Theorem dalem9

Description: Lemma for dath . Since -. C .<_ Y , the join Y .\/ C forms a 3-dimensional space. (Contributed by NM, 20-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem9.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem9.v	$$⊢V=\mathrm{LVols}\left(K\right)$$
	dalem9.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem9.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem9.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
Assertion	dalem9	$$⊢\left(\phi \wedge Y\ne Z\right)\to W\in V$$

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		dalem9.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalem9.v	$$⊢V=\mathrm{LVols}\left(K\right)$$
7		dalem9.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
8		dalem9.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
9		dalem9.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
10	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
11	10	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to K\in \mathrm{HL}$$
12	1	dalemyeo	$$⊢\phi \to Y\in O$$
13	12	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\in O$$
14	1 2 3 4 5 7	dalemcea	$$⊢\phi \to C\in A$$
15	14	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to C\in A$$
16	1 2 3 4 5 7 8	dalem-cly	$$⊢\left(\phi \wedge Y\ne Z\right)\to \neg C\underset{\dot{}}{\le }Y$$
17	2 3 4 5 6	lvoli3	$$⊢\left(\left(K\in \mathrm{HL}\wedge Y\in O\wedge C\in A\right)\wedge \neg C\underset{\dot{}}{\le }Y\right)\to Y\underset{\dot{}}{\vee }C\in V$$
18	11 13 15 16 17	syl31anc	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\underset{\dot{}}{\vee }C\in V$$
19	9 18	eqeltrid	$$⊢\left(\phi \wedge Y\ne Z\right)\to W\in V$$