cdlemefr27cl

Description: Part of proof of Lemma E in Crawley p. 113. Closure of N . (Contributed by NM, 23-Mar-2013)

	Ref	Expression
Hypotheses	cdlemefr27.b	$⊢ B = {Base}_{K}$
	cdlemefr27.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemefr27.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemefr27.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemefr27.a	$⊢ A = Atoms (K)$
	cdlemefr27.h	$⊢ H = LHyp (K)$
	cdlemefr27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdlemefr27.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdlemefr27.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
Assertion	cdlemefr27cl	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to N \in B$

Proof

Step	Hyp	Ref	Expression
1		cdlemefr27.b	$⊢ B = {Base}_{K}$
2		cdlemefr27.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemefr27.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemefr27.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemefr27.a	$⊢ A = Atoms (K)$
6		cdlemefr27.h	$⊢ H = LHyp (K)$
7		cdlemefr27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemefr27.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdlemefr27.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
10		simpr2	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
11	10	iffalsed	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C) = C$
12	9 11	syl5eq	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to N = C$
13		simpl1l	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to K \in HL$
14		simpl1r	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to W \in H$
15		simpl2	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to P \in A$
16		simpl3	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to Q \in A$
17		simpr1	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to s \in A$
18	2 3 4 5 6 7 8 1	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land s \in A)) \to C \in B$
19	13 14 15 16 17 18	syl23anc	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to C \in B$
20	12 19	eqeltrd	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to N \in B$

Metamath Proof Explorer

Theorem cdlemefr27cl

Proof