Theorem pjoccoi

Description: Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjidmco.1	⊢ 𝐻 ∈ Cℋ
Assertion	pjoccoi	⊢ ( ( projℎ ‘ 𝐻 ) ∘ ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ) = 0hop

Proof

Step	Hyp	Ref	Expression
1		pjidmco.1	⊢ 𝐻 ∈ Cℋ
2	1	chssii	⊢ 𝐻 ⊆ ℋ
3		ococss	⊢ ( 𝐻 ⊆ ℋ → 𝐻 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐻 ) ) )
4	1	choccli	⊢ ( ⊥ ‘ 𝐻 ) ∈ Cℋ
5	1 4	pjorthcoi	⊢ ( 𝐻 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ 𝐻 ) ∘ ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ) = 0hop )
6	2 3 5	mp2b	⊢ ( ( projℎ ‘ 𝐻 ) ∘ ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ) = 0hop