This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem chdmj2

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

Ref Expression
Assertion chdmj2 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 choccl ⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
2 chdmj1 ⊢ ( ( ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) )
3 1 2 sylan ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) )
4 ococ ⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 )
5 4 ineq1d ⊢ ( 𝐴 ∈ Cℋ → ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) = ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) )
6 5 adantr ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) = ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) )
7 3 6 eqtrd ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) )