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 ) ) )