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Metamath Proof Explorer

Theorem chdmj1

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

Ref Expression
Assertion chdmj1 
|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A vH B ) ) = ( ( _|_ ` A ) i^i ( _|_ ` B ) ) )

Proof

Step Hyp Ref Expression
1 chdmm4 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( ( _|_ ` A ) i^i ( _|_ ` B ) ) ) = ( A vH B ) )
2 1 fveq2d 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` ( ( _|_ ` A ) i^i ( _|_ ` B ) ) ) ) = ( _|_ ` ( A vH B ) ) )
3 choccl 
 |-  ( A e. CH -> ( _|_ ` A ) e. CH )
4 choccl 
 |-  ( B e. CH -> ( _|_ ` B ) e. CH )
5 chincl 
 |-  ( ( ( _|_ ` A ) e. CH /\ ( _|_ ` B ) e. CH ) -> ( ( _|_ ` A ) i^i ( _|_ ` B ) ) e. CH )
6 3 4 5 syl2an 
 |-  ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) i^i ( _|_ ` B ) ) e. CH )
7 ococ 
 |-  ( ( ( _|_ ` A ) i^i ( _|_ ` B ) ) e. CH -> ( _|_ ` ( _|_ ` ( ( _|_ ` A ) i^i ( _|_ ` B ) ) ) ) = ( ( _|_ ` A ) i^i ( _|_ ` B ) ) )
8 6 7 syl 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` ( ( _|_ ` A ) i^i ( _|_ ` B ) ) ) ) = ( ( _|_ ` A ) i^i ( _|_ ` B ) ) )
9 2 8 eqtr3d 
 |-  ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( A vH B ) ) = ( ( _|_ ` A ) i^i ( _|_ ` B ) ) )