Theorem ho0val

Description: Value of the zero Hilbert space operator (null projector). Remark in Beran p. 111. (Contributed by NM, 7-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	ho0val	⊢ ( 𝐴 ∈ ℋ → ( 0hop ‘ 𝐴 ) = 0ℎ )

Proof

Step	Hyp	Ref	Expression
1		choc1	⊢ ( ⊥ ‘ ℋ ) = 0ℋ
2	1	fveq2i	⊢ ( projℎ ‘ ( ⊥ ‘ ℋ ) ) = ( projℎ ‘ 0ℋ )
3		df-h0op	⊢ 0hop = ( projℎ ‘ 0ℋ )
4	2 3	eqtr4i	⊢ ( projℎ ‘ ( ⊥ ‘ ℋ ) ) = 0hop
5	4	fveq1i	⊢ ( ( projℎ ‘ ( ⊥ ‘ ℋ ) ) ‘ 𝐴 ) = ( 0hop ‘ 𝐴 )
6		helch	⊢ ℋ ∈ Cℋ
7		pjo	⊢ ( ( ℋ ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ ℋ ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) )
8	6 7	mpan	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ( ⊥ ‘ ℋ ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) )
9	5 8	syl5eqr	⊢ ( 𝐴 ∈ ℋ → ( 0hop ‘ 𝐴 ) = ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) )
10	6	pjhcli	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ∈ ℋ )
11		hvsubid	⊢ ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ∈ ℋ → ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) = 0ℎ )
12	10 11	syl	⊢ ( 𝐴 ∈ ℋ → ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) = 0ℎ )
13	9 12	eqtrd	⊢ ( 𝐴 ∈ ℋ → ( 0hop ‘ 𝐴 ) = 0ℎ )