Theorem 4atlem0ae

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	4atlem0ae	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)$$

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		simp3r	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
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		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\in A$$
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		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\in A$$
9		simp3l	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\ne Q$$
10	9	necomd	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\ne P$$
11	1 2 3	hlatexch1	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge Q\ne P\right)\to \left(Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
12	5 6 7 8 10 11	syl131anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
13	4 12	mtod	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)$$