Theorem rr3fv3cld

Description: Third component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 10-Aug-2026)

	Ref	Expression
Hypothesis	rr3fvd.1	$$⊢\phi \to A\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
Assertion	rr3fv3cld	$$⊢\phi \to A\left(3\right)\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		rr3fvd.1	$$⊢\phi \to A\in \left({ℝ}^{\left(1\dots 3\right)}\right)$$
2		rr3fvcl	$$⊢A\in \left({ℝ}^{\left(1\dots 3\right)}\right)\to \left(A\left(1\right)\in ℝ\wedge A\left(2\right)\in ℝ\wedge A\left(3\right)\in ℝ\right)$$
3	1 2	syl	$$⊢\phi \to \left(A\left(1\right)\in ℝ\wedge A\left(2\right)\in ℝ\wedge A\left(3\right)\in ℝ\right)$$
4	3	simp3d	$$⊢\phi \to A\left(3\right)\in ℝ$$