Theorem dalem27

Description: Lemma for dath . Show that the line G P intersects the dummy center of perspectivity c . (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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem23.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem23.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem23.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem23.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem23.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
Assertion	dalem27	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)$$

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		dalem23.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem23.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem23.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem23.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem23.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
11	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
12	11	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{Lat}$$
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	dalemccea	$$⊢\psi \to c\in A$$
16	15	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\in A$$
17	1	dalempea	$$⊢\phi \to P\in A$$
18	17	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\in A$$
19		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
20	19 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge c\in A\wedge P\in A\right)\to c\underset{\dot{}}{\vee }P\in {\mathrm{Base}}_K$$
21	14 16 18 20	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\vee }P\in {\mathrm{Base}}_K$$
22	5	dalemddea	$$⊢\psi \to d\in A$$
23	22	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\in A$$
24	1	dalemsea	$$⊢\phi \to S\in A$$
25	24	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to S\in A$$
26	19 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge d\in A\wedge S\in A\right)\to d\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
27	14 23 25 26	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
28	19 2 6	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge c\underset{\dot{}}{\vee }P\in {\mathrm{Base}}_K\wedge d\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }P\right)$$
29	12 21 27 28	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }P\right)$$
30	10 29	eqbrtrid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }P\right)$$
31	1 2 3 4 5 6 7 8 9 10	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$
32	1 2 3 4 7 8	dalemply	$$⊢\phi \to P\underset{\dot{}}{\le }Y$$
33	32	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\underset{\dot{}}{\le }Y$$
34	1 2 3 4 5 6 7 8 9 10	dalem24	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg G\underset{\dot{}}{\le }Y$$
35		nbrne2	$$⊢\left(P\underset{\dot{}}{\le }Y\wedge \neg G\underset{\dot{}}{\le }Y\right)\to P\ne G$$
36	35	necomd	$$⊢\left(P\underset{\dot{}}{\le }Y\wedge \neg G\underset{\dot{}}{\le }Y\right)\to G\ne P$$
37	33 34 36	syl2anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\ne P$$
38	2 3 4	hlatexch2	$$⊢\left(K\in \mathrm{HL}\wedge \left(G\in A\wedge c\in A\wedge P\in A\right)\wedge G\ne P\right)\to \left(G\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }P\right)\to c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\right)$$
39	14 31 16 18 37 38	syl131anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }P\right)\to c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\right)$$
40	30 39	mpd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)$$