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Metamath Proof Explorer

Theorem chdmm2

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

Ref Expression
Assertion chdmm2 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 choccl ⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
2 chdmm1 ⊢ ( ( ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) )
3 1 2 sylan ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) )
4 ococ ⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 )
5 4 adantr ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 )
6 5 oveq1d ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) = ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) )
7 3 6 eqtrd ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 ) ) = ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) )