Theorem dalem13

Description: Lemma for dalem14 . (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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem13.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem13.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem13.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem13.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
Assertion	dalem13	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\underset{\dot{}}{\vee }Z=W$$

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		dalem13.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalem13.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7		dalem13.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
8		dalem13.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
9	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
10	9	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to K\in \mathrm{HL}$$
11	1	dalemyeo	$$⊢\phi \to Y\in O$$
12	11	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\in O$$
13	1	dalemzeo	$$⊢\phi \to Z\in O$$
14	13	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to Z\in O$$
15		eqid	$$⊢\mathrm{LVols}\left(K\right)=\mathrm{LVols}\left(K\right)$$
16	1 2 3 4 5 15 6 7 8	dalem9	$$⊢\left(\phi \wedge Y\ne Z\right)\to W\in \mathrm{LVols}\left(K\right)$$
17	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
18	1 5	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
19	1 4	dalemceb	$$⊢\phi \to C\in {\mathrm{Base}}_K$$
20		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
21	20 2 3	latlej1	$$⊢\left(K\in \mathrm{Lat}\wedge Y\in {\mathrm{Base}}_K\wedge C\in {\mathrm{Base}}_K\right)\to Y\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }C\right)$$
22	17 18 19 21	syl3anc	$$⊢\phi \to Y\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }C\right)$$
23	22 8	breqtrrdi	$$⊢\phi \to Y\underset{\dot{}}{\le }W$$
24	23	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\underset{\dot{}}{\le }W$$
25	1 2 3 4 5 6 7 8	dalem8	$$⊢\phi \to Z\underset{\dot{}}{\le }W$$
26	25	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to Z\underset{\dot{}}{\le }W$$
27		simpr	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\ne Z$$
28	2 3 5 15	2lplnj	$$⊢\left(K\in \mathrm{HL}\wedge \left(Y\in O\wedge Z\in O\wedge W\in \mathrm{LVols}\left(K\right)\right)\wedge \left(Y\underset{\dot{}}{\le }W\wedge Z\underset{\dot{}}{\le }W\wedge Y\ne Z\right)\right)\to Y\underset{\dot{}}{\vee }Z=W$$
29	10 12 14 16 24 26 27 28	syl133anc	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\underset{\dot{}}{\vee }Z=W$$