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	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chdmj1i	$$⊢\perp \left(A{\vee }_{\mathscr{H}}B\right)=\perp \left(A\right)\cap \perp \left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	chdmm4i	$$⊢\perp \left(\perp \left(A\right)\cap \perp \left(B\right)\right)=A{\vee }_{\mathscr{H}}B$$
4	3	fveq2i	$$⊢\perp \left(\perp \left(\perp \left(A\right)\cap \perp \left(B\right)\right)\right)=\perp \left(A{\vee }_{\mathscr{H}}B\right)$$
5	1	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
6	2	choccli	$$⊢\perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
7	5 6	chincli	$$⊢\perp \left(A\right)\cap \perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
8	7	pjococi	$$⊢\perp \left(\perp \left(\perp \left(A\right)\cap \perp \left(B\right)\right)\right)=\perp \left(A\right)\cap \perp \left(B\right)$$
9	4 8	eqtr3i	$$⊢\perp \left(A{\vee }_{\mathscr{H}}B\right)=\perp \left(A\right)\cap \perp \left(B\right)$$