Theorem 4atlem0be

Description: Lemma for 4at . (Contributed by NM, 10-Jul-2012)

	Ref	Expression
Hypotheses	4at.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	4at.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	4at.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	4atlem0be	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\ne R$$

Proof

Step	Hyp	Ref	Expression
1		4at.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		4at.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		4at.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		simp1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to K\in \mathrm{HL}$$
5		simp23	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to R\in A$$
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		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to Q\in A$$
8		simp3	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
9	1 2 3	atnlej1	$$⊢\left(K\in \mathrm{HL}\wedge \left(R\in A\wedge P\in A\wedge Q\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to R\ne P$$
10	9	necomd	$$⊢\left(K\in \mathrm{HL}\wedge \left(R\in A\wedge P\in A\wedge Q\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\ne R$$
11	4 5 6 7 8 10	syl131anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\ne R$$