Theorem dalem16

Description: Lemma for dath . The atoms D , E , and F form a line of perspectivity. This is Desargues's theorem for the special case where planes Y and Z are different. (Contributed by NM, 7-Aug-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem16.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem16.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem16.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem16.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem16.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
	dalem16.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
	dalem16.f	$$⊢F=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)$$
Assertion	dalem16	$$⊢\left(\phi \wedge Y\ne Z\right)\to F\underset{\dot{}}{\le }\left(D\underset{\dot{}}{\vee }E\right)$$

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		dalem16.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
6		dalem16.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
7		dalem16.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
8		dalem16.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
9		dalem16.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
10		dalem16.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
11		dalem16.f	$$⊢F=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)$$
12		eqid	$$⊢Y\underset{\dot{}}{\wedge }Z=Y\underset{\dot{}}{\wedge }Z$$
13	1 2 3 4 5 6 7 8 12 11	dalem12	$$⊢\phi \to F\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)$$
14	13	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to F\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)$$
15	1 2 3 4 5 6 7 8 12 9	dalem10	$$⊢\phi \to D\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)$$
16	1 2 3 4 5 6 7 8 12 10	dalem11	$$⊢\phi \to E\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)$$
17	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
18	1 2 3 4 5 6 7 8 9	dalemdea	$$⊢\phi \to D\in A$$
19		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
20	19 4	atbase	$$⊢D\in A\to D\in {\mathrm{Base}}_K$$
21	18 20	syl	$$⊢\phi \to D\in {\mathrm{Base}}_K$$
22	1 2 3 4 5 6 7 8 10	dalemeea	$$⊢\phi \to E\in A$$
23	19 4	atbase	$$⊢E\in A\to E\in {\mathrm{Base}}_K$$
24	22 23	syl	$$⊢\phi \to E\in {\mathrm{Base}}_K$$
25	1 6	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
26	1	dalemzeo	$$⊢\phi \to Z\in O$$
27	19 6	lplnbase	$$⊢Z\in O\to Z\in {\mathrm{Base}}_K$$
28	26 27	syl	$$⊢\phi \to Z\in {\mathrm{Base}}_K$$
29	19 5	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge Y\in {\mathrm{Base}}_K\wedge Z\in {\mathrm{Base}}_K\right)\to Y\underset{\dot{}}{\wedge }Z\in {\mathrm{Base}}_K$$
30	17 25 28 29	syl3anc	$$⊢\phi \to Y\underset{\dot{}}{\wedge }Z\in {\mathrm{Base}}_K$$
31	19 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(D\in {\mathrm{Base}}_K\wedge E\in {\mathrm{Base}}_K\wedge Y\underset{\dot{}}{\wedge }Z\in {\mathrm{Base}}_K\right)\right)\to \left(\left(D\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)\wedge E\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)\right)\leftrightarrow \left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)\right)$$
32	17 21 24 30 31	syl13anc	$$⊢\phi \to \left(\left(D\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)\wedge E\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)\right)\leftrightarrow \left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)\right)$$
33	15 16 32	mpbi2and	$$⊢\phi \to \left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)$$
34	33	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to \left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)$$
35	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
36	35	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to K\in \mathrm{HL}$$
37	1 2 3 4 5 6 7 8 9 10	dalemdnee	$$⊢\phi \to D\ne E$$
38		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
39	3 4 38	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	35 18 22 37 39	syl31anc	$$⊢\phi \to D\underset{\dot{}}{\vee }E\in \mathrm{LLines}\left(K\right)$$
41	40	adantr	$$⊢\left(\phi \wedge Y\ne Z\right)\to D\underset{\dot{}}{\vee }E\in \mathrm{LLines}\left(K\right)$$
42	1 2 3 4 5 38 6 7 8 12	dalem15	$$⊢\left(\phi \wedge Y\ne Z\right)\to Y\underset{\dot{}}{\wedge }Z\in \mathrm{LLines}\left(K\right)$$
43	2 38	llncmp	$$⊢\left(K\in \mathrm{HL}\wedge D\underset{\dot{}}{\vee }E\in \mathrm{LLines}\left(K\right)\wedge Y\underset{\dot{}}{\wedge }Z\in \mathrm{LLines}\left(K\right)\right)\to \left(\left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)\leftrightarrow D\underset{\dot{}}{\vee }E=Y\underset{\dot{}}{\wedge }Z\right)$$
44	36 41 42 43	syl3anc	$$⊢\left(\phi \wedge Y\ne Z\right)\to \left(\left(D\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\le }\left(Y\underset{\dot{}}{\wedge }Z\right)\leftrightarrow D\underset{\dot{}}{\vee }E=Y\underset{\dot{}}{\wedge }Z\right)$$
45	34 44	mpbid	$$⊢\left(\phi \wedge Y\ne Z\right)\to D\underset{\dot{}}{\vee }E=Y\underset{\dot{}}{\wedge }Z$$
46	14 45	breqtrrd	$$⊢\left(\phi \wedge Y\ne Z\right)\to F\underset{\dot{}}{\le }\left(D\underset{\dot{}}{\vee }E\right)$$