cdleme32fvcl

Description: Part of proof of Lemma D in Crawley p. 113. Closure of the function F . (Contributed by NM, 10-Feb-2013)

	Ref	Expression
Hypotheses	cdleme32.b	$⊢ B = {Base}_{K}$
	cdleme32.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme32.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme32.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme32.a	$⊢ A = Atoms (K)$
	cdleme32.h	$⊢ H = LHyp (K)$
	cdleme32.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme32.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme32.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme32.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme32.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
	cdleme32.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
	cdleme32.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
	cdleme32.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
Assertion	cdleme32fvcl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \to F (X) \in B$

Proof

Step	Hyp	Ref	Expression
1		cdleme32.b	$⊢ B = {Base}_{K}$
2		cdleme32.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme32.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme32.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme32.a	$⊢ A = Atoms (K)$
6		cdleme32.h	$⊢ H = LHyp (K)$
7		cdleme32.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme32.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme32.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdleme32.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
11		cdleme32.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
12		cdleme32.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
13		cdleme32.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
14		cdleme32.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
15		eqid	$⊢ (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to z = N \underset{˙}{\lor} (X \underset{˙}{\land} W))) = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to z = N \underset{˙}{\lor} (X \underset{˙}{\land} W)))$
16	13 14 15	cdleme31fv1	$⊢ (X \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to z = N \underset{˙}{\lor} (X \underset{˙}{\land} W)))$
17	16	adantll	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to z = N \underset{˙}{\lor} (X \underset{˙}{\land} W)))$
18		simpll1	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
19		simpll2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
20		simpll3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
21		simprl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to P \neq Q$
22		simplr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to X \in B$
23		simprr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to \neg X \underset{˙}{\leq} W$
24	1 2 3 4 5 6 7 8 9 10 11 12 15	cdleme29cl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to z = N \underset{˙}{\lor} (X \underset{˙}{\land} W))) \in B$
25	18 19 20 21 22 23 24	syl312anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to z = N \underset{˙}{\lor} (X \underset{˙}{\land} W))) \in B$
26	17 25	eqeltrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) \in B$
27	14	cdleme31fv2	$⊢ (X \in B \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) = X$
28		simpl	$⊢ (X \in B \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to X \in B$
29	27 28	eqeltrd	$⊢ (X \in B \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) \in B$
30	29	adantll	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) \in B$
31	26 30	pm2.61dan	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \to F (X) \in B$

Metamath Proof Explorer

Theorem cdleme32fvcl

Proof