4atexlemu

	Ref	Expression
Hypotheses	4thatlem.ph	$⊢ φ \leftrightarrow (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (S \in A \land (R \in A \land \neg R \underset{˙}{\leq} W \land P \underset{˙}{\lor} R = Q \underset{˙}{\lor} R) \land (T \in A \land U \underset{˙}{\lor} T = V \underset{˙}{\lor} T)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))$
	4thatlem0.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	4thatlem0.j	$⊢ \underset{˙}{\lor} = join (K)$
	4thatlem0.m	$⊢ \underset{˙}{\land} = meet (K)$
	4thatlem0.a	$⊢ A = Atoms (K)$
	4thatlem0.h	$⊢ H = LHyp (K)$
	4thatlem0.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
Assertion	4atexlemu	$⊢ φ \to U \in A$

Step	Hyp	Ref	Expression
1		4thatlem.ph	$⊢ φ \leftrightarrow (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (S \in A \land (R \in A \land \neg R \underset{˙}{\leq} W \land P \underset{˙}{\lor} R = Q \underset{˙}{\lor} R) \land (T \in A \land U \underset{˙}{\lor} T = V \underset{˙}{\lor} T)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))$
2		4thatlem0.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		4thatlem0.j	$⊢ \underset{˙}{\lor} = join (K)$
4		4thatlem0.m	$⊢ \underset{˙}{\land} = meet (K)$
5		4thatlem0.a	$⊢ A = Atoms (K)$
6		4thatlem0.h	$⊢ H = LHyp (K)$
7		4thatlem0.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8	1	4atexlemk	$⊢ φ \to K \in HL$
9	1	4atexlemw	$⊢ φ \to W \in H$
10	1	4atexlempw	$⊢ φ \to (P \in A \land \neg P \underset{˙}{\leq} W)$
11	1	4atexlemq	$⊢ φ \to Q \in A$
12	1	4atexlempnq	$⊢ φ \to P \neq Q$
13	2 3 4 5 6 7	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
14	8 9 10 11 12 13	syl212anc	$⊢ φ \to U \in A$

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