Theorem cvlatexch2

Description: Atom exchange property. (Contributed by NM, 5-Nov-2012)

	Ref	Expression
Hypotheses	cvlatexch.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cvlatexch.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cvlatexch.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	cvlatexch2	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\to Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cvlatexch.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cvlatexch.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cvlatexch.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4	1 2 3	cvlatexch1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\to Q\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$
5		cvllat	$$⊢K\in \mathrm{CvLat}\to K\in \mathrm{Lat}$$
6	5	3ad2ant1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to K\in \mathrm{Lat}$$
7		simp22	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to Q\in A$$
8		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
9	8 3	atbase	$$⊢Q\in A\to Q\in {\mathrm{Base}}_K$$
10	7 9	syl	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to Q\in {\mathrm{Base}}_K$$
11		simp23	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to R\in A$$
12	8 3	atbase	$$⊢R\in A\to R\in {\mathrm{Base}}_K$$
13	11 12	syl	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to R\in {\mathrm{Base}}_K$$
14	8 2	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in {\mathrm{Base}}_K\wedge R\in {\mathrm{Base}}_K\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
15	6 10 13 14	syl3anc	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
16	15	breq2d	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\leftrightarrow P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)$$
17		simp21	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to P\in A$$
18	8 3	atbase	$$⊢P\in A\to P\in {\mathrm{Base}}_K$$
19	17 18	syl	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to P\in {\mathrm{Base}}_K$$
20	8 2	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge P\in {\mathrm{Base}}_K\wedge R\in {\mathrm{Base}}_K\right)\to P\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }P$$
21	6 19 13 20	syl3anc	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to P\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }P$$
22	21	breq2d	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\leftrightarrow Q\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$
23	4 16 22	3imtr4d	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne R\right)\to \left(P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\to Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\right)$$