Theorem 2llnma2rN

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013) (New usage is discouraged.)

	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	2llnma2rN	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)=R$$

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	2 4	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge R\in A\right)\to P\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }P$$
9	5 6 7 8	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }P$$
10		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\in A$$
11	2 4	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge R\in A\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
12	5 10 7 11	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
13	9 12	oveq12d	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }Q\right)$$
14	1 2 3 4	2llnma2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(R\underset{\dot{}}{\vee }Q\right)=R$$
15	13 14	eqtrd	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)=R$$