Theorem dalem61

Description: Lemma for dath . Show that atoms D , E , and F lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms c and d . (Contributed by NM, 11-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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem61.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem61.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem61.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem61.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem61.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
	dalem61.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
	dalem61.f	$$⊢F=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)$$
Assertion	dalem61	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to F\underset{\dot{}}{\le }\left(D\underset{\dot{}}{\vee }E\right)$$

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		dalem61.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem61.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem61.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem61.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem61.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
11		dalem61.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
12		dalem61.f	$$⊢F=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)$$
13		eqid	$$⊢\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
14		eqid	$$⊢\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
15		eqid	$$⊢\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
16		eqid	$$⊢\left(\left(\left(\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\vee }\left(\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)\right)\right)\underset{\dot{}}{\wedge }Y=\left(\left(\left(\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\vee }\left(\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)\right)\right)\underset{\dot{}}{\wedge }Y$$
17	1 2 3 4 5 6 7 8 9 12 13 14 15 16	dalem59	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to F\underset{\dot{}}{\le }\left(\left(\left(\left(\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\vee }\left(\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)\right)\right)\underset{\dot{}}{\wedge }Y\right)$$
18	1 2 3 4 5 6 7 8 9 10 11 13 14 15 16	dalem60	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to D\underset{\dot{}}{\vee }E=\left(\left(\left(\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\vee }\left(\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)\right)\right)\underset{\dot{}}{\wedge }Y$$
19	17 18	breqtrrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to F\underset{\dot{}}{\le }\left(D\underset{\dot{}}{\vee }E\right)$$