Theorem dalem8

Description: Lemma for dath . Plane Z belongs to the 3-dimensional space. (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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem6.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem6.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem6.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem6.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
Assertion	dalem8	$$⊢\phi \to Z\underset{\dot{}}{\le }W$$

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		dalem6.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalem6.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7		dalem6.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
8		dalem6.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
9	1 2 3 4 5 6 7 8	dalem6	$$⊢\phi \to S\underset{\dot{}}{\le }W$$
10	1 2 3 4 5 6 7 8	dalem7	$$⊢\phi \to T\underset{\dot{}}{\le }W$$
11	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
12	1 4	dalemseb	$$⊢\phi \to S\in {\mathrm{Base}}_K$$
13	1 4	dalemteb	$$⊢\phi \to T\in {\mathrm{Base}}_K$$
14	1 5	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
15	1 4	dalemceb	$$⊢\phi \to C\in {\mathrm{Base}}_K$$
16		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
17	16 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge Y\in {\mathrm{Base}}_K\wedge C\in {\mathrm{Base}}_K\right)\to Y\underset{\dot{}}{\vee }C\in {\mathrm{Base}}_K$$
18	11 14 15 17	syl3anc	$$⊢\phi \to Y\underset{\dot{}}{\vee }C\in {\mathrm{Base}}_K$$
19	8 18	eqeltrid	$$⊢\phi \to W\in {\mathrm{Base}}_K$$
20	16 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(S\in {\mathrm{Base}}_K\wedge T\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\right)\to \left(\left(S\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }W\right)\leftrightarrow \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W\right)$$
21	11 12 13 19 20	syl13anc	$$⊢\phi \to \left(\left(S\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }W\right)\leftrightarrow \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W\right)$$
22	9 10 21	mpbi2and	$$⊢\phi \to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W$$
23	1 2 3 4 5 6 8	dalem5	$$⊢\phi \to U\underset{\dot{}}{\le }W$$
24	1 3 4	dalemsjteb	$$⊢\phi \to S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
25	1 4	dalemueb	$$⊢\phi \to U\in {\mathrm{Base}}_K$$
26	16 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge U\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\right)\to \left(\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W\wedge U\underset{\dot{}}{\le }W\right)\leftrightarrow \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }W\right)$$
27	11 24 25 19 26	syl13anc	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W\wedge U\underset{\dot{}}{\le }W\right)\leftrightarrow \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }W\right)$$
28	22 23 27	mpbi2and	$$⊢\phi \to \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }W$$
29	7 28	eqbrtrid	$$⊢\phi \to Z\underset{\dot{}}{\le }W$$