Theorem pjchi

Description: Projection of a vector in the projection subspace. Lemma 4.4(ii) of Beran p. 111. (Contributed by NM, 27-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjop.1	⊢ 𝐻 ∈ Cℋ
	pjop.2	⊢ 𝐴 ∈ ℋ
Assertion	pjchi	⊢ ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		pjop.1	⊢ 𝐻 ∈ Cℋ
2		pjop.2	⊢ 𝐴 ∈ ℋ
3	1 2	pjhclii	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ
4		ax-hvaddid	⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ 0ℎ ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) )
5	3 4	ax-mp	⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ 0ℎ ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 )
6	1 2	pjpji	⊢ 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) )
7	1 2	pjoc1i	⊢ ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0ℎ )
8	7	biimpi	⊢ ( 𝐴 ∈ 𝐻 → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0ℎ )
9	8	oveq2d	⊢ ( 𝐴 ∈ 𝐻 → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ 0ℎ ) )
10	6 9	eqtr2id	⊢ ( 𝐴 ∈ 𝐻 → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ 0ℎ ) = 𝐴 )
11	5 10	eqtr3id	⊢ ( 𝐴 ∈ 𝐻 → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 )
12	1 2	pjclii	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻
13		eleq1	⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 ↔ 𝐴 ∈ 𝐻 ) )
14	12 13	mpbii	⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 → 𝐴 ∈ 𝐻 )
15	11 14	impbii	⊢ ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 )