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Metamath Proof Explorer

Theorem chdmm3

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

Ref Expression
Assertion chdmm3 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 choccl ⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
2 chdmm1 ⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
3 1 2 sylan2 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
4 ococ ⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
5 4 adantl ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
6 5 oveq2d ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) )
7 3 6 eqtrd ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) )