Theorem dalem54

Description: Lemma for dath . Line G H intersects the auxiliary axis of perspectivity B . (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	dalem54	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in A$$

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		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	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
15	14	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
16	1 2 3 4 5 6 7 8 9 10	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$
17	1 2 3 4 5 6 7 8 9 11	dalem29	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in A$$
18	1 2 3 4 5 6 7 8 9 10 11 12	dalem41	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\ne H$$
19		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
20	3 4 19	llni2	$$⊢\left(\left(K\in \mathrm{HL}\wedge G\in A\wedge H\in A\right)\wedge G\ne H\right)\to G\underset{\dot{}}{\vee }H\in \mathrm{LLines}\left(K\right)$$
21	15 16 17 18 20	syl31anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }H\in \mathrm{LLines}\left(K\right)$$
22	1 2 3 4 5 6 19 7 8 9 10 11 12 13	dalem53	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in \mathrm{LLines}\left(K\right)$$
23	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
24	23	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{Lat}$$
25		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
26	25 19	llnbase	$$⊢G\underset{\dot{}}{\vee }H\in \mathrm{LLines}\left(K\right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
27	21 26	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
28	1 2 3 4 5 6 7 8 9 12	dalem34	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in A$$
29	25 4	atbase	$$⊢I\in A\to I\in {\mathrm{Base}}_K$$
30	28 29	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in {\mathrm{Base}}_K$$
31	25 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge I\in {\mathrm{Base}}_K\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K$$
32	24 27 30 31	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K$$
33	1 7	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
34	33	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to Y\in {\mathrm{Base}}_K$$
35	25 2 6	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in {\mathrm{Base}}_K\wedge Y\in {\mathrm{Base}}_K\right)\to \left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\le }Y$$
36	24 32 34 35	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\le }Y$$
37	13 36	eqbrtrid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\underset{\dot{}}{\le }Y$$
38	1 2 3 4 5 6 7 8 9 10	dalem24	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg G\underset{\dot{}}{\le }Y$$
39	25 4	atbase	$$⊢G\in A\to G\in {\mathrm{Base}}_K$$
40	16 39	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in {\mathrm{Base}}_K$$
41	25 4	atbase	$$⊢H\in A\to H\in {\mathrm{Base}}_K$$
42	17 41	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in {\mathrm{Base}}_K$$
43	25 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(G\in {\mathrm{Base}}_K\wedge H\in {\mathrm{Base}}_K\wedge Y\in {\mathrm{Base}}_K\right)\right)\to \left(\left(G\underset{\dot{}}{\le }Y\wedge H\underset{\dot{}}{\le }Y\right)\leftrightarrow \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }Y\right)$$
44	24 40 42 34 43	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\le }Y\wedge H\underset{\dot{}}{\le }Y\right)\leftrightarrow \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }Y\right)$$
45		simpl	$$⊢\left(G\underset{\dot{}}{\le }Y\wedge H\underset{\dot{}}{\le }Y\right)\to G\underset{\dot{}}{\le }Y$$
46	44 45	biimtrrdi	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }Y\to G\underset{\dot{}}{\le }Y\right)$$
47	38 46	mtod	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }Y$$
48		nbrne2	$$⊢\left(B\underset{\dot{}}{\le }Y\wedge \neg \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\le }Y\right)\to B\ne G\underset{\dot{}}{\vee }H$$
49	37 47 48	syl2anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\ne G\underset{\dot{}}{\vee }H$$
50	49	necomd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }H\ne B$$
51		hlatl	$$⊢K\in \mathrm{HL}\to K\in \mathrm{AtLat}$$
52	15 51	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{AtLat}$$
53	25 19	llnbase	$$⊢B\in \mathrm{LLines}\left(K\right)\to B\in {\mathrm{Base}}_K$$
54	22 53	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in {\mathrm{Base}}_K$$
55	25 6	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge B\in {\mathrm{Base}}_K\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in {\mathrm{Base}}_K$$
56	24 27 54 55	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in {\mathrm{Base}}_K$$
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 3 4	dalempjqeb	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
59	58	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
60	25 2 6	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)$$
61	24 27 59 60	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)$$
62	1 2 3 4 5 6 7 8 9 10 11 12	dalem51	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(\left(K\in \mathrm{HL}\wedge c\in A\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\ne Y\right)$$
63	62	simpld	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(K\in \mathrm{HL}\wedge c\in A\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)$$
64	25 4	atbase	$$⊢c\in A\to c\in {\mathrm{Base}}_K$$
65	64	anim2i	$$⊢\left(K\in \mathrm{HL}\wedge c\in A\right)\to \left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)$$
66	65	3anim1i	$$⊢\left(\left(K\in \mathrm{HL}\wedge c\in A\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)$$
67		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)$$
68		eqid	$$⊢\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I$$
69		eqid	$$⊢\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)$$
70	67 2 3 4 6 7 68 8 13 69	dalem10	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }B$$
71	66 70	syl3an1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge c\in A\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }B$$
72	63 71	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }B$$
73	25 6	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in {\mathrm{Base}}_K$$
74	24 27 59 73	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in {\mathrm{Base}}_K$$
75	25 2 6	latlem12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in {\mathrm{Base}}_K\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge B\in {\mathrm{Base}}_K\right)\right)\to \left(\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }B\right)\leftrightarrow \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\right)$$
76	24 74 27 54 75	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }B\right)\leftrightarrow \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\right)$$
77	61 72 76	mpbi2and	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)$$
78		eqid	$$⊢\mathrm{0.}\left(K\right)=\mathrm{0.}\left(K\right)$$
79	25 2 78 4	atlen0	$$⊢\left(\left(K\in \mathrm{AtLat}\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in {\mathrm{Base}}_K\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\in A\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\ne \mathrm{0.}\left(K\right)$$
80	52 56 57 77 79	syl31anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\ne \mathrm{0.}\left(K\right)$$
81	6 78 4 19	2llnmat	$$⊢\left(\left(K\in \mathrm{HL}\wedge G\underset{\dot{}}{\vee }H\in \mathrm{LLines}\left(K\right)\wedge B\in \mathrm{LLines}\left(K\right)\right)\wedge \left(G\underset{\dot{}}{\vee }H\ne B\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\ne \mathrm{0.}\left(K\right)\right)\right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in A$$
82	15 21 22 50 80 81	syl32anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in A$$