Theorem 4atlem0a

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	4atlem0a	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }S\right)$$

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		simprr	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
5		simpl1	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to K\in \mathrm{HL}$$
6		simpl3l	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to R\in A$$
7		simpl3r	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to S\in A$$
8		simpl2l	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to P\in A$$
9		simpl2r	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to Q\in A$$
10		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
11	10 2 3	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
12	5 8 9 11	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
13		simprl	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
14	10 1 2 3	hlexch1	$$⊢\left(K\in \mathrm{HL}\wedge \left(R\in A\wedge S\in A\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }S\right)\to S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)$$
15	5 6 7 12 13 14	syl131anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }S\right)\to S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)$$
16	4 15	mtod	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }S\right)$$