Theorem pjocvec

Description: The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of Halmos p. 45. (Contributed by NM, 24-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	pjocvec	⊢ ( 𝐻 ∈ Cℋ → ( ⊥ ‘ 𝐻 ) = { 𝑥 ∈ ℋ ∣ ( ( projℎ ‘ 𝐻 ) ‘ 𝑥 ) = 0ℎ } )

Proof

Step	Hyp	Ref	Expression
1		choccl	⊢ ( 𝐻 ∈ Cℋ → ( ⊥ ‘ 𝐻 ) ∈ Cℋ )
2		chss	⊢ ( ( ⊥ ‘ 𝐻 ) ∈ Cℋ → ( ⊥ ‘ 𝐻 ) ⊆ ℋ )
3	1 2	syl	⊢ ( 𝐻 ∈ Cℋ → ( ⊥ ‘ 𝐻 ) ⊆ ℋ )
4		sseqin2	⊢ ( ( ⊥ ‘ 𝐻 ) ⊆ ℋ ↔ ( ℋ ∩ ( ⊥ ‘ 𝐻 ) ) = ( ⊥ ‘ 𝐻 ) )
5	3 4	sylib	⊢ ( 𝐻 ∈ Cℋ → ( ℋ ∩ ( ⊥ ‘ 𝐻 ) ) = ( ⊥ ‘ 𝐻 ) )
6		pjoc2	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝑥 ∈ ℋ ) → ( 𝑥 ∈ ( ⊥ ‘ 𝐻 ) ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝑥 ) = 0ℎ ) )
7	6	rabbi2dva	⊢ ( 𝐻 ∈ Cℋ → ( ℋ ∩ ( ⊥ ‘ 𝐻 ) ) = { 𝑥 ∈ ℋ ∣ ( ( projℎ ‘ 𝐻 ) ‘ 𝑥 ) = 0ℎ } )
8	5 7	eqtr3d	⊢ ( 𝐻 ∈ Cℋ → ( ⊥ ‘ 𝐻 ) = { 𝑥 ∈ ℋ ∣ ( ( projℎ ‘ 𝐻 ) ‘ 𝑥 ) = 0ℎ } )