Theorem 4atexlemv

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

	Ref	Expression
Hypotheses	4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
	4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
Assertion	4atexlemv	$$⊢\phi \to V\in A$$

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		4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
9	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
10	1	4atexlemw	$$⊢\phi \to W\in H$$
11	1	4atexlempw	$$⊢\phi \to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
12	1	4atexlems	$$⊢\phi \to S\in A$$
13	1 2 3 5	4atexlempns	$$⊢\phi \to P\ne S$$
14	2 3 4 5 6 8	lhpat2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge P\ne S\right)\right)\to V\in A$$
15	9 10 11 12 13 14	syl212anc	$$⊢\phi \to V\in A$$