Theorem 2llnma1

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-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	2llnma1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)=Q$$

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	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to K\in \mathrm{HL}$$
6		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\in A$$
7		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
8	7 4	atbase	$$⊢P\in A\to P\in {\mathrm{Base}}_K$$
9	6 8	syl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\in {\mathrm{Base}}_K$$
10		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to Q\in A$$
11		simp23	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to R\in A$$
12		simp3	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
13	2 4	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }P$$
14	5 6 10 13	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }P$$
15	14	breq2d	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)$$
16	12 15	mtbid	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \neg R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)$$
17	7 1 2 3 4	2llnma1b	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in {\mathrm{Base}}_K\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to \left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)=Q$$
18	5 9 10 11 16 17	syl131anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)=Q$$