Theorem shlej2i

Description: Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	|- A e. SH
	shincl.2	|- B e. SH
	shless.1	|- C e. SH
Assertion	shlej2i	|- ( A C_ B -> ( C vH A ) C_ ( C vH B ) )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	|- A e. SH
2		shincl.2	|- B e. SH
3		shless.1	|- C e. SH
4	1 2 3	shlej1i	|- ( A C_ B -> ( A vH C ) C_ ( B vH C ) )
5	3 1	shjcomi	|- ( C vH A ) = ( A vH C )
6	3 2	shjcomi	|- ( C vH B ) = ( B vH C )
7	4 5 6	3sstr4g	|- ( A C_ B -> ( C vH A ) C_ ( C vH B ) )