Theorem atnlej2

Description: If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012)

	Ref	Expression
Hypotheses	atnlej.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	atnlej.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	atnlej.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	atnlej2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to P\ne R$$

Proof

Step	Hyp	Ref	Expression
1		atnlej.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		atnlej.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		atnlej.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		hllat	$$⊢K\in \mathrm{HL}\to K\in \mathrm{Lat}$$
5	4	3ad2ant1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to K\in \mathrm{Lat}$$
6		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to P\in A$$
7		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
8	7 3	atbase	$$⊢P\in A\to P\in {\mathrm{Base}}_K$$
9	6 8	syl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to P\in {\mathrm{Base}}_K$$
10		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to Q\in A$$
11	7 3	atbase	$$⊢Q\in A\to Q\in {\mathrm{Base}}_K$$
12	10 11	syl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to Q\in {\mathrm{Base}}_K$$
13		simp23	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to R\in A$$
14	7 3	atbase	$$⊢R\in A\to R\in {\mathrm{Base}}_K$$
15	13 14	syl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to R\in {\mathrm{Base}}_K$$
16		simp3	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)$$
17	7 1 2	latnlej1r	$$⊢\left(K\in \mathrm{Lat}\wedge \left(P\in {\mathrm{Base}}_K\wedge Q\in {\mathrm{Base}}_K\wedge R\in {\mathrm{Base}}_K\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to P\ne R$$
18	5 9 12 15 16 17	syl131anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to P\ne R$$