4atexlemqtb

	Ref	Expression
Hypotheses	4thatlem.ph	$⊢ φ \leftrightarrow (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (S \in A \land (R \in A \land \neg R \underset{˙}{\leq} W \land P \underset{˙}{\lor} R = Q \underset{˙}{\lor} R) \land (T \in A \land U \underset{˙}{\lor} T = V \underset{˙}{\lor} T)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))$
	4thatlempqb.j	$⊢ \underset{˙}{\lor} = join (K)$
	4thatlempqb.a	$⊢ A = Atoms (K)$
Assertion	4atexlemqtb	$⊢ φ \to Q \underset{˙}{\lor} T \in {Base}_{K}$

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		4thatlempqb.j	$⊢ \underset{˙}{\lor} = join (K)$
3		4thatlempqb.a	$⊢ A = Atoms (K)$
4	1	4atexlemk	$⊢ φ \to K \in HL$
5	1	4atexlemq	$⊢ φ \to Q \in A$
6	1	4atexlemt	$⊢ φ \to T \in A$
7		eqid	$⊢ {Base}_{K} = {Base}_{K}$
8	7 2 3	hlatjcl	$⊢ (K \in HL \land Q \in A \land T \in A) \to Q \underset{˙}{\lor} T \in {Base}_{K}$
9	4 5 6 8	syl3anc	$⊢ φ \to Q \underset{˙}{\lor} T \in {Base}_{K}$

Metamath Proof Explorer