Theorem dalem22

Description: Lemma for dath . Show that lines c d and P S determine a plane. (Contributed by NM, 2-Aug-2012)

	Ref	Expression
Hypotheses	dalem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
	dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem22.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem22.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem22.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
Assertion	dalem22	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\in O$$

Proof

Step	Hyp	Ref	Expression
1		dalem.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		dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
6		dalem22.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
7		dalem22.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
8		dalem22.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
9		eqid	$$⊢\mathrm{meet}\left(K\right)=\mathrm{meet}\left(K\right)$$
10	1 2 3 4 5 9 6 7 8	dalem21	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }d\right)\mathrm{meet}\left(K\right)\left(P\underset{\dot{}}{\vee }S\right)\in A$$
11	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
12	11	adantr	$$⊢\left(\phi \wedge \psi \right)\to K\in \mathrm{HL}$$
13	1 2 3 4 5	dalemcjden	$$⊢\left(\phi \wedge \psi \right)\to c\underset{\dot{}}{\vee }d\in \mathrm{LLines}\left(K\right)$$
14	1 2 3 4 6 7	dalempjsen	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)$$
15	14	adantr	$$⊢\left(\phi \wedge \psi \right)\to P\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)$$
16		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
17	3 9 4 16 6	2llnmj	$$⊢\left(K\in \mathrm{HL}\wedge c\underset{\dot{}}{\vee }d\in \mathrm{LLines}\left(K\right)\wedge P\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)\right)\to \left(\left(c\underset{\dot{}}{\vee }d\right)\mathrm{meet}\left(K\right)\left(P\underset{\dot{}}{\vee }S\right)\in A\leftrightarrow \left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\in O\right)$$
18	12 13 15 17	syl3anc	$$⊢\left(\phi \wedge \psi \right)\to \left(\left(c\underset{\dot{}}{\vee }d\right)\mathrm{meet}\left(K\right)\left(P\underset{\dot{}}{\vee }S\right)\in A\leftrightarrow \left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\in O\right)$$
19	18	3adant2	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(c\underset{\dot{}}{\vee }d\right)\mathrm{meet}\left(K\right)\left(P\underset{\dot{}}{\vee }S\right)\in A\leftrightarrow \left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\in O\right)$$
20	10 19	mpbid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\in O$$