cdleme20k

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, antepenultimate line. D , F , Y , G represent s_2, f(s), t_2, f(t). (Contributed by NM, 20-Nov-2012)

	Ref	Expression
Hypotheses	cdleme19.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme19.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme19.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme19.a	$⊢ A = Atoms (K)$
	cdleme19.h	$⊢ H = LHyp (K)$
	cdleme19.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme19.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme19.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
	cdleme19.d	$⊢ D = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
	cdleme19.y	$⊢ Y = (R \underset{˙}{\lor} T) \underset{˙}{\land} W$
	cdleme20.v	$⊢ V = (S \underset{˙}{\lor} T) \underset{˙}{\land} W$
Assertion	cdleme20k	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \underset{˙}{\lor} D \neq P \underset{˙}{\lor} Q$

Proof

Step	Hyp	Ref	Expression
1		cdleme19.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme19.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme19.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme19.a	$⊢ A = Atoms (K)$
5		cdleme19.h	$⊢ H = LHyp (K)$
6		cdleme19.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme19.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme19.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
9		cdleme19.d	$⊢ D = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
10		cdleme19.y	$⊢ Y = (R \underset{˙}{\lor} T) \underset{˙}{\land} W$
11		cdleme20.v	$⊢ V = (S \underset{˙}{\lor} T) \underset{˙}{\land} W$
12		simp11	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (K \in HL \land W \in H)$
13		simp12	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
14		simp13	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
15		simp2r	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
16		simp2l	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
17		simp3r	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
18		simp3l	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
19	1 2 3 4 5 9	cdlemednpq	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
20	12 13 14 15 16 17 18 19	syl133anc	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
21		simp11l	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
22	21	hllatd	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in Lat$
23		simp11r	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
24		simp2ll	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \in A$
25		eqid	$⊢ {Base}_{K} = {Base}_{K}$
26	1 2 3 4 5 6 7 25	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land S \in A)) \to F \in {Base}_{K}$
27	21 23 13 14 24 26	syl23anc	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in {Base}_{K}$
28		simp2rl	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \in A$
29	1 2 3 4 5 9 25	cdlemedb	$⊢ ((K \in HL \land W \in H) \land (R \in A \land S \in A)) \to D \in {Base}_{K}$
30	21 23 28 24 29	syl22anc	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \in {Base}_{K}$
31	25 1 2	latlej2	$⊢ (K \in Lat \land F \in {Base}_{K} \land D \in {Base}_{K}) \to D \underset{˙}{\leq} (F \underset{˙}{\lor} D)$
32	22 27 30 31	syl3anc	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \underset{˙}{\leq} (F \underset{˙}{\lor} D)$
33		breq2	$⊢ F \underset{˙}{\lor} D = P \underset{˙}{\lor} Q \to (D \underset{˙}{\leq} (F \underset{˙}{\lor} D) \leftrightarrow D \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
34	32 33	syl5ibcom	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (F \underset{˙}{\lor} D = P \underset{˙}{\lor} Q \to D \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
35	34	necon3bd	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (\neg D \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to F \underset{˙}{\lor} D \neq P \underset{˙}{\lor} Q)$
36	20 35	mpd	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \underset{˙}{\lor} D \neq P \underset{˙}{\lor} Q$

Metamath Proof Explorer

Theorem cdleme20k

Proof