Theorem dalem1

Description: Lemma for dath . Show the lines P S and Q T are different. (Contributed by NM, 9-Aug-2012)

	Ref	Expression
Hypotheses	dalema.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
	dalemc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalemc.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalemc.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem1.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem1.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
Assertion	dalem1	$$⊢\phi \to P\underset{\dot{}}{\vee }S\ne Q\underset{\dot{}}{\vee }T$$

Proof

Step	Hyp	Ref	Expression
1		dalema.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
2		dalemc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalemc.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalemc.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalem1.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalem1.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7	1	dalemclpjs	$$⊢\phi \to C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
8	1	dalem-clpjq	$$⊢\phi \to \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
9	8	adantr	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
10	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
11	1	dalempea	$$⊢\phi \to P\in A$$
12	1	dalemsea	$$⊢\phi \to S\in A$$
13	2 3 4	hlatlej1	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge S\in A\right)\to P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
14	10 11 12 13	syl3anc	$$⊢\phi \to P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
15	14	adantr	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
16	1	dalemqea	$$⊢\phi \to Q\in A$$
17	1	dalemtea	$$⊢\phi \to T\in A$$
18	2 3 4	hlatlej1	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge T\in A\right)\to Q\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
19	10 16 17 18	syl3anc	$$⊢\phi \to Q\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
20	19	adantr	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to Q\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
21		simpr	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T$$
22	20 21	breqtrrd	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
23	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
24	1 4	dalempeb	$$⊢\phi \to P\in {\mathrm{Base}}_K$$
25	1 4	dalemqeb	$$⊢\phi \to Q\in {\mathrm{Base}}_K$$
26		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
27	26 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge S\in A\right)\to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
28	10 11 12 27	syl3anc	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
29	26 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(P\in {\mathrm{Base}}_K\wedge Q\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\right)\to \left(\left(P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)$$
30	23 24 25 28 29	syl13anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)$$
31	30	adantr	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to \left(\left(P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)$$
32	15 22 31	mpbi2and	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
33	1	dalemrea	$$⊢\phi \to R\in A$$
34	1	dalemyeo	$$⊢\phi \to Y\in O$$
35	3 4 5 6	lplnri1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge Y\in O\right)\to P\ne Q$$
36	10 11 16 33 34 35	syl131anc	$$⊢\phi \to P\ne Q$$
37	2 3 4	ps-1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge P\ne Q\right)\wedge \left(P\in A\wedge S\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\leftrightarrow P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }S\right)$$
38	10 11 16 36 11 12 37	syl132anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\leftrightarrow P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }S\right)$$
39	38	adantr	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\leftrightarrow P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }S\right)$$
40	32 39	mpbid	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to P\underset{\dot{}}{\vee }Q=P\underset{\dot{}}{\vee }S$$
41	40	breq2d	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)$$
42	9 41	mtbid	$$⊢\left(\phi \wedge P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\right)\to \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
43	42	ex	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }S=Q\underset{\dot{}}{\vee }T\to \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)$$
44	43	necon2ad	$$⊢\phi \to \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\to P\underset{\dot{}}{\vee }S\ne Q\underset{\dot{}}{\vee }T\right)$$
45	7 44	mpd	$$⊢\phi \to P\underset{\dot{}}{\vee }S\ne Q\underset{\dot{}}{\vee }T$$