Theorem dalem18

Description: Lemma for dath . Show that a dummy atom c exists outside of the Y and Z planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargues's theorem does not always hold in 2 dimensions.) (Contributed by NM, 29-Jul-2012)

	Ref	Expression
Hypotheses	dalema.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
	dalemc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalemc.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalemc.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem18.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
Assertion	dalem18	$$⊢\phi \to \exists c\in A\neg c\underset{\dot{}}{\le }Y$$

Proof

Step	Hyp	Ref	Expression
1		dalema.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		dalemc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalemc.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalemc.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalem18.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
6	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
7	1	dalempea	$$⊢\phi \to P\in A$$
8	1	dalemqea	$$⊢\phi \to Q\in A$$
9	1	dalemrea	$$⊢\phi \to R\in A$$
10	3 2 4	3dim3	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \exists c\in A\neg c\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
11	6 7 8 9 10	syl13anc	$$⊢\phi \to \exists c\in A\neg c\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
12	5	breq2i	$$⊢c\underset{\dot{}}{\le }Y\leftrightarrow c\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
13	12	notbii	$$⊢\neg c\underset{\dot{}}{\le }Y\leftrightarrow \neg c\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
14	13	rexbii	$$⊢\exists c\in A\neg c\underset{\dot{}}{\le }Y\leftrightarrow \exists c\in A\neg c\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
15	11 14	sylibr	$$⊢\phi \to \exists c\in A\neg c\underset{\dot{}}{\le }Y$$