ( 𝜑 → 0 ≤ ( 𝐴 ↑ 2 ) )
5	3	oveq1d	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) )
6	2	recnd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
7	6	sqvald	⊢ ( 𝜑 → ( 𝐵 ↑ 2 ) = ( 𝐵 · 𝐵 ) )
8		eqcom	⊢ ( 𝐴 = 𝐵 ↔ 𝐵 = 𝐴 )
9	8	imbi2i	⊢ ( ( 𝜑 → 𝐴 = 𝐵 ) ↔ ( 𝜑 → 𝐵 = 𝐴 ) )
10	3 9	mpbi	⊢ ( 𝜑 → 𝐵 = 𝐴 )
11	10	oveq1d	⊢ ( 𝜑 → ( 𝐵 · 𝐵 ) = ( 𝐴 · 𝐵 ) )
12	7 11	eqtrd	⊢ ( 𝜑 → ( 𝐵 ↑ 2 ) = ( 𝐴 · 𝐵 ) )
13	5 12	eqtrd	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐵 ) )
14	4 13	breqtrd	⊢ ( 𝜑 → 0 ≤ ( 𝐴 · 𝐵 ) )