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Metamath Proof Explorer

Theorem cdleme25cl

Description: Show closure of the unique element in cdleme25c . (Contributed by NM, 2-Feb-2013)

		Ref	Expression
	Hypotheses	cdleme24.b	$⊢ B = {Base}_{K}$
		cdleme24.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdleme24.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdleme24.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdleme24.a	$⊢ A = Atoms (K)$
		cdleme24.h	$⊢ H = LHyp (K)$
		cdleme24.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdleme24.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
		cdleme24.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} s) \underset{˙}{\land} W))$
		cdleme25cl.i	$⊢ I = (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N))$
	Assertion	cdleme25cl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to I \in B$

Proof

Step	Hyp	Ref	Expression
1		cdleme24.b	$⊢ B = {Base}_{K}$
2		cdleme24.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme24.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme24.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme24.a	$⊢ A = Atoms (K)$
6		cdleme24.h	$⊢ H = LHyp (K)$
7		cdleme24.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme24.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme24.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} s) \underset{˙}{\land} W))$
10		cdleme25cl.i	$⊢ I = (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N))$
11	1 2 3 4 5 6 7 8 9	cdleme25c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \exists! u \in B \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N)$
12		riotacl	$⊢ \exists! u \in B \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N) \to (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N)) \in B$
13	11 12	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N)) \in B$
14	10 13	eqeltrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to I \in B$