Theorem dalem11

Description: Lemma for dath . Analogue of dalem10 for E . (Contributed by NM, 23-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem11.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem11.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem11.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem11.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem11.x	$$⊢X=Y\underset{\dot{}}{\wedge }Z$$
	dalem11.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
Assertion	dalem11	$$⊢\phi \to E\underset{\dot{}}{\le }X$$

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		dalem11.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
6		dalem11.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
7		dalem11.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
8		dalem11.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
9		dalem11.x	$$⊢X=Y\underset{\dot{}}{\wedge }Z$$
10		dalem11.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
11	1 2 3 4 7 8	dalemrot	$$⊢\phi \to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
12		biid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\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)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\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)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)\leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\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)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\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)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
13		eqid	$$⊢\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
14		eqid	$$⊢\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
15		eqid	$$⊢\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\right)=\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\right)$$
16	12 2 3 4 5 6 13 14 15 10	dalem10	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)\to E\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\right)\right)$$
17	11 16	syl	$$⊢\phi \to E\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\right)\right)$$
18	1 3 4	dalemqrprot	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
19	7 18	eqtr4id	$$⊢\phi \to Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
20	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
21	1	dalemtea	$$⊢\phi \to T\in A$$
22	1	dalemuea	$$⊢\phi \to U\in A$$
23	1	dalemsea	$$⊢\phi \to S\in A$$
24	3 4	hlatjrot	$$⊢\left(K\in \mathrm{HL}\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\to \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
25	20 21 22 23 24	syl13anc	$$⊢\phi \to \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
26	8 25	eqtr4id	$$⊢\phi \to Z=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
27	19 26	oveq12d	$$⊢\phi \to Y\underset{\dot{}}{\wedge }Z=\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\right)$$
28	9 27	eqtrid	$$⊢\phi \to X=\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\right)$$
29	17 28	breqtrrd	$$⊢\phi \to E\underset{\dot{}}{\le }X$$