Theorem pjoml2i

Description: Variation of orthomodular law. Definition in Kalmbach p. 22. (Contributed by NM, 31-Oct-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	|- A e. CH
	pjoml2.2	|- B e. CH
Assertion	pjoml2i	|- ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	|- A e. CH
2		pjoml2.2	|- B e. CH
3		inss2	|- ( ( _|_ ` A ) i^i B ) C_ B
4	1	choccli	|- ( _|_ ` A ) e. CH
5	4 2	chincli	|- ( ( _|_ ` A ) i^i B ) e. CH
6	1 5 2	chlubii	|- ( ( A C_ B /\ ( ( _|_ ` A ) i^i B ) C_ B ) -> ( A vH ( ( _|_ ` A ) i^i B ) ) C_ B )
7	3 6	mpan2	|- ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) C_ B )
8	1 5	chdmj1i	|- ( _|_ ` ( A vH ( ( _|_ ` A ) i^i B ) ) ) = ( ( _|_ ` A ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) )
9	8	ineq2i	|- ( B i^i ( _|_ ` ( A vH ( ( _|_ ` A ) i^i B ) ) ) ) = ( B i^i ( ( _|_ ` A ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) )
10		incom	|- ( B i^i ( _|_ ` A ) ) = ( ( _|_ ` A ) i^i B )
11	10	ineq1i	|- ( ( B i^i ( _|_ ` A ) ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) = ( ( ( _|_ ` A ) i^i B ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) )
12		inass	|- ( ( B i^i ( _|_ ` A ) ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) = ( B i^i ( ( _|_ ` A ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) )
13	5	chocini	|- ( ( ( _|_ ` A ) i^i B ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) = 0H
14	11 12 13	3eqtr3i	|- ( B i^i ( ( _|_ ` A ) i^i ( _|_ ` ( ( _|_ ` A ) i^i B ) ) ) ) = 0H
15	9 14	eqtri	|- ( B i^i ( _|_ ` ( A vH ( ( _|_ ` A ) i^i B ) ) ) ) = 0H
16	1 5	chjcli	|- ( A vH ( ( _|_ ` A ) i^i B ) ) e. CH
17	2	chshii	|- B e. SH
18	16 17	pjomli	|- ( ( ( A vH ( ( _|_ ` A ) i^i B ) ) C_ B /\ ( B i^i ( _|_ ` ( A vH ( ( _|_ ` A ) i^i B ) ) ) ) = 0H ) -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )
19	7 15 18	sylancl	|- ( A C_ B -> ( A vH ( ( _|_ ` A ) i^i B ) ) = B )