Theorem int-sqdefd

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

	Ref	Expression
Hypotheses	int-sqdefd.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	int-sqdefd.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	int-sqdefd	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = ( 𝐴 ↑ 2 ) )

Proof

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