Metamath Proof Explorer

Theorem osumcllem5N

Description: Lemma for osumclN . (Contributed by NM, 24-Mar-2012) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	osumcllem.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		osumcllem.j	$⊢ \underset{˙}{\lor} = join (K)$
		osumcllem.a	$⊢ A = Atoms (K)$
		osumcllem.p	$⊢ \underset{˙}{+} = +_{𝑃} (K)$
		osumcllem.o	$⊢ \underset{˙}{⊥} = ⊥_{𝑃} (K)$
		osumcllem.c	$⊢ C = PSubCl (K)$
		osumcllem.m	$⊢ M = X \underset{˙}{+} \{p\}$
		osumcllem.u	$⊢ U = \underset{˙}{⊥} (\underset{˙}{⊥} (X \underset{˙}{+} Y))$
	Assertion	osumcllem5N	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to p \in (X \underset{˙}{+} Y)$

Proof

Step	Hyp	Ref	Expression
1		osumcllem.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		osumcllem.j	$⊢ \underset{˙}{\lor} = join (K)$
3		osumcllem.a	$⊢ A = Atoms (K)$
4		osumcllem.p	$⊢ \underset{˙}{+} = +_{𝑃} (K)$
5		osumcllem.o	$⊢ \underset{˙}{⊥} = ⊥_{𝑃} (K)$
6		osumcllem.c	$⊢ C = PSubCl (K)$
7		osumcllem.m	$⊢ M = X \underset{˙}{+} \{p\}$
8		osumcllem.u	$⊢ U = \underset{˙}{⊥} (\underset{˙}{⊥} (X \underset{˙}{+} Y))$
9		simp11	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to K \in HL$
10	9	hllatd	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to K \in Lat$
11		simp12	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to X \subseteq A$
12		simp13	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to Y \subseteq A$
13		simp31	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to r \in X$
14		simp32	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to q \in Y$
15		simp2	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to p \in A$
16		simp33	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to p \underset{˙}{\leq} (r \underset{˙}{\lor} q)$
17	1 2 3 4	elpaddri	$⊢ ((K \in Lat \land X \subseteq A \land Y \subseteq A) \land (r \in X \land q \in Y) \land (p \in A \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to p \in (X \underset{˙}{+} Y)$
18	10 11 12 13 14 15 16 17	syl322anc	$⊢ ((K \in HL \land X \subseteq A \land Y \subseteq A) \land p \in A \land (r \in X \land q \in Y \land p \underset{˙}{\leq} (r \underset{˙}{\lor} q))) \to p \in (X \underset{˙}{+} Y)$