Theorem pjcli

Description: Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjcl.1	⊢ 𝐻 ∈ Cℋ
Assertion	pjcli	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		pjcl.1	⊢ 𝐻 ∈ Cℋ
2		axpjcl	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 )