lhp2atne

Description: Inequality for joins with 2 different atoms under co-atom W . (Contributed by NM, 22-Jul-2013)

	Ref	Expression
Hypotheses	lhp2atnle.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	lhp2atnle.j	$⊢ \underset{˙}{\lor} = join (K)$
	lhp2atnle.a	$⊢ A = Atoms (K)$
	lhp2atnle.h	$⊢ H = LHyp (K)$
Assertion	lhp2atne	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to P \underset{˙}{\lor} U \neq Q \underset{˙}{\lor} V$

Proof

Step	Hyp	Ref	Expression
1		lhp2atnle.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		lhp2atnle.j	$⊢ \underset{˙}{\lor} = join (K)$
3		lhp2atnle.a	$⊢ A = Atoms (K)$
4		lhp2atnle.h	$⊢ H = LHyp (K)$
5		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to (K \in HL \land W \in H)$
6		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
7		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to U \neq V$
8		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to (U \in A \land U \underset{˙}{\leq} W)$
9		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to (V \in A \land V \underset{˙}{\leq} W)$
10	1 2 3 4	lhp2atnle	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land U \neq V) \land (U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \to \neg V \underset{˙}{\leq} (P \underset{˙}{\lor} U)$
11	5 6 7 8 9 10	syl311anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to \neg V \underset{˙}{\leq} (P \underset{˙}{\lor} U)$
12		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to K \in HL$
13		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to Q \in A$
14		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to V \in A$
15	1 2 3	hlatlej2	$⊢ (K \in HL \land Q \in A \land V \in A) \to V \underset{˙}{\leq} (Q \underset{˙}{\lor} V)$
16	12 13 14 15	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to V \underset{˙}{\leq} (Q \underset{˙}{\lor} V)$
17	16	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \land P \underset{˙}{\lor} U = Q \underset{˙}{\lor} V) \to V \underset{˙}{\leq} (Q \underset{˙}{\lor} V)$
18		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \land P \underset{˙}{\lor} U = Q \underset{˙}{\lor} V) \to P \underset{˙}{\lor} U = Q \underset{˙}{\lor} V$
19	17 18	breqtrrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \land P \underset{˙}{\lor} U = Q \underset{˙}{\lor} V) \to V \underset{˙}{\leq} (P \underset{˙}{\lor} U)$
20	19	ex	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to (P \underset{˙}{\lor} U = Q \underset{˙}{\lor} V \to V \underset{˙}{\leq} (P \underset{˙}{\lor} U))$
21	20	necon3bd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to (\neg V \underset{˙}{\leq} (P \underset{˙}{\lor} U) \to P \underset{˙}{\lor} U \neq Q \underset{˙}{\lor} V)$
22	11 21	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((U \in A \land U \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land U \neq V) \to P \underset{˙}{\lor} U \neq Q \underset{˙}{\lor} V$

Metamath Proof Explorer

Theorem lhp2atne

Proof