Theorem meetcomALT

Description: The meet of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 17-Sep-2011) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		meetcom.b	$$⊢B={\mathrm{Base}}_K$$
2		meetcom.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
3		prcom	$$⊢\left\{Y,X\right\}=\left\{X,Y\right\}$$
4	3	fveq2i	$$⊢\mathrm{glb}\left(K\right)\left(\left\{Y,X\right\}\right)=\mathrm{glb}\left(K\right)\left(\left\{X,Y\right\}\right)$$
5	4	a1i	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to \mathrm{glb}\left(K\right)\left(\left\{Y,X\right\}\right)=\mathrm{glb}\left(K\right)\left(\left\{X,Y\right\}\right)$$
6		eqid	$$⊢\mathrm{glb}\left(K\right)=\mathrm{glb}\left(K\right)$$
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	meetval	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to Y\underset{\dot{}}{\wedge }X=\mathrm{glb}\left(K\right)\left(\left\{Y,X\right\}\right)$$
11	6 2 7 9 8	meetval	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\wedge }Y=\mathrm{glb}\left(K\right)\left(\left\{X,Y\right\}\right)$$
12	5 10 11	3eqtr4rd	$$⊢\left(K\in V\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\wedge }Y=Y\underset{\dot{}}{\wedge }X$$