Theorem dalem15

Description: Lemma for dath . The axis of perspectivity X is a line. (Contributed by NM, 21-Jul-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem15.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem15.n	$$⊢N=\mathrm{LLines}\left(K\right)$$
	dalem15.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem15.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem15.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem15.x	$$⊢X=Y\underset{\dot{}}{\wedge }Z$$
Assertion	dalem15	$$⊢\left(\phi \wedge Y\ne Z\right)\to X\in N$$

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		dalem15.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
6		dalem15.n	$$⊢N=\mathrm{LLines}\left(K\right)$$
7		dalem15.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem15.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem15.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem15.x	$$⊢X=Y\underset{\dot{}}{\wedge }Z$$
11		eqid	$$⊢\mathrm{LVols}\left(K\right)=\mathrm{LVols}\left(K\right)$$
12		eqid	$$⊢Y\underset{\dot{}}{\vee }C=Y\underset{\dot{}}{\vee }C$$
13	1 2 3 4 7 11 8 9 12	dalem14	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\underset{\dot{}}{\vee }Z\in \mathrm{LVols}\left(K\right)$$
14	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
15	1	dalemyeo	$$⊢\phi \to Y\in O$$
16	1	dalemzeo	$$⊢\phi \to Z\in O$$
17	3 5 6 7 11	2lplnmj	$$⊢\left(K\in \mathrm{HL}\wedge Y\in O\wedge Z\in O\right)\to \left(Y\underset{\dot{}}{\wedge }Z\in N\leftrightarrow Y\underset{\dot{}}{\vee }Z\in \mathrm{LVols}\left(K\right)\right)$$
18	14 15 16 17	syl3anc	$$⊢\phi \to \left(Y\underset{\dot{}}{\wedge }Z\in N\leftrightarrow Y\underset{\dot{}}{\vee }Z\in \mathrm{LVols}\left(K\right)\right)$$
19	18	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to \left(Y\underset{\dot{}}{\wedge }Z\in N\leftrightarrow Y\underset{\dot{}}{\vee }Z\in \mathrm{LVols}\left(K\right)\right)$$
20	13 19	mpbird	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\underset{\dot{}}{\wedge }Z\in N$$
21	10 20	eqeltrid	$$⊢\left(\phi \wedge Y\ne Z\right)\to X\in N$$