Theorem 4noncolr1

Description: A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012)

	Ref	Expression
Hypotheses	3noncol.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	3noncol.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	3noncol.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	4noncolr1	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to \left(S\ne P\wedge \neg Q\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge \neg R\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3noncol.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		3noncol.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		3noncol.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		simp11	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to K\in \mathrm{HL}$$
5		simp13	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to Q\in A$$
6		simp2l	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to R\in A$$
7		simp2r	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to S\in A$$
8		simp12	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to P\in A$$
9	1 2 3	4noncolr3	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to \left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)$$
10	1 2 3	4noncolr2	$$⊢\left(\left(K\in \mathrm{HL}\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge P\in A\right)\wedge \left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(S\ne P\wedge \neg Q\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge \neg R\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q\right)\right)$$
11	4 5 6 7 8 9 10	syl321anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\to \left(S\ne P\wedge \neg Q\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge \neg R\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q\right)\right)$$