Theorem 4atexlempsb

Description: Lemma for 4atexlem7 . (Contributed by NM, 23-Nov-2012)

	Ref	Expression
Hypotheses	4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
	4thatlempqb.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	4thatlempqb.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	4atexlempsb	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$

Proof

Step	Hyp	Ref	Expression
1		4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
2		4thatlempqb.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		4thatlempqb.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
5	1	4atexlemp	$$⊢\phi \to P\in A$$
6	1	4atexlems	$$⊢\phi \to S\in A$$
7		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
8	7 2 3	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge S\in A\right)\to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
9	4 5 6 8	syl3anc	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$