Metamath Proof Explorer


Theorem chdmm2i

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 chdmm2i
|- ( _|_ ` ( ( _|_ ` 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 3 2 chdmm1i
 |-  ( _|_ ` ( ( _|_ ` A ) i^i B ) ) = ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` B ) )
5 1 pjococi
 |-  ( _|_ ` ( _|_ ` A ) ) = A
6 5 oveq1i
 |-  ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` B ) ) = ( A vH ( _|_ ` B ) )
7 4 6 eqtri
 |-  ( _|_ ` ( ( _|_ ` A ) i^i B ) ) = ( A vH ( _|_ ` B ) )