Theorem dalem57

Description: Lemma for dath . Axis of perspectivity point D is on the auxiliary line B . (Contributed by NM, 9-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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem57.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem57.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem57.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem57.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem57.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
	dalem57.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
	dalem57.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
	dalem57.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
	dalem57.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
Assertion	dalem57	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\underset{\dot{}}{\le }B$$

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		dalem57.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem57.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem57.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem57.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem57.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
11		dalem57.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
12		dalem57.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
13		dalem57.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
14		dalem57.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
15	1 2 3 4 5 6 7 8 9 11 12 13 14	dalem55	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B$$
16	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
17	16	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{Lat}$$
18	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
19	18	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
20	1 2 3 4 5 6 7 8 9 11	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$
21	1 2 3 4 5 6 7 8 9 12	dalem29	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in A$$
22		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
23	22 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge G\in A\wedge H\in A\right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
24	19 20 21 23	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K$$
25	1 3 4	dalempjqeb	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
26	25	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
27	22 2 6	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
28	17 24 26 27	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
29	15 28	eqbrtrrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
30	1 2 3 4 5 6 7 8 9 11 12 13 14	dalem56	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B$$
31	1 3 4	dalemsjteb	$$⊢\phi \to S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
32	31	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
33	22 2 6	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
34	17 24 32 33	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
35	30 34	eqbrtrrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
36	1 2 3 4 5 6 7 8 9 11 12 13 14	dalem54	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in A$$
37	22 4	atbase	$$⊢\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in A\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in {\mathrm{Base}}_K$$
38	36 37	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in {\mathrm{Base}}_K$$
39	22 2 6	latlem12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\right)\right)\to \left(\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\leftrightarrow \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)$$
40	17 38 26 32 39	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\leftrightarrow \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)$$
41	29 35 40	mpbi2and	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)$$
42	41 10	breqtrrdi	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }D$$
43		hlatl	$$⊢K\in \mathrm{HL}\to K\in \mathrm{AtLat}$$
44	19 43	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{AtLat}$$
45	1 2 3 4 6 7 8 9 10	dalemdea	$$⊢\phi \to D\in A$$
46	45	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\in A$$
47	2 4	atcmp	$$⊢\left(K\in \mathrm{AtLat}\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\in A\wedge D\in A\right)\to \left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }D\leftrightarrow \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B=D\right)$$
48	44 36 46 47	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }D\leftrightarrow \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B=D\right)$$
49	42 48	mpbid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B=D$$
50		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
51	1 2 3 4 5 6 50 7 8 9 11 12 13 14	dalem53	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in \mathrm{LLines}\left(K\right)$$
52	22 50	llnbase	$$⊢B\in \mathrm{LLines}\left(K\right)\to B\in {\mathrm{Base}}_K$$
53	51 52	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in {\mathrm{Base}}_K$$
54	22 2 6	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge G\underset{\dot{}}{\vee }H\in {\mathrm{Base}}_K\wedge B\in {\mathrm{Base}}_K\right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }B$$
55	17 24 53 54	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\wedge }B\right)\underset{\dot{}}{\le }B$$
56	49 55	eqbrtrrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\underset{\dot{}}{\le }B$$