Theorem dalem38

Description: Lemma for dath . Plane Y belongs to the 3-dimensional volume G H I c . (Contributed by NM, 5-Aug-2012)

	Ref	Expression
Hypotheses	dalem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
	dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem38.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem38.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem38.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem38.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem38.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
	dalem38.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
	dalem38.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
Assertion	dalem38	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)$$

Proof

Step	Hyp	Ref	Expression
1		dalem.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		dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
6		dalem38.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem38.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem38.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem38.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem38.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
11		dalem38.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
12		dalem38.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
13	1 2 3 4 5 6 7 8 9 10	dalem28	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }c\right)$$
14	1 2 3 4 5 6 7 8 9 11	dalem33	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Q\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }c\right)$$
15	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
16	15	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{Lat}$$
17	1 4	dalempeb	$$⊢\phi \to P\in {\mathrm{Base}}_K$$
18	17	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\in {\mathrm{Base}}_K$$
19	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
20	19	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
21	1 2 3 4 5 6 7 8 9 10	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$
22	5	dalemccea	$$⊢\psi \to c\in A$$
23	22	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\in A$$
24		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
25	24 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge G\in A\wedge c\in A\right)\to G\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
26	20 21 23 25	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
27	1 4	dalemqeb	$$⊢\phi \to Q\in {\mathrm{Base}}_K$$
28	27	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Q\in {\mathrm{Base}}_K$$
29	1 2 3 4 5 6 7 8 9 11	dalem29	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in A$$
30	24 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge H\in A\wedge c\in A\right)\to H\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
31	20 29 23 30	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
32	24 2 3	latjlej12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(P\in {\mathrm{Base}}_K\wedge G\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K\right)\wedge \left(Q\in {\mathrm{Base}}_K\wedge H\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K\right)\right)\to \left(\left(P\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }c\right)\wedge Q\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }c\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(H\underset{\dot{}}{\vee }c\right)\right)\right)$$
33	16 18 26 28 31 32	syl122anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(P\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }c\right)\wedge Q\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }c\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(H\underset{\dot{}}{\vee }c\right)\right)\right)$$
34	13 14 33	mp2and	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(H\underset{\dot{}}{\vee }c\right)\right)$$
35	24 4	atbase	$$⊢G\in A\to G\in {\mathrm{Base}}_K$$
36	21 35	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in {\mathrm{Base}}_K$$
37	24 4	atbase	$$⊢H\in A\to H\in {\mathrm{Base}}_K$$
38	29 37	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in {\mathrm{Base}}_K$$
39	5 4	dalemcceb	$$⊢\psi \to c\in {\mathrm{Base}}_K$$
40	39	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\in {\mathrm{Base}}_K$$
41	24 3	latjjdir	$$⊢\left(K\in \mathrm{Lat}\wedge \left(G\in {\mathrm{Base}}_K\wedge H\in {\mathrm{Base}}_K\wedge c\in {\mathrm{Base}}_K\right)\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c=\left(G\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(H\underset{\dot{}}{\vee }c\right)$$
42	16 36 38 40 41	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c=\left(G\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(H\underset{\dot{}}{\vee }c\right)$$
43	34 42	breqtrrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)$$
44	1 2 3 4 5 6 7 8 9 12	dalem37	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to R\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }c\right)$$
45	1 3 4	dalempjqeb	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
46	45	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
47	24 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge G\in A\wedge H\in A\right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
48	20 21 29 47	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
49	24 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge c\in {\mathrm{Base}}_K\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
50	16 48 40 49	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
51	1 4	dalemreb	$$⊢\phi \to R\in {\mathrm{Base}}_K$$
52	51	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to R\in {\mathrm{Base}}_K$$
53	1 2 3 4 5 6 7 8 9 12	dalem34	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in A$$
54	24 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge I\in A\wedge c\in A\right)\to I\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
55	20 53 23 54	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
56	24 2 3	latjlej12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K\right)\wedge \left(R\in {\mathrm{Base}}_K\wedge I\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)\wedge R\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }c\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(I\underset{\dot{}}{\vee }c\right)\right)\right)$$
57	16 46 50 52 55 56	syl122anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)\wedge R\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }c\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(I\underset{\dot{}}{\vee }c\right)\right)\right)$$
58	43 44 57	mp2and	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(I\underset{\dot{}}{\vee }c\right)\right)$$
59	24 4	atbase	$$⊢I\in A\to I\in {\mathrm{Base}}_K$$
60	53 59	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in {\mathrm{Base}}_K$$
61	24 3	latjjdir	$$⊢\left(K\in \mathrm{Lat}\wedge \left(G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge I\in {\mathrm{Base}}_K\wedge c\in {\mathrm{Base}}_K\right)\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(I\underset{\dot{}}{\vee }c\right)$$
62	16 48 60 40 61	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\vee }\left(I\underset{\dot{}}{\vee }c\right)$$
63	58 62	breqtrrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)$$
64	8 63	eqbrtrid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)$$