Theorem chlej12i

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

	Ref	Expression
Hypotheses	ch0le.1	|- A e. CH
	chjcl.2	|- B e. CH
	chlub.1	|- C e. CH
	chlej12.4	|- D e. CH
Assertion	chlej12i	|- ( ( A C_ B /\ C C_ D ) -> ( A vH C ) C_ ( B vH D ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	|- A e. CH
2		chjcl.2	|- B e. CH
3		chlub.1	|- C e. CH
4		chlej12.4	|- D e. CH
5	1 2 3	chlej1i	|- ( A C_ B -> ( A vH C ) C_ ( B vH C ) )
6	3 4 2	chlej2i	|- ( C C_ D -> ( B vH C ) C_ ( B vH D ) )
7	5 6	sylan9ss	|- ( ( A C_ B /\ C C_ D ) -> ( A vH C ) C_ ( B vH D ) )