cdleme11a

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 12-Jun-2012)

	Ref	Expression
Hypotheses	cdleme11.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme11.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme11.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme11.a	$⊢ A = Atoms (K)$
	cdleme11.h	$⊢ H = LHyp (K)$
	cdleme11.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
Assertion	cdleme11a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to S \underset{˙}{\lor} U = S \underset{˙}{\lor} T$

Proof

Step	Hyp	Ref	Expression
1		cdleme11.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme11.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme11.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme11.a	$⊢ A = Atoms (K)$
5		cdleme11.h	$⊢ H = LHyp (K)$
6		cdleme11.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		simp3rr	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to U \underset{˙}{\leq} (S \underset{˙}{\lor} T)$
8		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to K \in HL$
9		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (K \in HL \land W \in H)$
10		simp2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
11		simp2r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (Q \in A \land P \neq Q)$
12	1 2 3 4 5 6	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
13	9 10 11 12	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to U \in A$
14		simp3rl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to T \in A$
15		simp3ll	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to S \in A$
16		simp2ll	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to P \in A$
17		simp2rl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to Q \in A$
18		simp3l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
19	1 2 3 4 5 6	cdleme0c	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \to U \neq S$
20	9 16 17 18 19	syl121anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to U \neq S$
21	1 2 4	hlatexchb1	$⊢ (K \in HL \land (U \in A \land T \in A \land S \in A) \land U \neq S) \to (U \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow S \underset{˙}{\lor} U = S \underset{˙}{\lor} T)$
22	8 13 14 15 20 21	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (U \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow S \underset{˙}{\lor} U = S \underset{˙}{\lor} T)$
23	7 22	mpbid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to S \underset{˙}{\lor} U = S \underset{˙}{\lor} T$

Metamath Proof Explorer

Theorem cdleme11a

Proof