Theorem 3noncolr1N

Description: Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012) (New usage is discouraged.)

	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	3noncolr1N	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\ne P\wedge \neg Q\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$

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		simp1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{HL}$$
5		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\in A$$
6		simp23	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in A$$
7		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to P\in A$$
8	1 2 3	3noncolr2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(Q\ne R\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
9	1 2 3	3noncolr2	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(Q\ne R\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to \left(R\ne P\wedge \neg Q\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$
10	4 5 6 7 8 9	syl131anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\ne P\wedge \neg Q\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$