Theorem dalem2

Description: Lemma for dath . Show the lines P Q and S T form a plane. (Contributed by NM, 11-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	dalem2	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(S\underset{\dot{}}{\vee }T\right)\in O$$

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	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
8	1	dalempea	$$⊢\phi \to P\in A$$
9	1	dalemqea	$$⊢\phi \to Q\in A$$
10	1	dalemsea	$$⊢\phi \to S\in A$$
11	1	dalemtea	$$⊢\phi \to T\in A$$
12	3 4	hlatj4	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\right)\wedge \left(S\in A\wedge T\in A\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(S\underset{\dot{}}{\vee }T\right)=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }T\right)$$
13	7 8 9 10 11 12	syl122anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(S\underset{\dot{}}{\vee }T\right)=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }T\right)$$
14	1 2 3 4 5 6	dalempjsen	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)$$
15	1 2 3 4 5 6	dalemqnet	$$⊢\phi \to Q\ne T$$
16		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
17	3 4 16	llni2	$$⊢\left(\left(K\in \mathrm{HL}\wedge Q\in A\wedge T\in A\right)\wedge Q\ne T\right)\to Q\underset{\dot{}}{\vee }T\in \mathrm{LLines}\left(K\right)$$
18	7 9 11 15 17	syl31anc	$$⊢\phi \to Q\underset{\dot{}}{\vee }T\in \mathrm{LLines}\left(K\right)$$
19	1 2 3 4 5 6	dalem1	$$⊢\phi \to P\underset{\dot{}}{\vee }S\ne Q\underset{\dot{}}{\vee }T$$
20	1 2 3 4 5 6	dalemcea	$$⊢\phi \to C\in A$$
21	1	dalemclpjs	$$⊢\phi \to C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
22	1	dalemclqjt	$$⊢\phi \to C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
23		eqid	$$⊢\mathrm{meet}\left(K\right)=\mathrm{meet}\left(K\right)$$
24		eqid	$$⊢\mathrm{0.}\left(K\right)=\mathrm{0.}\left(K\right)$$
25	2 23 24 4 16	2llnm4	$$⊢\left(K\in \mathrm{HL}\wedge \left(C\in A\wedge P\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)\wedge Q\underset{\dot{}}{\vee }T\in \mathrm{LLines}\left(K\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }S\right)\mathrm{meet}\left(K\right)\left(Q\underset{\dot{}}{\vee }T\right)\ne \mathrm{0.}\left(K\right)$$
26	7 20 14 18 21 22 25	syl132anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }S\right)\mathrm{meet}\left(K\right)\left(Q\underset{\dot{}}{\vee }T\right)\ne \mathrm{0.}\left(K\right)$$
27	23 24 4 16	2llnmat	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)\wedge Q\underset{\dot{}}{\vee }T\in \mathrm{LLines}\left(K\right)\right)\wedge \left(P\underset{\dot{}}{\vee }S\ne Q\underset{\dot{}}{\vee }T\wedge \left(P\underset{\dot{}}{\vee }S\right)\mathrm{meet}\left(K\right)\left(Q\underset{\dot{}}{\vee }T\right)\ne \mathrm{0.}\left(K\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }S\right)\mathrm{meet}\left(K\right)\left(Q\underset{\dot{}}{\vee }T\right)\in A$$
28	7 14 18 19 26 27	syl32anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }S\right)\mathrm{meet}\left(K\right)\left(Q\underset{\dot{}}{\vee }T\right)\in A$$
29	3 23 4 16 5	2llnmj	$$⊢\left(K\in \mathrm{HL}\wedge P\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)\wedge Q\underset{\dot{}}{\vee }T\in \mathrm{LLines}\left(K\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\mathrm{meet}\left(K\right)\left(Q\underset{\dot{}}{\vee }T\right)\in A\leftrightarrow \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }T\right)\in O\right)$$
30	7 14 18 29	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }S\right)\mathrm{meet}\left(K\right)\left(Q\underset{\dot{}}{\vee }T\right)\in A\leftrightarrow \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }T\right)\in O\right)$$
31	28 30	mpbid	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }T\right)\in O$$
32	13 31	eqeltrd	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(S\underset{\dot{}}{\vee }T\right)\in O$$