Theorem chabs1

Description: Hilbert lattice absorption law. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 15-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chabs1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\left(A\cap B\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		ssid	$$⊢A\subseteq A$$
2		inss1	$$⊢A\cap B\subseteq A$$
3	1 2	pm3.2i	$$⊢\left(A\subseteq A\wedge A\cap B\subseteq A\right)$$
4		simpl	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A\in {\mathbf{C}}_{\mathscr{H}}$$
5		chincl	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A\cap B\in {\mathbf{C}}_{\mathscr{H}}$$
6		chlub	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge A\cap B\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A\subseteq A\wedge A\cap B\subseteq A\right)\leftrightarrow A{\vee }_{\mathscr{H}}\left(A\cap B\right)\subseteq A\right)$$
7	4 5 4 6	syl3anc	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A\subseteq A\wedge A\cap B\subseteq A\right)\leftrightarrow A{\vee }_{\mathscr{H}}\left(A\cap B\right)\subseteq A\right)$$
8	3 7	mpbii	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\left(A\cap B\right)\subseteq A$$
9		chub1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge A\cap B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A\subseteq A{\vee }_{\mathscr{H}}\left(A\cap B\right)$$
10	5 9	syldan	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A\subseteq A{\vee }_{\mathscr{H}}\left(A\cap B\right)$$
11	8 10	eqssd	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\left(A\cap B\right)=A$$