Theorem pjoml6i

Description: An equivalent of the orthomodular law. Theorem 29.13(e) of MaedaMaeda p. 132. (Contributed by NM, 30-May-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	|- A e. CH
	pjoml2.2	|- B e. CH
Assertion	pjoml6i	|- ( A C_ B -> E. x e. CH ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	|- A e. CH
2		pjoml2.2	|- B e. CH
3	1	choccli	|- ( _|_ ` A ) e. CH
4	3 2	chincli	|- ( ( _|_ ` A ) i^i B ) e. CH
5	1 2	pjoml2i	|- ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )
6	2	choccli	|- ( _|_ ` B ) e. CH
7	1 6	chub1i	|- A C_ ( A vH ( _|_ ` B ) )
8	1 2	chdmm2i	|- ( _|_ ` ( ( _|_ ` A ) i^i B ) ) = ( A vH ( _|_ ` B ) )
9	7 8	sseqtrri	|- A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) )
10	5 9	jctil	|- ( A C_ B -> ( A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) ) /\ ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) )
11		fveq2	|- ( x = ( ( _|_ ` A ) i^i B ) -> ( _|_ ` x ) = ( _|_ ` ( ( _|_ ` A ) i^i B ) ) )
12	11	sseq2d	|- ( x = ( ( _|_ ` A ) i^i B ) -> ( A C_ ( _|_ ` x ) <-> A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) )
13		oveq2	|- ( x = ( ( _|_ ` A ) i^i B ) -> ( A vH x ) = ( A vH ( ( _|_ ` A ) i^i B ) ) )
14	13	eqeq1d	|- ( x = ( ( _|_ ` A ) i^i B ) -> ( ( A vH x ) = B <-> ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) )
15	12 14	anbi12d	|- ( x = ( ( _|_ ` A ) i^i B ) -> ( ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) <-> ( A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) ) /\ ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) ) )
16	15	rspcev	|- ( ( ( ( _|_ ` A ) i^i B ) e. CH /\ ( A C_ ( _|_ ` ( ( _|_ ` A ) i^i B ) ) /\ ( A vH ( ( _|_ ` A ) i^i B ) ) = B ) ) -> E. x e. CH ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) )
17	4 10 16	sylancr	|- ( A C_ B -> E. x e. CH ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) )