Theorem dalem5

Description: Lemma for dath . Atom U (in plane Z = S T U ) belongs to the 3-dimensional volume formed by Y and C . (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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem5.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem5.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem5.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
Assertion	dalem5	$$⊢\phi \to U\underset{\dot{}}{\le }W$$

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		dalem5.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalem5.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7		dalem5.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
8		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
9	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
10	1 4	dalemueb	$$⊢\phi \to U\in {\mathrm{Base}}_K$$
11	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
12	1	dalemrea	$$⊢\phi \to R\in A$$
13	1 2 3 4 5 6	dalemcea	$$⊢\phi \to C\in A$$
14	8 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge C\in A\right)\to R\underset{\dot{}}{\vee }C\in {\mathrm{Base}}_K$$
15	11 12 13 14	syl3anc	$$⊢\phi \to R\underset{\dot{}}{\vee }C\in {\mathrm{Base}}_K$$
16	1 5	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
17	1 4	dalemceb	$$⊢\phi \to C\in {\mathrm{Base}}_K$$
18	8 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge Y\in {\mathrm{Base}}_K\wedge C\in {\mathrm{Base}}_K\right)\to Y\underset{\dot{}}{\vee }C\in {\mathrm{Base}}_K$$
19	9 16 17 18	syl3anc	$$⊢\phi \to Y\underset{\dot{}}{\vee }C\in {\mathrm{Base}}_K$$
20	7 19	eqeltrid	$$⊢\phi \to W\in {\mathrm{Base}}_K$$
21	1	dalemclrju	$$⊢\phi \to C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)$$
22	1	dalemuea	$$⊢\phi \to U\in A$$
23	1	dalempea	$$⊢\phi \to P\in A$$
24		simp313	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)\to \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)$$
25	1 24	sylbi	$$⊢\phi \to \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)$$
26	2 3 4	atnlej1	$$⊢\left(K\in \mathrm{HL}\wedge \left(C\in A\wedge R\in A\wedge P\in A\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\to C\ne R$$
27	11 13 12 23 25 26	syl131anc	$$⊢\phi \to C\ne R$$
28	2 3 4	hlatexch1	$$⊢\left(K\in \mathrm{HL}\wedge \left(C\in A\wedge U\in A\wedge R\in A\right)\wedge C\ne R\right)\to \left(C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to U\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }C\right)\right)$$
29	11 13 22 12 27 28	syl131anc	$$⊢\phi \to \left(C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to U\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }C\right)\right)$$
30	21 29	mpd	$$⊢\phi \to U\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }C\right)$$
31	1 3 4	dalempjqeb	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
32	1 4	dalemreb	$$⊢\phi \to R\in {\mathrm{Base}}_K$$
33	8 2 3	latlej2	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge R\in {\mathrm{Base}}_K\right)\to R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
34	9 31 32 33	syl3anc	$$⊢\phi \to R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
35	34 6	breqtrrdi	$$⊢\phi \to R\underset{\dot{}}{\le }Y$$
36	8 2 3	latjlej1	$$⊢\left(K\in \mathrm{Lat}\wedge \left(R\in {\mathrm{Base}}_K\wedge Y\in {\mathrm{Base}}_K\wedge C\in {\mathrm{Base}}_K\right)\right)\to \left(R\underset{\dot{}}{\le }Y\to \left(R\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }C\right)\right)$$
37	9 32 16 17 36	syl13anc	$$⊢\phi \to \left(R\underset{\dot{}}{\le }Y\to \left(R\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }C\right)\right)$$
38	35 37	mpd	$$⊢\phi \to \left(R\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\vee }C\right)$$
39	38 7	breqtrrdi	$$⊢\phi \to \left(R\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\le }W$$
40	8 2 9 10 15 20 30 39	lattrd	$$⊢\phi \to U\underset{\dot{}}{\le }W$$