Theorem 2llnne2N

Description: Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-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	2llnne2N	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)\to R\underset{\dot{}}{\vee }P\ne R\underset{\dot{}}{\vee }Q$$

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		simpl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\right)\to K\in \mathrm{HL}$$
5		simprr	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\right)\to R\in A$$
6		simprl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\right)\to P\in A$$
7	1 2 3	hlatlej2	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge P\in A\right)\to P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)$$
8	4 5 6 7	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\right)\to P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)$$
9		breq2	$$⊢R\underset{\dot{}}{\vee }P=R\underset{\dot{}}{\vee }Q\to \left(P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\leftrightarrow P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)$$
10	8 9	syl5ibcom	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\right)\to \left(R\underset{\dot{}}{\vee }P=R\underset{\dot{}}{\vee }Q\to P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)$$
11	10	necon3bd	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\right)\to \left(\neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\to R\underset{\dot{}}{\vee }P\ne R\underset{\dot{}}{\vee }Q\right)$$
12	11	3impia	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)\to R\underset{\dot{}}{\vee }P\ne R\underset{\dot{}}{\vee }Q$$