Theorem hlatexch1

Description: Atom exchange property. (Contributed by NM, 7-Jan-2012)

	Ref	Expression
Hypotheses	hlatexchb.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	hlatexchb.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	hlatexchb.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	hlatexch1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\to Q\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		hlatexchb.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		hlatexchb.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		hlatexchb.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		hlcvl	$$⊢K\in \mathrm{HL}\to K\in \mathrm{CvLat}$$
5	1 2 3	cvlatexch1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\to Q\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$
6	4 5	syl3an1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\to Q\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$