Theorem dalem39

Description: Lemma for dath . Auxiliary atoms G , H , and I are not colinear. (Contributed by NM, 4-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	dalem39	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg H\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)$$

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	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
14	13	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
15	1	dalemyeo	$$⊢\phi \to Y\in O$$
16	15	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\in O$$
17	5	dalemccea	$$⊢\psi \to c\in A$$
18	17	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\in A$$
19	5	dalem-ccly	$$⊢\psi \to \neg c\underset{\dot{}}{\le }Y$$
20	19	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg c\underset{\dot{}}{\le }Y$$
21		eqid	$$⊢\mathrm{LVols}\left(K\right)=\mathrm{LVols}\left(K\right)$$
22	2 3 4 7 21	lvoli3	$$⊢\left(\left(K\in \mathrm{HL}\wedge Y\in O\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to Y\underset{\dot{}}{\vee }c\in \mathrm{LVols}\left(K\right)$$
23	14 16 18 20 22	syl31anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\underset{\dot{}}{\vee }c\in \mathrm{LVols}\left(K\right)$$
24	1 2 3 4 5 6 7 8 9 12	dalem34	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in A$$
25	1 2 3 4 5 6 7 8 9 10	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$
26	2 3 4 21	lvolnle3at	$$⊢\left(\left(K\in \mathrm{HL}\wedge Y\underset{\dot{}}{\vee }c\in \mathrm{LVols}\left(K\right)\right)\wedge \left(I\in A\wedge G\in A\wedge c\in A\right)\right)\to \neg \left(Y\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\le }\left(\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }c\right)$$
27	14 23 24 25 18 26	syl23anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg \left(Y\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\le }\left(\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }c\right)$$
28	1 2 3 4 5 6 7 8 9 10 11 12	dalem38	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)$$
29	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
30	29	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{Lat}$$
31	1 2 3 4 5 6 7 8 9 11	dalem29	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in A$$
32		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
33	32 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge G\in A\wedge H\in A\right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
34	14 25 31 33	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
35	32 4	atbase	$$⊢I\in A\to I\in {\mathrm{Base}}_K$$
36	24 35	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in {\mathrm{Base}}_K$$
37	32 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge I\in {\mathrm{Base}}_K\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K$$
38	30 34 36 37	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K$$
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	32 2 3	latlej2	$$⊢\left(K\in \mathrm{Lat}\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K\wedge c\in {\mathrm{Base}}_K\right)\to c\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)$$
42	30 38 40 41	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)$$
43	1 7	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
44	43	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\in {\mathrm{Base}}_K$$
45	32 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K\wedge c\in {\mathrm{Base}}_K\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
46	30 38 40 45	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K$$
47	32 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(Y\in {\mathrm{Base}}_K\wedge c\in {\mathrm{Base}}_K\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\in {\mathrm{Base}}_K\right)\right)\to \left(\left(Y\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)\wedge c\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)\right)\leftrightarrow \left(Y\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)\right)$$
48	30 44 40 46 47	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(Y\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)\wedge c\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)\right)\leftrightarrow \left(Y\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)\right)$$
49	28 42 48	mpbi2and	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(Y\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c\right)$$
50	3 4	hlatjrot	$$⊢\left(K\in \mathrm{HL}\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I=\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H$$
51	14 25 31 24 50	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I=\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H$$
52	51	oveq1d	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }c=\left(\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c$$
53	49 52	breqtrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(Y\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\le }\left(\left(\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)$$
54	53	adantr	$$⊢\left(\left(\phi \wedge Y=Z\wedge \psi \right)\wedge H\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\to \left(Y\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\le }\left(\left(\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c\right)$$
55	32 4	atbase	$$⊢H\in A\to H\in {\mathrm{Base}}_K$$
56	31 55	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in {\mathrm{Base}}_K$$
57	32 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge I\in A\wedge G\in A\right)\to I\underset{\dot{}}{\vee }G\in {\mathrm{Base}}_K$$
58	14 24 25 57	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\underset{\dot{}}{\vee }G\in {\mathrm{Base}}_K$$
59	32 2 3	latleeqj2	$$⊢\left(K\in \mathrm{Lat}\wedge H\in {\mathrm{Base}}_K\wedge I\underset{\dot{}}{\vee }G\in {\mathrm{Base}}_K\right)\to \left(H\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\leftrightarrow \left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H=I\underset{\dot{}}{\vee }G\right)$$
60	30 56 58 59	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(H\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\leftrightarrow \left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H=I\underset{\dot{}}{\vee }G\right)$$
61	60	biimpa	$$⊢\left(\left(\phi \wedge Y=Z\wedge \psi \right)\wedge H\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\to \left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H=I\underset{\dot{}}{\vee }G$$
62	61	oveq1d	$$⊢\left(\left(\phi \wedge Y=Z\wedge \psi \right)\wedge H\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\to \left(\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }c=\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }c$$
63	54 62	breqtrd	$$⊢\left(\left(\phi \wedge Y=Z\wedge \psi \right)\wedge H\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\to \left(Y\underset{\dot{}}{\vee }c\right)\underset{\dot{}}{\le }\left(\left(I\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\vee }c\right)$$
64	27 63	mtand	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg H\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)$$