Theorem posjidm

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

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

Proof

Step	Hyp	Ref	Expression
1		posjidm.b	$$⊢B={\mathrm{Base}}_K$$
2		posjidm.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		eqid	$$⊢\mathrm{lub}\left(K\right)=\mathrm{lub}\left(K\right)$$
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	joinval	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\underset{\dot{}}{\vee }X=\mathrm{lub}\left(K\right)\left(\left\{X,X\right\}\right)$$
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	lubpr	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to \mathrm{lub}\left(K\right)\left(\left\{X,X\right\}\right)=X$$
11	6 10	eqtrd	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\underset{\dot{}}{\vee }X=X$$