Theorem lplnribN

Description: Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	islpln2a.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	islpln2a.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	islpln2a.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	islpln2a.p	$$⊢P=\mathrm{LPlanes}\left(K\right)$$
	islpln2a.y	$$⊢Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S$$
Assertion	lplnribN	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\wedge Y\in P\right)\to \neg R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }S\right)$$

Proof

Step	Hyp	Ref	Expression
1		islpln2a.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		islpln2a.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		islpln2a.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		islpln2a.p	$$⊢P=\mathrm{LPlanes}\left(K\right)$$
5		islpln2a.y	$$⊢Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S$$
6	1 2 3	3noncolr1N	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to \left(S\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }Q\right)\right)$$
7	6	simprd	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }Q\right)$$
8	7	3expia	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(\left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\to \neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }Q\right)\right)$$
9	1 2 3 4 5	islpln2ah	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(Y\in P\leftrightarrow \left(Q\ne R\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$
10	2 3	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge S\in A\right)\to Q\underset{\dot{}}{\vee }S=S\underset{\dot{}}{\vee }Q$$
11	10	3adant3r2	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to Q\underset{\dot{}}{\vee }S=S\underset{\dot{}}{\vee }Q$$
12	11	breq2d	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }S\right)\leftrightarrow R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }Q\right)\right)$$
13	12	notbid	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(\neg R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }S\right)\leftrightarrow \neg R\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }Q\right)\right)$$
14	8 9 13	3imtr4d	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(Y\in P\to \neg R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }S\right)\right)$$
15	14	3impia	$$⊢\left(K\in \mathrm{HL}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\wedge Y\in P\right)\to \neg R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }S\right)$$