Theorem dalem60

Description: Lemma for dath . B is an axis of perspectivity (almost). (Contributed by NM, 11-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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem60.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem60.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem60.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem60.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem60.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
	dalem60.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
	dalem60.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
	dalem60.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
	dalem60.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
	dalem60.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
Assertion	dalem60	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\underset{\dot{}}{\vee }E=B$$

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		dalem60.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem60.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem60.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem60.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem60.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
11		dalem60.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
12		dalem60.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
13		dalem60.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
14		dalem60.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
15		dalem60.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
16	1 2 3 4 5 6 7 8 9 10 12 13 14 15	dalem57	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\underset{\dot{}}{\le }B$$
17	1 2 3 4 5 6 7 8 9 11 12 13 14 15	dalem58	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to E\underset{\dot{}}{\le }B$$
18	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
19	18	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{Lat}$$
20	1 2 3 4 6 7 8 9 10	dalemdea	$$⊢\phi \to D\in A$$
21		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
22	21 4	atbase	$$⊢D\in A\to D\in {\mathrm{Base}}_K$$
23	20 22	syl	$$⊢\phi \to D\in {\mathrm{Base}}_K$$
24	23	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\in {\mathrm{Base}}_K$$
25	1 2 3 4 6 7 8 9 11	dalemeea	$$⊢\phi \to E\in A$$
26	21 4	atbase	$$⊢E\in A\to E\in {\mathrm{Base}}_K$$
27	25 26	syl	$$⊢\phi \to E\in {\mathrm{Base}}_K$$
28	27	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to E\in {\mathrm{Base}}_K$$
29		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
30	1 2 3 4 5 6 29 7 8 9 12 13 14 15	dalem53	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in \mathrm{LLines}\left(K\right)$$
31	21 29	llnbase	$$⊢B\in \mathrm{LLines}\left(K\right)\to B\in {\mathrm{Base}}_K$$
32	30 31	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in {\mathrm{Base}}_K$$
33	21 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(D\in {\mathrm{Base}}_K\wedge E\in {\mathrm{Base}}_K\wedge B\in {\mathrm{Base}}_K\right)\right)\to \left(\left(D\underset{\dot{}}{\le }B\wedge E\underset{\dot{}}{\le }B\right)\leftrightarrow \left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }B\right)$$
34	19 24 28 32 33	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(D\underset{\dot{}}{\le }B\wedge E\underset{\dot{}}{\le }B\right)\leftrightarrow \left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }B\right)$$
35	16 17 34	mpbi2and	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }B$$
36	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
37	36	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
38	1 2 3 4 6 7 8 9 10 11	dalemdnee	$$⊢\phi \to D\ne E$$
39	3 4 29	llni2	$$⊢\left(\left(K\in \mathrm{HL}\wedge D\in A\wedge E\in A\right)\wedge D\ne E\right)\to D\underset{\dot{}}{\vee }E\in \mathrm{LLines}\left(K\right)$$
40	36 20 25 38 39	syl31anc	$$⊢\phi \to D\underset{\dot{}}{\vee }E\in \mathrm{LLines}\left(K\right)$$
41	40	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\underset{\dot{}}{\vee }E\in \mathrm{LLines}\left(K\right)$$
42	2 29	llncmp	$$⊢\left(K\in \mathrm{HL}\wedge D\underset{\dot{}}{\vee }E\in \mathrm{LLines}\left(K\right)\wedge B\in \mathrm{LLines}\left(K\right)\right)\to \left(\left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }B\leftrightarrow D\underset{\dot{}}{\vee }E=B\right)$$
43	37 41 30 42	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }B\leftrightarrow D\underset{\dot{}}{\vee }E=B\right)$$
44	35 43	mpbid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\underset{\dot{}}{\vee }E=B$$