cdlemd1

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 29-May-2012)

	Ref	Expression
Hypotheses	cdlemd1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemd1.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemd1.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemd1.a	$⊢ A = Atoms (K)$
	cdlemd1.h	$⊢ H = LHyp (K)$
Assertion	cdlemd1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to R = (P \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\land} (Q \underset{˙}{\lor} ((Q \underset{˙}{\lor} R) \underset{˙}{\land} W))$

Proof

Step	Hyp	Ref	Expression
1		cdlemd1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemd1.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemd1.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemd1.a	$⊢ A = Atoms (K)$
5		cdlemd1.h	$⊢ H = LHyp (K)$
6		simpll	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to K \in HL$
7		simpr1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to P \in A$
8		simpr2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to Q \in A$
9		simpr31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to R \in A$
10		simpr32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to P \neq Q$
11		simpr33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
12	1 2 3 4	2llnma2	$⊢ (K \in HL \land (P \in A \land Q \in A \land R \in A) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} P) \underset{˙}{\land} (R \underset{˙}{\lor} Q) = R$
13	6 7 8 9 10 11 12	syl132anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to (R \underset{˙}{\lor} P) \underset{˙}{\land} (R \underset{˙}{\lor} Q) = R$
14		hllat	$⊢ K \in HL \to K \in Lat$
15	14	ad2antrr	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to K \in Lat$
16		eqid	$⊢ {Base}_{K} = {Base}_{K}$
17	16 4	atbase	$⊢ R \in A \to R \in {Base}_{K}$
18	9 17	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to R \in {Base}_{K}$
19	16 4	atbase	$⊢ P \in A \to P \in {Base}_{K}$
20	7 19	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to P \in {Base}_{K}$
21	16 2	latjcom	$⊢ (K \in Lat \land R \in {Base}_{K} \land P \in {Base}_{K}) \to R \underset{˙}{\lor} P = P \underset{˙}{\lor} R$
22	15 18 20 21	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to R \underset{˙}{\lor} P = P \underset{˙}{\lor} R$
23		simpl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to (K \in HL \land W \in H)$
24		simpr1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
25	16 1 2 3 4 5	cdlemc1	$⊢ ((K \in HL \land W \in H) \land R \in {Base}_{K} \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to P \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) = P \underset{˙}{\lor} R$
26	23 18 24 25	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to P \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) = P \underset{˙}{\lor} R$
27	22 26	eqtr4d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to R \underset{˙}{\lor} P = P \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)$
28	16 4	atbase	$⊢ Q \in A \to Q \in {Base}_{K}$
29	8 28	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to Q \in {Base}_{K}$
30	16 2	latjcom	$⊢ (K \in Lat \land R \in {Base}_{K} \land Q \in {Base}_{K}) \to R \underset{˙}{\lor} Q = Q \underset{˙}{\lor} R$
31	15 18 29 30	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to R \underset{˙}{\lor} Q = Q \underset{˙}{\lor} R$
32		simpr2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
33	16 1 2 3 4 5	cdlemc1	$⊢ ((K \in HL \land W \in H) \land R \in {Base}_{K} \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to Q \underset{˙}{\lor} ((Q \underset{˙}{\lor} R) \underset{˙}{\land} W) = Q \underset{˙}{\lor} R$
34	23 18 32 33	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to Q \underset{˙}{\lor} ((Q \underset{˙}{\lor} R) \underset{˙}{\land} W) = Q \underset{˙}{\lor} R$
35	31 34	eqtr4d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to R \underset{˙}{\lor} Q = Q \underset{˙}{\lor} ((Q \underset{˙}{\lor} R) \underset{˙}{\land} W)$
36	27 35	oveq12d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to (R \underset{˙}{\lor} P) \underset{˙}{\land} (R \underset{˙}{\lor} Q) = (P \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\land} (Q \underset{˙}{\lor} ((Q \underset{˙}{\lor} R) \underset{˙}{\land} W))$
37	13 36	eqtr3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to R = (P \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\land} (Q \underset{˙}{\lor} ((Q \underset{˙}{\lor} R) \underset{˙}{\land} W))$

Metamath Proof Explorer

Theorem cdlemd1

Proof