Theorem joincomALT

Description: The join of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 16-Sep-2011) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		joincom.b	$$⊢B={\mathrm{Base}}_K$$
2		joincom.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		prcom	$$⊢\left\{Y,X\right\}=\left\{X,Y\right\}$$
4	3	fveq2i	$$⊢\mathrm{lub}\left(K\right)\left(\left\{Y,X\right\}\right)=\mathrm{lub}\left(K\right)\left(\left\{X,Y\right\}\right)$$
5	4	a1i	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to \mathrm{lub}\left(K\right)\left(\left\{Y,X\right\}\right)=\mathrm{lub}\left(K\right)\left(\left\{X,Y\right\}\right)$$
6		eqid	$$⊢\mathrm{lub}\left(K\right)=\mathrm{lub}\left(K\right)$$
7		simp1	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to K\in V$$
8		simp3	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to Y\in B$$
9		simp2	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to X\in B$$
10	6 2 7 8 9	joinval	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to Y\underset{\dot{}}{\vee }X=\mathrm{lub}\left(K\right)\left(\left\{Y,X\right\}\right)$$
11	6 2 7 9 8	joinval	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\vee }Y=\mathrm{lub}\left(K\right)\left(\left\{X,Y\right\}\right)$$
12	5 10 11	3eqtr4rd	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\vee }Y=Y\underset{\dot{}}{\vee }X$$