cdleme21f

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 29-Nov-2012)

	Ref	Expression
Hypotheses	cdleme21.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme21.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme21.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme21.a	$⊢ A = Atoms (K)$
	cdleme21.h	$⊢ H = LHyp (K)$
	cdleme21.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme21.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme21.b	$⊢ B = (z \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme21.d	$⊢ D = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
	cdleme21.e	$⊢ E = (R \underset{˙}{\lor} z) \underset{˙}{\land} W$
	cdleme21d.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} D)$
	cdleme21d.z	$⊢ Z = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (B \underset{˙}{\lor} E)$
	cdleme21.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
	cdleme21.y	$⊢ Y = (R \underset{˙}{\lor} T) \underset{˙}{\land} W$
	cdleme21.o	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} Y)$
Assertion	cdleme21f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to N = O$

Proof

Step	Hyp	Ref	Expression
1		cdleme21.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme21.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme21.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme21.a	$⊢ A = Atoms (K)$
5		cdleme21.h	$⊢ H = LHyp (K)$
6		cdleme21.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme21.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme21.b	$⊢ B = (z \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} z) \underset{˙}{\land} W))$
9		cdleme21.d	$⊢ D = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
10		cdleme21.e	$⊢ E = (R \underset{˙}{\lor} z) \underset{˙}{\land} W$
11		cdleme21d.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} D)$
12		cdleme21d.z	$⊢ Z = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (B \underset{˙}{\lor} E)$
13		cdleme21.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
14		cdleme21.y	$⊢ Y = (R \underset{˙}{\lor} T) \underset{˙}{\land} W$
15		cdleme21.o	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} Y)$
16		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to (K \in HL \land W \in H)$
17		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
19		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
20		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
21		simp231	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to P \neq Q$
22		simp232	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
23		simp32l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
24	22 23	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
25		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z)$
26	1 2 3 4 5 6 7 8 9 10 11 12	cdleme21d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to N = Z$
27	16 17 18 19 20 21 24 25 26	syl323anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to N = Z$
28	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	cdleme21e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to O = Z$
29	27 28	eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land P \underset{˙}{\lor} z = S \underset{˙}{\lor} z))) \to N = O$

Metamath Proof Explorer

Theorem cdleme21f

Proof