Theorem pj0i

Description: The projection of the zero vector. (Contributed by NM, 31-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pj0.1	⊢ 𝐻 ∈ Cℋ
Assertion	pj0i	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 0ℎ ) = 0ℎ

Proof

Step	Hyp	Ref	Expression
1		pj0.1	⊢ 𝐻 ∈ Cℋ
2	1	chshii	⊢ 𝐻 ∈ Sℋ
3		oc0	⊢ ( 𝐻 ∈ Sℋ → 0ℎ ∈ ( ⊥ ‘ 𝐻 ) )
4	2 3	ax-mp	⊢ 0ℎ ∈ ( ⊥ ‘ 𝐻 )
5		ax-hv0cl	⊢ 0ℎ ∈ ℋ
6	1 5	pjoc2i	⊢ ( 0ℎ ∈ ( ⊥ ‘ 𝐻 ) ↔ ( ( projℎ ‘ 𝐻 ) ‘ 0ℎ ) = 0ℎ )
7	4 6	mpbi	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 0ℎ ) = 0ℎ