Theorem chdmj4

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

	Ref	Expression
Assertion	chdmj4	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∩ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
2		chdmj2	⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
4		ococ	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
5	4	adantl	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
6	5	ineq2d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∩ 𝐵 ) )
7	3 6	eqtrd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∩ 𝐵 ) )