Theorem 2llnma2

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012)

	Ref	Expression
Hypotheses	2llnm.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	2llnm.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	2llnm.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	2llnm.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	2llnma2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }Q\right)=R$$

Proof

Step	Hyp	Ref	Expression
1		2llnm.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		2llnm.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		2llnm.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		2llnm.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		simp1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{HL}$$
6		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\in A$$
7		simp23	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in A$$
8		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\in A$$
9	1 2 4	4atlem0ae	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)$$
10	1 2 3 4	2llnma1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\wedge Q\in A\right)\wedge \neg Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\right)\to \left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }Q\right)=R$$
11	5 6 7 8 9 10	syl131anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }Q\right)=R$$