Theorem latmass

Description: Lattice meet is associative. (Contributed by Stefan O'Rear, 30-Jan-2015)

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

Proof

Step	Hyp	Ref	Expression
1		latmass.b	$$⊢B={\mathrm{Base}}_K$$
2		latmass.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
3		eqid	$$⊢\mathrm{ODual}\left(K\right)=\mathrm{ODual}\left(K\right)$$
4	3	odulat	$$⊢K\in \mathrm{Lat}\to \mathrm{ODual}\left(K\right)\in \mathrm{Lat}$$
5	3 1	odubas	$$⊢B={\mathrm{Base}}_{\mathrm{ODual}\left(K\right)}$$
6	3 2	odujoin	$$⊢\underset{\dot{}}{\wedge }=\mathrm{join}\left(\mathrm{ODual}\left(K\right)\right)$$
7	5 6	latjass	$$⊢\left(\mathrm{ODual}\left(K\right)\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\wedge }Z=X\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\wedge }Z\right)$$
8	4 7	sylan	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\wedge }Z=X\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\wedge }Z\right)$$