Theorem ho0subi

Description: Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hodseq.2	|- S : ~H --> ~H
	hodseq.3	|- T : ~H --> ~H
Assertion	ho0subi	|- ( S -op T ) = ( S +op ( 0hop -op T ) )

Proof

Step	Hyp	Ref	Expression
1		hodseq.2	|- S : ~H --> ~H
2		hodseq.3	|- T : ~H --> ~H
3		ho0f	|- 0hop : ~H --> ~H
4	3 2	hosubcli	|- ( 0hop -op T ) : ~H --> ~H
5	2 1 4	hoadd12i	|- ( T +op ( S +op ( 0hop -op T ) ) ) = ( S +op ( T +op ( 0hop -op T ) ) )
6	2 3	hodseqi	|- ( T +op ( 0hop -op T ) ) = 0hop
7	6	oveq2i	|- ( S +op ( T +op ( 0hop -op T ) ) ) = ( S +op 0hop )
8	1	hoaddid1i	|- ( S +op 0hop ) = S
9	5 7 8	3eqtri	|- ( T +op ( S +op ( 0hop -op T ) ) ) = S
10	1 4	hoaddcli	|- ( S +op ( 0hop -op T ) ) : ~H --> ~H
11	1 2 10	hodsi	|- ( ( S -op T ) = ( S +op ( 0hop -op T ) ) <-> ( T +op ( S +op ( 0hop -op T ) ) ) = S )
12	9 11	mpbir	|- ( S -op T ) = ( S +op ( 0hop -op T ) )