Theorem dalem52

Description: Lemma for dath . Lines G H and P Q intersect at an atom. (Contributed by NM, 8-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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem44.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem44.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem44.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem44.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem44.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
	dalem44.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
	dalem44.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
Assertion	dalem52	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in A$$

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		dalem44.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem44.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem44.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem44.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem44.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
11		dalem44.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
12		dalem44.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	5 4	dalemcceb	$$⊢\psi \to c\in {\mathrm{Base}}_K$$
16	15	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\in {\mathrm{Base}}_K$$
17	14 16	jca	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)$$
18	1 2 3 4 5 6 7 8 9 10	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$
19	1 2 3 4 5 6 7 8 9 11	dalem29	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in A$$
20	1 2 3 4 5 6 7 8 9 12	dalem34	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in A$$
21	18 19 20	3jca	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\in A\wedge H\in A\wedge I\in A\right)$$
22	1	dalempea	$$⊢\phi \to P\in A$$
23	1	dalemqea	$$⊢\phi \to Q\in A$$
24	1	dalemrea	$$⊢\phi \to R\in A$$
25	22 23 24	3jca	$$⊢\phi \to \left(P\in A\wedge Q\in A\wedge R\in A\right)$$
26	25	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(P\in A\wedge Q\in A\wedge R\in A\right)$$
27	1 2 3 4 5 6 7 8 9 10 11 12	dalem42	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O$$
28	1	dalemyeo	$$⊢\phi \to Y\in O$$
29	28	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\in O$$
30	1 2 3 4 5 6 7 8 9 10 11 12	dalem45	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)$$
31	1 2 3 4 5 6 7 8 9 10 11 12	dalem46	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)$$
32	1 2 3 4 5 6 7 8 9 10 11 12	dalem47	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)$$
33	30 31 32	3jca	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)$$
34	1 2 3 4 5 6 7 8 9 10 11 12	dalem48	$$⊢\left(\phi \wedge \psi \right)\to \neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
35	1 2 3 4 5 6 7 8 9 10 11 12	dalem49	$$⊢\left(\phi \wedge \psi \right)\to \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)$$
36	1 2 3 4 5 6 7 8 9 10 11 12	dalem50	$$⊢\left(\phi \wedge \psi \right)\to \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)$$
37	34 35 36	3jca	$$⊢\left(\phi \wedge \psi \right)\to \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$
38	37	3adant2	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)$$
39	1 2 3 4 5 6 7 8 9 10	dalem27	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)$$
40	1 2 3 4 5 6 7 8 9 11	dalem32	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)$$
41	1 2 3 4 5 6 7 8 9 12	dalem36	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)$$
42	39 40 41	3jca	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)$$
43		biid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \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(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)\leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \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(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)$$
44		eqid	$$⊢\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I$$
45		eqid	$$⊢\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)$$
46	43 2 3 4 6 7 44 8 45	dalemdea	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in A$$
47	17 21 26 27 29 33 38 42 46	syl323anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in A$$