Theorem latjlej2

Description: Add join to both sides of a lattice ordering. ( chlej2i analog.) (Contributed by NM, 8-Nov-2011)

	Ref	Expression
Hypotheses	latlej.b	$$⊢B={\mathrm{Base}}_K$$
	latlej.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	latlej.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
Assertion	latjlej2	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{\le }Y\to \left(Z\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\le }\left(Z\underset{\dot{}}{\vee }Y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		latlej.b	$$⊢B={\mathrm{Base}}_K$$
2		latlej.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		latlej.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4	1 2 3	latjlej1	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{\le }Y\to \left(X\underset{\dot{}}{\vee }Z\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }Z\right)\right)$$
5	1 3	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Z\in B\right)\to X\underset{\dot{}}{\vee }Z=Z\underset{\dot{}}{\vee }X$$
6	5	3adant3r2	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to X\underset{\dot{}}{\vee }Z=Z\underset{\dot{}}{\vee }X$$
7	1 3	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge Y\in B\wedge Z\in B\right)\to Y\underset{\dot{}}{\vee }Z=Z\underset{\dot{}}{\vee }Y$$
8	7	3adant3r1	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to Y\underset{\dot{}}{\vee }Z=Z\underset{\dot{}}{\vee }Y$$
9	6 8	breq12d	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\vee }Z\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }Z\right)\leftrightarrow \left(Z\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\le }\left(Z\underset{\dot{}}{\vee }Y\right)\right)$$
10	4 9	sylibd	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{\le }Y\to \left(Z\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\le }\left(Z\underset{\dot{}}{\vee }Y\right)\right)$$