Theorem chincl

Description: Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chincl	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A\cap B\in {\mathbf{C}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		ineq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to A\cap B=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\cap B$$
2	1	eleq1d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(A\cap B\in {\mathbf{C}}_{\mathscr{H}}\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\cap B\in {\mathbf{C}}_{\mathscr{H}}\right)$$
3		ineq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\cap B=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\cap if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)$$
4	3	eleq1d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\cap B\in {\mathbf{C}}_{\mathscr{H}}\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\cap if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}\right)$$
5		ifchhv	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
6		ifchhv	$$⊢if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
7	5 6	chincli	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\cap if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
8	2 4 7	dedth2h	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A\cap B\in {\mathbf{C}}_{\mathscr{H}}$$