Theorem pjpji

Description: Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjpj.1	⊢ 𝐻 ∈ Cℋ
	pjpj.2	⊢ 𝐴 ∈ ℋ
Assertion	pjpji	⊢ 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		pjpj.1	⊢ 𝐻 ∈ Cℋ
2		pjpj.2	⊢ 𝐴 ∈ ℋ
3	1 2	pjpj0i	⊢ 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) )