cdleme20d

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, second line. D , F , Y , G represent s_2, f(s), t_2, f(t). (Contributed by NM, 17-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	cdleme20d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (F \underset{˙}{\lor} G) \underset{˙}{\land} (D \underset{˙}{\lor} Y) = V$

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		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
13		hlol	$⊢ K \in HL \to K \in OL$
14	12 13	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in OL$
15	12	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in Lat$
16		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
17		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
18		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
19		simp21l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \in A$
20		eqid	$⊢ {Base}_{K} = {Base}_{K}$
21	1 2 3 4 5 6 7 20	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land S \in A)) \to F \in {Base}_{K}$
22	12 16 17 18 19 21	syl23anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in {Base}_{K}$
23		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to T \in A$
24	1 2 3 4 5 6 8 20	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land T \in A)) \to G \in {Base}_{K}$
25	12 16 17 18 23 24	syl23anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G \in {Base}_{K}$
26	20 2	latjcl	$⊢ (K \in Lat \land F \in {Base}_{K} \land G \in {Base}_{K}) \to F \underset{˙}{\lor} G \in {Base}_{K}$
27	15 22 25 26	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \underset{˙}{\lor} G \in {Base}_{K}$
28	20 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
29	16 28	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in {Base}_{K}$
30		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \in A$
31	20 2 4	hlatjcl	$⊢ (K \in HL \land R \in A \land S \in A) \to R \underset{˙}{\lor} S \in {Base}_{K}$
32	12 30 19 31	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\lor} S \in {Base}_{K}$
33	20 4	atbase	$⊢ T \in A \to T \in {Base}_{K}$
34	23 33	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to T \in {Base}_{K}$
35	20 2	latjcl	$⊢ (K \in Lat \land R \underset{˙}{\lor} S \in {Base}_{K} \land T \in {Base}_{K}) \to (R \underset{˙}{\lor} S) \underset{˙}{\lor} T \in {Base}_{K}$
36	15 32 34 35	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} S) \underset{˙}{\lor} T \in {Base}_{K}$
37	20 3	latmassOLD	$⊢ (K \in OL \land (F \underset{˙}{\lor} G \in {Base}_{K} \land W \in {Base}_{K} \land (R \underset{˙}{\lor} S) \underset{˙}{\lor} T \in {Base}_{K})) \to ((F \underset{˙}{\lor} G) \underset{˙}{\land} W) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) = (F \underset{˙}{\lor} G) \underset{˙}{\land} (W \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T))$
38	14 27 29 36 37	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((F \underset{˙}{\lor} G) \underset{˙}{\land} W) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) = (F \underset{˙}{\lor} G) \underset{˙}{\land} (W \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T))$
39	1 2 4	hlatlej2	$⊢ (K \in HL \land R \in A \land S \in A) \to S \underset{˙}{\leq} (R \underset{˙}{\lor} S)$
40	12 30 19 39	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \underset{˙}{\leq} (R \underset{˙}{\lor} S)$
41	20 4	atbase	$⊢ S \in A \to S \in {Base}_{K}$
42	19 41	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \in {Base}_{K}$
43	20 1 2	latjlej1	$⊢ (K \in Lat \land (S \in {Base}_{K} \land R \underset{˙}{\lor} S \in {Base}_{K} \land T \in {Base}_{K})) \to (S \underset{˙}{\leq} (R \underset{˙}{\lor} S) \to (S \underset{˙}{\lor} T) \underset{˙}{\leq} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T))$
44	15 42 32 34 43	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \underset{˙}{\leq} (R \underset{˙}{\lor} S) \to (S \underset{˙}{\lor} T) \underset{˙}{\leq} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T))$
45	40 44	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \underset{˙}{\lor} T) \underset{˙}{\leq} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
46	20 2 4	hlatjcl	$⊢ (K \in HL \land S \in A \land T \in A) \to S \underset{˙}{\lor} T \in {Base}_{K}$
47	12 19 23 46	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \underset{˙}{\lor} T \in {Base}_{K}$
48	20 1 3	latleeqm1	$⊢ (K \in Lat \land S \underset{˙}{\lor} T \in {Base}_{K} \land (R \underset{˙}{\lor} S) \underset{˙}{\lor} T \in {Base}_{K}) \to ((S \underset{˙}{\lor} T) \underset{˙}{\leq} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) \leftrightarrow (S \underset{˙}{\lor} T) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) = S \underset{˙}{\lor} T)$
