cdleme22aa

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 3rd line on p. 115. Show that t \/ v = p \/ q implies v = u. (Contributed by NM, 2-Dec-2012)

	Ref	Expression
Hypotheses	cdleme22.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme22.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme22.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme22.a	$⊢ A = Atoms (K)$
	cdleme22.h	$⊢ H = LHyp (K)$
	cdleme22.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
Assertion	cdleme22aa	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V = U$

Proof

Step	Hyp	Ref	Expression
1		cdleme22.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme22.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme22.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme22.a	$⊢ A = Atoms (K)$
5		cdleme22.h	$⊢ H = LHyp (K)$
6		cdleme22.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
8		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \underset{˙}{\leq} W$
9		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
10	9	hllatd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in Lat$
11		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \in A$
12		eqid	$⊢ {Base}_{K} = {Base}_{K}$
13	12 4	atbase	$⊢ V \in A \to V \in {Base}_{K}$
14	11 13	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \in {Base}_{K}$
15		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
16		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
17	12 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
18	9 15 16 17	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
19		simp1r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
20	12 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
21	19 20	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in {Base}_{K}$
22	12 1 3	latlem12	$⊢ (K \in Lat \land (V \in {Base}_{K} \land P \underset{˙}{\lor} Q \in {Base}_{K} \land W \in {Base}_{K})) \to ((V \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land V \underset{˙}{\leq} W) \leftrightarrow V \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
23	10 14 18 21 22	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((V \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land V \underset{˙}{\leq} W) \leftrightarrow V \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
24	7 8 23	mpbi2and	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
25	24 6	breqtrrdi	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \underset{˙}{\leq} U$
26		hlatl	$⊢ K \in HL \to K \in AtLat$
27	9 26	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in AtLat$
28		simp21r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg P \underset{˙}{\leq} W$
29		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
30	1 2 3 4 5 6	cdleme0a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
31	9 19 15 28 16 29 30	syl222anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \in A$
32	1 4	atcmp	$⊢ (K \in AtLat \land V \in A \land U \in A) \to (V \underset{˙}{\leq} U \leftrightarrow V = U)$
33	27 11 31 32	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (V \underset{˙}{\leq} U \leftrightarrow V = U)$
34	25 33	mpbid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land (V \in A \land V \underset{˙}{\leq} W \land V \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V = U$

Metamath Proof Explorer

Theorem cdleme22aa

Proof