Theorem crosspcle3d

Description: Closure of the third component of the cross product's coordinate formula. (Contributed by Jiamin Zhao, 11-Aug-2026)

	Ref	Expression
Hypotheses	crosspcled.1	⊢ ( 𝜑 → 𝐴 ∈ ( ℝ ↑m ( 1 ... 3 ) ) )
	crosspcled.2	⊢ ( 𝜑 → 𝐵 ∈ ( ℝ ↑m ( 1 ... 3 ) ) )
Assertion	crosspcle3d	⊢ ( 𝜑 → ( ( ( 𝐴 ‘ 1 ) · ( 𝐵 ‘ 2 ) ) − ( ( 𝐴 ‘ 2 ) · ( 𝐵 ‘ 1 ) ) ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		crosspcled.1	⊢ ( 𝜑 → 𝐴 ∈ ( ℝ ↑m ( 1 ... 3 ) ) )
2		crosspcled.2	⊢ ( 𝜑 → 𝐵 ∈ ( ℝ ↑m ( 1 ... 3 ) ) )
3	1	rr3fv1cld	⊢ ( 𝜑 → ( 𝐴 ‘ 1 ) ∈ ℝ )
4	2	rr3fv2cld	⊢ ( 𝜑 → ( 𝐵 ‘ 2 ) ∈ ℝ )
5	3 4	remulcld	⊢ ( 𝜑 → ( ( 𝐴 ‘ 1 ) · ( 𝐵 ‘ 2 ) ) ∈ ℝ )
6	1	rr3fv2cld	⊢ ( 𝜑 → ( 𝐴 ‘ 2 ) ∈ ℝ )
7	2	rr3fv1cld	⊢ ( 𝜑 → ( 𝐵 ‘ 1 ) ∈ ℝ )
8	6 7	remulcld	⊢ ( 𝜑 → ( ( 𝐴 ‘ 2 ) · ( 𝐵 ‘ 1 ) ) ∈ ℝ )
9	5 8	resubcld	⊢ ( 𝜑 → ( ( ( 𝐴 ‘ 1 ) · ( 𝐵 ‘ 2 ) ) − ( ( 𝐴 ‘ 2 ) · ( 𝐵 ‘ 1 ) ) ) ∈ ℝ )