Theorem dalem6

Description: Lemma for dath . Analogue of dalem5 for S . (Contributed by NM, 21-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem6.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem6.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem6.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem6.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
Assertion	dalem6	$$⊢\phi \to S\underset{\dot{}}{\le }W$$

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		dalem6.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalem6.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7		dalem6.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
8		dalem6.w	$$⊢W=Y\underset{\dot{}}{\vee }C$$
9	1 2 3 4 6 7	dalemrot	$$⊢\phi \to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
10		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)$$
11		eqid	$$⊢\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
12		eqid	$$⊢\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }C=\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }C$$
13	10 2 3 4 5 11 12	dalem5	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)\to S\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }C\right)$$
14	9 13	syl	$$⊢\phi \to S\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }C\right)$$
15	1 3 4	dalemqrprot	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
16	6 15	eqtr4id	$$⊢\phi \to Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
17	16	oveq1d	$$⊢\phi \to Y\underset{\dot{}}{\vee }C=\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }C$$
18	8 17	syl5eq	$$⊢\phi \to W=\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }C$$
19	14 18	breqtrrd	$$⊢\phi \to S\underset{\dot{}}{\le }W$$