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 e. ~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 e. CH
7		pjo	|- ( ( ~H e. CH /\ A e. ~H ) -> ( ( projh ` ( _|_ ` ~H ) ) ` A ) = ( ( ( projh ` ~H ) ` A ) -h ( ( projh ` ~H ) ` A ) ) )
8	6 7	mpan	|- ( A e. ~H -> ( ( projh ` ( _|_ ` ~H ) ) ` A ) = ( ( ( projh ` ~H ) ` A ) -h ( ( projh ` ~H ) ` A ) ) )
9	5 8	syl5eqr	|- ( A e. ~H -> ( 0hop ` A ) = ( ( ( projh ` ~H ) ` A ) -h ( ( projh ` ~H ) ` A ) ) )
10	6	pjhcli	|- ( A e. ~H -> ( ( projh ` ~H ) ` A ) e. ~H )
11		hvsubid	|- ( ( ( projh ` ~H ) ` A ) e. ~H -> ( ( ( projh ` ~H ) ` A ) -h ( ( projh ` ~H ) ` A ) ) = 0h )
12	10 11	syl	|- ( A e. ~H -> ( ( ( projh ` ~H ) ` A ) -h ( ( projh ` ~H ) ` A ) ) = 0h )
13	9 12	eqtrd	|- ( A e. ~H -> ( 0hop ` A ) = 0h )