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 ) ) )