Theorem 3noncolr2

Description: Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012)

	Ref	Expression
Hypotheses	3noncol.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	3noncol.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	3noncol.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	3noncolr2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(Q\ne R\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3noncol.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		3noncol.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		3noncol.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		hllat	$$⊢K\in \mathrm{HL}\to K\in \mathrm{Lat}$$
5	4	3ad2ant1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{Lat}$$
6		simp23	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in A$$
7		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
8	7 3	atbase	$$⊢R\in A\to R\in {\mathrm{Base}}_K$$
9	6 8	syl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in {\mathrm{Base}}_K$$
10		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\in A$$
11	7 3	atbase	$$⊢P\in A\to P\in {\mathrm{Base}}_K$$
12	10 11	syl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\in {\mathrm{Base}}_K$$
13		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\in A$$
14	7 3	atbase	$$⊢Q\in A\to Q\in {\mathrm{Base}}_K$$
15	13 14	syl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\in {\mathrm{Base}}_K$$
16		simp3r	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
17	7 1 2	latnlej1r	$$⊢\left(K\in \mathrm{Lat}\wedge \left(R\in {\mathrm{Base}}_K\wedge P\in {\mathrm{Base}}_K\wedge Q\in {\mathrm{Base}}_K\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to R\ne Q$$
18	5 9 12 15 16 17	syl131anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\ne Q$$
19	18	necomd	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\ne R$$
20		simp1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{HL}$$
21		simp3l	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\ne Q$$
22	1 2 3	hlatexch1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to \left(P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\to R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)$$
23	20 10 6 13 21 22	syl131anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\to R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)$$
24	2 3	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }P$$
25	20 10 13 24	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }P$$
26	25	breq2d	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)$$
27	23 26	sylibrd	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
28	16 27	mtod	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)$$
29	19 28	jca	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(Q\ne R\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$