Theorem chdmj3i

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	chdmj3i	⊢ ( ⊥ ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3	2	choccli	⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ
4	1 3	chdmj1i	⊢ ( ⊥ ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) )
5	2	pjococi	⊢ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵
6	5	ineq2i	⊢ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 )
7	4 6	eqtri	⊢ ( ⊥ ‘ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐵 ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ 𝐵 )