Theorem chdmm1i

Description: De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	|- A e. CH
	chjcl.2	|- B e. CH
Assertion	chdmm1i	|- ( _|_ ` ( A i^i B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	|- A e. CH
2		chjcl.2	|- B e. CH
3	1	choccli	|- ( _|_ ` A ) e. CH
4	2	choccli	|- ( _|_ ` B ) e. CH
5	3 4	chub1i	|- ( _|_ ` A ) C_ ( ( _|_ ` A ) vH ( _|_ ` B ) )
6	3 4	chjcli	|- ( ( _|_ ` A ) vH ( _|_ ` B ) ) e. CH
7	1 6	chsscon1i	|- ( ( _|_ ` A ) C_ ( ( _|_ ` A ) vH ( _|_ ` B ) ) <-> ( _|_ ` ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) C_ A )
8	5 7	mpbi	|- ( _|_ ` ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) C_ A
9	4 3	chub2i	|- ( _|_ ` B ) C_ ( ( _|_ ` A ) vH ( _|_ ` B ) )
10	2 6	chsscon1i	|- ( ( _|_ ` B ) C_ ( ( _|_ ` A ) vH ( _|_ ` B ) ) <-> ( _|_ ` ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) C_ B )
11	9 10	mpbi	|- ( _|_ ` ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) C_ B
12	8 11	ssini	|- ( _|_ ` ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) C_ ( A i^i B )
13	1 2	chincli	|- ( A i^i B ) e. CH
14	6 13	chsscon1i	|- ( ( _|_ ` ( ( _|_ ` A ) vH ( _|_ ` B ) ) ) C_ ( A i^i B ) <-> ( _|_ ` ( A i^i B ) ) C_ ( ( _|_ ` A ) vH ( _|_ ` B ) ) )
15	12 14	mpbi	|- ( _|_ ` ( A i^i B ) ) C_ ( ( _|_ ` A ) vH ( _|_ ` B ) )
16		inss1	|- ( A i^i B ) C_ A
17	13 1	chsscon3i	|- ( ( A i^i B ) C_ A <-> ( _|_ ` A ) C_ ( _|_ ` ( A i^i B ) ) )
18	16 17	mpbi	|- ( _|_ ` A ) C_ ( _|_ ` ( A i^i B ) )
19		inss2	|- ( A i^i B ) C_ B
20	13 2	chsscon3i	|- ( ( A i^i B ) C_ B <-> ( _|_ ` B ) C_ ( _|_ ` ( A i^i B ) ) )
21	19 20	mpbi	|- ( _|_ ` B ) C_ ( _|_ ` ( A i^i B ) )
22	13	choccli	|- ( _|_ ` ( A i^i B ) ) e. CH
23	3 4 22	chlubii	|- ( ( ( _|_ ` A ) C_ ( _|_ ` ( A i^i B ) ) /\ ( _|_ ` B ) C_ ( _|_ ` ( A i^i B ) ) ) -> ( ( _|_ ` A ) vH ( _|_ ` B ) ) C_ ( _|_ ` ( A i^i B ) ) )
24	18 21 23	mp2an	|- ( ( _|_ ` A ) vH ( _|_ ` B ) ) C_ ( _|_ ` ( A i^i B ) )
25	15 24	eqssi	|- ( _|_ ` ( A i^i B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) )