Theorem dalem55

Description: Lemma for dath . Lines G H and P Q intersect at the auxiliary line B (later shown to be an axis of perspectivity; see dalem60 ). (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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem54.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem54.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem54.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem54.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem54.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
	dalem54.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
	dalem54.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
	dalem54.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
Assertion	dalem55	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B$$

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		dalem54.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem54.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem54.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem54.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem54.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
11		dalem54.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
12		dalem54.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
13		dalem54.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
14	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
15	14	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{Lat}$$
16	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
17	16	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
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		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
21	20 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge G\in A\wedge H\in A\right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
22	17 18 19 21	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
23	1 3 4	dalempjqeb	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
24	23	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
25	20 2 6	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)$$
26	15 22 24 25	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)$$
27	1 2 3 4 5 6 7 8 9 12	dalem34	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in A$$
28	20 4	atbase	$$⊢I\in A\to I\in {\mathrm{Base}}_K$$
29	27 28	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in {\mathrm{Base}}_K$$
30	20 2 3	latlej1	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge I\in {\mathrm{Base}}_K\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)$$
31	15 22 29 30	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)$$
32	1 4	dalemreb	$$⊢\phi \to R\in {\mathrm{Base}}_K$$
33	20 2 3	latlej1	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge R\in {\mathrm{Base}}_K\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
34	14 23 32 33	syl3anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
35	34 8	breqtrrdi	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }Y$$
36	35	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }Y$$
37	1 2 3 4 5 6 7 8 9 10 11 12	dalem42	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O$$
38	20 7	lplnbase	$$⊢\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K$$
39	37 38	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K$$
40	1 7	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
41	40	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\in {\mathrm{Base}}_K$$
42	20 2 6	latmlem12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K\right)\wedge \left(P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge Y\in {\mathrm{Base}}_K\right)\right)\to \left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\wedge \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }Y\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y\right)\right)$$
43	15 22 39 24 41 42	syl122anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\wedge \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }Y\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y\right)\right)$$
44	31 36 43	mp2and	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y\right)$$
45	44 13	breqtrrdi	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }B$$
46	20 6	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in {\mathrm{Base}}_K$$
47	15 22 24 46	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in {\mathrm{Base}}_K$$
48		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
49	1 2 3 4 5 6 48 7 8 9 10 11 12 13	dalem53	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in \mathrm{LLines}\left(K\right)$$
50	20 48	llnbase	$$⊢B\in \mathrm{LLines}\left(K\right)\to B\in {\mathrm{Base}}_K$$
51	49 50	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in {\mathrm{Base}}_K$$
52	20 2 6	latlem12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in {\mathrm{Base}}_K\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge B\in {\mathrm{Base}}_K\right)\right)\to \left(\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }B\right)\leftrightarrow \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\right)$$
53	15 47 22 51 52	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }B\right)\leftrightarrow \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\right)$$
54	26 45 53	mpbi2and	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)$$
55		hlatl	$$⊢K\in \mathrm{HL}\to K\in \mathrm{AtLat}$$
56	17 55	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{AtLat}$$
57	1 2 3 4 5 6 7 8 9 10 11 12	dalem52	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in A$$
58	1 2 3 4 5 6 7 8 9 10 11 12 13	dalem54	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in A$$
59	2 4	atcmp	$$⊢\left(K\in \mathrm{AtLat}\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in A\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in A\right)\to \left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\leftrightarrow \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)$$
60	56 57 58 59	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\leftrightarrow \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)$$
61	54 60	mpbid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B$$