Theorem dalemply

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalempnes.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalempnes.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
Assertion	dalemply	$$⊢\phi \to P\underset{\dot{}}{\le }Y$$

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		dalempnes.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalempnes.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
8	1 4	dalempeb	$$⊢\phi \to P\in {\mathrm{Base}}_K$$
9	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
10	1	dalemqea	$$⊢\phi \to Q\in A$$
11	1	dalemrea	$$⊢\phi \to R\in A$$
12		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
13	12 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge R\in A\right)\to Q\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K$$
14	9 10 11 13	syl3anc	$$⊢\phi \to Q\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K$$
15	12 2 3	latlej1	$$⊢\left(K\in \mathrm{Lat}\wedge P\in {\mathrm{Base}}_K\wedge Q\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K\right)\to P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
16	7 8 14 15	syl3anc	$$⊢\phi \to P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
17	1	dalempea	$$⊢\phi \to P\in A$$
18	3 4	hlatjass	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R=P\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }R\right)$$
19	9 17 10 11 18	syl13anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R=P\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }R\right)$$
20	16 19	breqtrrd	$$⊢\phi \to P\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
21	20 6	breqtrrdi	$$⊢\phi \to P\underset{\dot{}}{\le }Y$$