Metamath Proof Explorer


Theorem chdmj3

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

Ref Expression
Assertion chdmj3 ( ( 𝐴C𝐵C ) → ( ⊥ ‘ ( 𝐴 ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 choccl ( 𝐵C → ( ⊥ ‘ 𝐵 ) ∈ C )
2 chdmj1 ( ( 𝐴C ∧ ( ⊥ ‘ 𝐵 ) ∈ C ) → ( ⊥ ‘ ( 𝐴 ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
3 1 2 sylan2 ( ( 𝐴C𝐵C ) → ( ⊥ ‘ ( 𝐴 ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
4 ococ ( 𝐵C → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
5 4 adantl ( ( 𝐴C𝐵C ) → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
6 5 ineq2d ( ( 𝐴C𝐵C ) → ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) )
7 3 6 eqtrd ( ( 𝐴C𝐵C ) → ( ⊥ ‘ ( 𝐴 ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) )