Theorem posmidm

Description: Poset meet is idempotent. latmidm could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024)

	Ref	Expression
Hypotheses	posjidm.b	$$⊢B={\mathrm{Base}}_K$$
	posmidm.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
Assertion	posmidm	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\underset{\dot{}}{\wedge }X=X$$

Proof

Step	Hyp	Ref	Expression
1		posjidm.b	$$⊢B={\mathrm{Base}}_K$$
2		posmidm.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
3		eqid	$$⊢\mathrm{glb}\left(K\right)=\mathrm{glb}\left(K\right)$$
4		simpl	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to K\in \mathrm{Poset}$$
5		simpr	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\in B$$
6	3 2 4 5 5	meetval	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\underset{\dot{}}{\wedge }X=\mathrm{glb}\left(K\right)\left(\left\{X,X\right\}\right)$$
7		eqid	$$⊢{\le }_K={\le }_K$$
8	1 7	posref	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X{\le }_{K}X$$
9		eqidd	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to \left\{X,X\right\}=\left\{X,X\right\}$$
10	4 1 5 5 7 8 9 3	glbpr	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to \mathrm{glb}\left(K\right)\left(\left\{X,X\right\}\right)=X$$
11	6 10	eqtrd	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\underset{\dot{}}{\wedge }X=X$$