49	15 47 36 48	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((S \underset{˙}{\lor} T) \underset{˙}{\leq} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) \leftrightarrow (S \underset{˙}{\lor} T) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) = S \underset{˙}{\lor} T)$
50	45 49	mpbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \underset{˙}{\lor} T) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) = S \underset{˙}{\lor} T$
51	50	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((S \underset{˙}{\lor} T) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)) \underset{˙}{\land} W = (S \underset{˙}{\lor} T) \underset{˙}{\land} W$
52	11 51	eqtr4id	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V = ((S \underset{˙}{\lor} T) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)) \underset{˙}{\land} W$
53	20 3	latm32	$⊢ (K \in OL \land (S \underset{˙}{\lor} T \in {Base}_{K} \land (R \underset{˙}{\lor} S) \underset{˙}{\lor} T \in {Base}_{K} \land W \in {Base}_{K})) \to ((S \underset{˙}{\lor} T) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)) \underset{˙}{\land} W = ((S \underset{˙}{\lor} T) \underset{˙}{\land} W) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
54	14 47 36 29 53	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((S \underset{˙}{\lor} T) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)) \underset{˙}{\land} W = ((S \underset{˙}{\lor} T) \underset{˙}{\land} W) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
55		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
56		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
57		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (T \in A \land \neg T \underset{˙}{\leq} W)$
58		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \neq Q \land S \neq T)$
59		simp32l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
60		simp32r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
61	1 2 3 4 5 6 7 8	cdleme16	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \underset{˙}{\lor} T) \underset{˙}{\land} W = (F \underset{˙}{\lor} G) \underset{˙}{\land} W$
62	55 56 57 58 59 60 61	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \underset{˙}{\lor} T) \underset{˙}{\land} W = (F \underset{˙}{\lor} G) \underset{˙}{\land} W$
63	62	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((S \underset{˙}{\lor} T) \underset{˙}{\land} W) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) = ((F \underset{˙}{\lor} G) \underset{˙}{\land} W) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
64	54 63	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((S \underset{˙}{\lor} T) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)) \underset{˙}{\land} W = ((F \underset{˙}{\lor} G) \underset{˙}{\land} W) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
65	52 64	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V = ((F \underset{˙}{\lor} G) \underset{˙}{\land} W) \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
66		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
67		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
68	1 2 3 4 5 6 7 8 9 10 11	cdleme20c	$⊢ ((K \in HL \land W \in H) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land T \in A) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \underset{˙}{\lor} Y = ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) \underset{˙}{\land} W$
69	12 16 66 56 23 59 67 68	syl232anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \underset{˙}{\lor} Y = ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) \underset{˙}{\land} W$
70	20 3	latmcom	$⊢ (K \in Lat \land (R \underset{˙}{\lor} S) \underset{˙}{\lor} T \in {Base}_{K} \land W \in {Base}_{K}) \to ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) \underset{˙}{\land} W = W \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
71	15 36 29 70	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T) \underset{˙}{\land} W = W \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
72	69 71	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \underset{˙}{\lor} Y = W \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T)$
73	72	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (F \underset{˙}{\lor} G) \underset{˙}{\land} (D \underset{˙}{\lor} Y) = (F \underset{˙}{\lor} G) \underset{˙}{\land} (W \underset{˙}{\land} ((R \underset{˙}{\lor} S) \underset{˙}{\lor} T))$
74	38 65 73	3eqtr4rd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (F \underset{˙}{\lor} G) \underset{˙}{\land} (D \underset{˙}{\lor} Y) = V$

Metamath Proof Explorer

Theorem cdleme20d

Proof