Theorem pjpj0i

Description: Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjcli.1	⊢ 𝐻 ∈ Cℋ
	pjcli.2	⊢ 𝐴 ∈ ℋ
Assertion	pjpj0i	⊢ 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		pjcli.1	⊢ 𝐻 ∈ Cℋ
2		pjcli.2	⊢ 𝐴 ∈ ℋ
3		axpjpj	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )
4	1 2 3	mp2an	⊢ 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) )