Theorem chdmj2i

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

	Ref	Expression
Hypotheses	ch0le.1	⊢ 𝐴 ∈ Cℋ
	chjcl.2	⊢ 𝐵 ∈ Cℋ
Assertion	chdmj2i	⊢ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3	1	choccli	⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ
4	3 2	chdmj1i	⊢ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) )
5	1	pjococi	⊢ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴
6	5	ineq1i	⊢ ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ∩ ( ⊥ ‘ 𝐵 ) ) = ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) )
7	4 6	eqtri	⊢ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) = ( 𝐴 ∩ ( ⊥ ‘ 𝐵 ) )