Theorem dalem10

Description: Lemma for dath . Atom D belongs to the axis of perspectivity X . (Contributed by NM, 19-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem10.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem10.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem10.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem10.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem10.x	$$⊢X=Y\underset{\dot{}}{\wedge }Z$$
	dalem10.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
Assertion	dalem10	$$⊢\phi \to D\underset{\dot{}}{\le }X$$

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		dalem10.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
6		dalem10.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
7		dalem10.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
8		dalem10.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
9		dalem10.x	$$⊢X=Y\underset{\dot{}}{\wedge }Z$$
10		dalem10.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
11	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
12	1 3 4	dalempjqeb	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
13	1 4	dalemreb	$$⊢\phi \to R\in {\mathrm{Base}}_K$$
14		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
15	14 2 3	latlej1	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge R\in {\mathrm{Base}}_K\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
16	11 12 13 15	syl3anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
17	1 3 4	dalemsjteb	$$⊢\phi \to S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
18	1 4	dalemueb	$$⊢\phi \to U\in {\mathrm{Base}}_K$$
19	14 2 3	latlej1	$$⊢\left(K\in \mathrm{Lat}\wedge S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge U\in {\mathrm{Base}}_K\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)$$
20	11 17 18 19	syl3anc	$$⊢\phi \to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)$$
21	1 6	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
22	7 21	eqeltrrid	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K$$
23	1	dalemzeo	$$⊢\phi \to Z\in O$$
24	14 6	lplnbase	$$⊢Z\in O\to Z\in {\mathrm{Base}}_K$$
25	23 24	syl	$$⊢\phi \to Z\in {\mathrm{Base}}_K$$
26	8 25	eqeltrrid	$$⊢\phi \to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K$$
27	14 2 5	latmlem12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K\right)\wedge \left(S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)\right)\right)$$
28	11 12 22 17 26 27	syl122anc	$$⊢\phi \to \left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)\right)\right)$$
29	16 20 28	mp2and	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)\right)$$
30	7 8	oveq12i	$$⊢Y\underset{\dot{}}{\wedge }Z=\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)$$
31	9 30	eqtri	$$⊢X=\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)$$
32	29 10 31	3brtr4g	$$⊢\phi \to D\underset{\dot{}}{\le }X$$