Theorem dalem53

Description: Lemma for dath . The auxliary axis of perspectivity B is a line (analogous to the actual axis of perspectivity X in dalem15 . (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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem53.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem53.n	$$⊢N=\mathrm{LLines}\left(K\right)$$
	dalem53.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem53.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem53.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem53.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
	dalem53.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
	dalem53.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
	dalem53.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
Assertion	dalem53	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in N$$

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		dalem53.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem53.n	$$⊢N=\mathrm{LLines}\left(K\right)$$
8		dalem53.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
9		dalem53.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
10		dalem53.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
11		dalem53.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
12		dalem53.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
13		dalem53.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
14		dalem53.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 8 9 10 11 12 13	dalem51	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(\left(K\in \mathrm{HL}\wedge c\in A\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\ne Y\right)$$
16		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
17	16 4	atbase	$$⊢c\in A\to c\in {\mathrm{Base}}_K$$
18	17	anim2i	$$⊢\left(K\in \mathrm{HL}\wedge c\in A\right)\to \left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)$$
19	18	3anim1i	$$⊢\left(\left(K\in \mathrm{HL}\wedge c\in A\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)$$
20		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)$$
21		eqid	$$⊢\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I$$
22	20 2 3 4 6 7 8 21 9 14	dalem15	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge c\in {\mathrm{Base}}_K\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\ne Y\right)\to B\in N$$
23	19 22	syl3anl1	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge c\in A\right)\wedge \left(G\in A\wedge H\in A\wedge I\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\in O\wedge Y\in O\right)\wedge \left(\left(\neg c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }H\right)\wedge \neg c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }I\right)\wedge \neg c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }G\right)\right)\wedge \left(\neg c\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg c\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg c\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\wedge c\underset{\dot{}}{\le }\left(H\underset{\dot{}}{\vee }Q\right)\wedge c\underset{\dot{}}{\le }\left(I\underset{\dot{}}{\vee }R\right)\right)\right)\right)\wedge \left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\ne Y\right)\to B\in N$$
24	15 23	syl	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to B\in N$$