Theorem latabs2

Description: Lattice absorption law. From definition of lattice in Kalmbach p. 14. ( chabs2 analog.) (Contributed by NM, 8-Nov-2011)

	Ref	Expression
Hypotheses	latabs1.b	$$⊢B={\mathrm{Base}}_K$$
	latabs1.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	latabs1.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
Assertion	latabs2	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\wedge }\left(X\underset{\dot{}}{\vee }Y\right)=X$$

Proof

Step	Hyp	Ref	Expression
1		latabs1.b	$$⊢B={\mathrm{Base}}_K$$
2		latabs1.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		latabs1.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		eqid	$$⊢{\le }_K={\le }_K$$
5	1 4 2	latlej1	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X{\le }_K\left(X\underset{\dot{}}{\vee }Y\right)$$
6	1 2	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\vee }Y\in B$$
7	1 4 3	latleeqm1	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge X\underset{\dot{}}{\vee }Y\in B\right)\to \left(X{\le }_K\left(X\underset{\dot{}}{\vee }Y\right)\leftrightarrow X\underset{\dot{}}{\wedge }\left(X\underset{\dot{}}{\vee }Y\right)=X\right)$$
8	6 7	syld3an3	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X{\le }_K\left(X\underset{\dot{}}{\vee }Y\right)\leftrightarrow X\underset{\dot{}}{\wedge }\left(X\underset{\dot{}}{\vee }Y\right)=X\right)$$
9	5 8	mpbid	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\wedge }\left(X\underset{\dot{}}{\vee }Y\right)=X$$