Theorem 2llnneN

Description: Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	2lnne.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	2lnne.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	2lnne.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	2llnneN	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\vee }P\ne R\underset{\dot{}}{\vee }Q$$

Proof

Step	Hyp	Ref	Expression
1		2lnne.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		2lnne.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		2lnne.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		simp1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{HL}$$
5		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\in A$$
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		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne Q\right)\to P\in A$$
8		simp23	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne Q\right)\to R\in A$$
9		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne Q\right)\to Q\in A$$
10	7 8 9	3jca	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne Q\right)\to \left(P\in A\wedge R\in A\wedge Q\in A\right)$$
11	1 2 3	hlatexch2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
12	10 11	syld3an2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne Q\right)\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
13	12	con3d	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P\ne Q\right)\to \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\to \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)$$
14	13	3exp	$$⊢K\in \mathrm{HL}\to \left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\to \left(P\ne Q\to \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\to \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)\right)\right)$$
15	14	imp4a	$$⊢K\in \mathrm{HL}\to \left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\to \left(\left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)\right)$$
16	15	3imp	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)$$
17	1 2 3	2llnne2N	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)\to R\underset{\dot{}}{\vee }P\ne R\underset{\dot{}}{\vee }Q$$
18	4 5 6 16 17	syl121anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\vee }P\ne R\underset{\dot{}}{\vee }Q$$