Theorem 3dimlem3

Description: Lemma for 3dim1 . (Contributed by NM, 25-Jul-2012)

	Ref	Expression
Hypotheses	3dim0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	3dim0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	3dim0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	3dimlem3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		3dim0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
2		3dim0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		3dim0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		simpr1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to P\ne Q$$
5		simpr2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)$$
6		simpl11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to K\in \mathrm{HL}$$
7		simpl2l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to R\in A$$
8		simpl12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to P\in A$$
9		simpl13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to Q\in A$$
10		simpl3l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to Q\ne R$$
11	10	necomd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to R\ne Q$$
12	2 1 3	hlatexch2	$$⊢\left(K\in \mathrm{HL}\wedge \left(R\in A\wedge P\in A\wedge Q\in A\right)\wedge R\ne Q\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\to P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)$$
13	6 7 8 9 11 12	syl131anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\to P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)$$
14	1 3	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge R\in A\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
15	6 9 7 14	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to Q\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }Q$$
16	15	breq2d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\leftrightarrow P\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }Q\right)\right)$$
17	13 16	sylibrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\to P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
18	5 17	mtod	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
19		simpl1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)$$
20		simpl2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(R\in A\wedge S\in A\right)$$
21		simpl3r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)$$
22		simpr3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)$$
23	1 2 3	3dimlem3a	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(\neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \neg T\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
24	19 20 21 5 22 23	syl113anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \neg T\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
25	4 18 24	3jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\right)\wedge \left(Q\ne R\wedge \neg T\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\wedge \left(P\ne Q\wedge \neg P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge P\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(P\ne Q\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)$$