cdleme02N

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 9-Nov-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme0.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme0.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme0.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme0.a	$⊢ A = Atoms (K)$
	cdleme0.h	$⊢ H = LHyp (K)$
	cdleme0.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
Assertion	cdleme02N	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (P \underset{˙}{\lor} U = Q \underset{˙}{\lor} U \land U \underset{˙}{\leq} W)$

Proof

Step	Hyp	Ref	Expression
1		cdleme0.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme0.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme0.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme0.a	$⊢ A = Atoms (K)$
5		cdleme0.h	$⊢ H = LHyp (K)$
6		cdleme0.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7	1 2 3 4 5 6	cdleme01N	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to ((U \neq P \land U \neq Q \land U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land U \underset{˙}{\leq} W)$
8		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to K \in HL$
9		hlcvl	$⊢ K \in HL \to K \in CvLat$
10	8 9	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to K \in CvLat$
11		simp2ll	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to P \in A$
12		simp2rl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to Q \in A$
13		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (K \in HL \land W \in H)$
14		simp2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
15		simp3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to P \neq Q$
16	1 2 3 4 5 6	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
17	13 14 12 15 16	syl112anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to U \in A$
18	4 1 2	cvlsupr2	$⊢ (K \in CvLat \land (P \in A \land Q \in A \land U \in A) \land P \neq Q) \to (P \underset{˙}{\lor} U = Q \underset{˙}{\lor} U \leftrightarrow (U \neq P \land U \neq Q \land U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))$
19	10 11 12 17 15 18	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (P \underset{˙}{\lor} U = Q \underset{˙}{\lor} U \leftrightarrow (U \neq P \land U \neq Q \land U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))$
20	19	anbi1d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to ((P \underset{˙}{\lor} U = Q \underset{˙}{\lor} U \land U \underset{˙}{\leq} W) \leftrightarrow ((U \neq P \land U \neq Q \land U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land U \underset{˙}{\leq} W))$
21	7 20	mpbird	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (P \underset{˙}{\lor} U = Q \underset{˙}{\lor} U \land U \underset{˙}{\leq} W)$

Metamath Proof Explorer

Theorem cdleme02N

Proof