Theorem chabs1i

Description: Hilbert lattice absorption law. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chabs.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chabs.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chabs1i	$$⊢A{\vee }_{\mathscr{H}}\left(A\cap B\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		chabs.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chabs.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		chabs1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\left(A\cap B\right)=A$$
4	1 2 3	mp2an	$$⊢A{\vee }_{\mathscr{H}}\left(A\cap B\right)=A$$