cdleme20i

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). We show (f(s) \/ s_2) /\ (f(t) \/ t_2) <_ p \/ q. (Contributed by NM, 18-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	cdleme20i	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to ((F \underset{˙}{\lor} D) \underset{˙}{\land} (G \underset{˙}{\lor} Y)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

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		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
13		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
14		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (T \in A \land \neg T \underset{˙}{\leq} W)$
15		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
16		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (P \neq Q \land S \neq T)$
17		simp321	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
18		simp322	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
19	17 18	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
20		simp323	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
21	16 19 20	3jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
22	1 2 3 4 5 6 7 8 9 10 11	cdleme20f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((F \underset{˙}{\lor} D) \underset{˙}{\land} (G \underset{˙}{\lor} Y)) \underset{˙}{\leq} (((D \underset{˙}{\lor} S) \underset{˙}{\land} (Y \underset{˙}{\lor} T)) \underset{˙}{\lor} ((S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G)))$
23	12 13 14 15 21 22	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to ((F \underset{˙}{\lor} D) \underset{˙}{\land} (G \underset{˙}{\lor} Y)) \underset{˙}{\leq} (((D \underset{˙}{\lor} S) \underset{˙}{\land} (Y \underset{˙}{\lor} T)) \underset{˙}{\lor} ((S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G)))$
24	1 2 3 4 5 6 7 8 9 10 11	cdleme20h	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to ((S \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} R)) \underset{˙}{\lor} ((S \underset{˙}{\lor} U) \underset{˙}{\land} (T \underset{˙}{\lor} U)) = R \underset{˙}{\lor} U$
25	1 2 3 4 5 6 7 8 9 10 11	cdleme20g	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((D \underset{˙}{\lor} S) \underset{˙}{\land} (Y \underset{˙}{\lor} T)) \underset{˙}{\lor} ((S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G)) = ((S \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} R)) \underset{˙}{\lor} ((S \underset{˙}{\lor} U) \underset{˙}{\land} (T \underset{˙}{\lor} U))$
26	12 13 14 15 21 25	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to ((D \underset{˙}{\lor} S) \underset{˙}{\land} (Y \underset{˙}{\lor} T)) \underset{˙}{\lor} ((S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G)) = ((S \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} R)) \underset{˙}{\lor} ((S \underset{˙}{\lor} U) \underset{˙}{\land} (T \underset{˙}{\lor} U))$
27		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (K \in HL \land W \in H)$
28		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to P \in A$
29		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to Q \in A$
30	1 2 3 4 5 6	cdleme4	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q = R \underset{˙}{\lor} U$
31	27 28 29 15 20 30	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to P \underset{˙}{\lor} Q = R \underset{˙}{\lor} U$
32	24 26 31	3eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to ((D \underset{˙}{\lor} S) \underset{˙}{\land} (Y \underset{˙}{\lor} T)) \underset{˙}{\lor} ((S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G)) = P \underset{˙}{\lor} Q$
33	23 32	breqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((P \neq Q \land S \neq T) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (\neg R \underset{˙}{\leq} (S \underset{˙}{\lor} T) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to ((F \underset{˙}{\lor} D) \underset{˙}{\land} (G \underset{˙}{\lor} Y)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

Metamath Proof Explorer

Theorem cdleme20i

Proof