Theorem latabs1

Description: Lattice absorption law. From definition of lattice in Kalmbach p. 14. ( chabs1 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	latabs1	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)=X$$

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 3	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\wedge }Y\right){\le }_{K}X$$
6	1 3	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\wedge }Y\in B$$
7	1 4 2	latleeqj2	$$⊢\left(K\in \mathrm{Lat}\wedge X\underset{\dot{}}{\wedge }Y\in B\wedge X\in B\right)\to \left(\left(X\underset{\dot{}}{\wedge }Y\right){\le }_{K}X\leftrightarrow X\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)=X\right)$$
8	7	3com23	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge X\underset{\dot{}}{\wedge }Y\in B\right)\to \left(\left(X\underset{\dot{}}{\wedge }Y\right){\le }_{K}X\leftrightarrow X\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)=X\right)$$
9	6 8	syld3an3	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{\wedge }Y\right){\le }_{K}X\leftrightarrow X\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)=X\right)$$
10	5 9	mpbid	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)=X$$