Theorem dalem19

Description: Lemma for dath . Show that a second dummy atom d exists outside of the Y and Z planes (when those planes are equal). (Contributed by NM, 15-Aug-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem19.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem19.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem19.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
Assertion	dalem19	$$⊢\left(\left(\left(\phi \wedge Y=Z\right)\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to \exists d\in A\left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)$$

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		dalem19.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalem19.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7		dalem19.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
8	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
9	8	ad3antrrr	$$⊢\left(\left(\left(\phi \wedge Y=Z\right)\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to K\in \mathrm{HL}$$
10	1 2 3 4 5 6	dalemcea	$$⊢\phi \to C\in A$$
11	10	ad3antrrr	$$⊢\left(\left(\left(\phi \wedge Y=Z\right)\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to C\in A$$
12		simplr	$$⊢\left(\left(\left(\phi \wedge Y=Z\right)\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to c\in A$$
13	1 5	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
14	13	ad3antrrr	$$⊢\left(\left(\left(\phi \wedge Y=Z\right)\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to Y\in {\mathrm{Base}}_K$$
15	1 2 3 4 5 6 7	dalem17	$$⊢\left(\phi \wedge Y=Z\right)\to C\underset{\dot{}}{\le }Y$$
16	15	ad2antrr	$$⊢\left(\left(\left(\phi \wedge Y=Z\right)\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to C\underset{\dot{}}{\le }Y$$
17		simpr	$$⊢\left(\left(\left(\phi \wedge Y=Z\right)\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to \neg c\underset{\dot{}}{\le }Y$$
18		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
19	18 2 3 4	atbtwnex	$$⊢\left(\left(K\in \mathrm{HL}\wedge C\in A\wedge c\in A\right)\wedge \left(Y\in {\mathrm{Base}}_K\wedge C\underset{\dot{}}{\le }Y\wedge \neg c\underset{\dot{}}{\le }Y\right)\right)\to \exists d\in A\left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)$$
20	9 11 12 14 16 17 19	syl33anc	$$⊢\left(\left(\left(\phi \wedge Y=Z\right)\wedge c\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\right)\to \exists d\in A\left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)$$