Theorem hlatcon2

Description: Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-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	hlatcon2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)$$

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	1 2 3	hlatcon3	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)$$
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	2 3	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge R\in A\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
9	5 6 7 8	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
10	9	breq2d	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\leftrightarrow P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)$$
11	4 10	mtbid	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)$$