Theorem chdmj1i

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	chdmj1i	⊢ ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3	1 2	chdmm4i	⊢ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ) = ( 𝐴 ∨ℋ 𝐵 )
4	3	fveq2i	⊢ ( ⊥ ‘ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) )
5	1	choccli	⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ
6	2	choccli	⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ
7	5 6	chincli	⊢ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ∈ Cℋ
8	7	pjococi	⊢ ( ⊥ ‘ ( ⊥ ‘ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) ) ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) )
9	4 8	eqtr3i	⊢ ( ⊥ ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ 𝐵 ) )