cdlemefr29clN

Description: Show closure of the unique element in cdleme29c . TODO fix comment. TODO Not needed? (Contributed by NM, 29-Mar-2013) (New usage is discouraged.)

	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	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
	cdlemefr29cl.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (R \underset{˙}{\land} W) = R) \to z = N \underset{˙}{\lor} (R \underset{˙}{\land} W)))$
Assertion	cdlemefr29clN	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to O \in B$

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		cdlemefr29cl.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (R \underset{˙}{\land} W) = R) \to z = N \underset{˙}{\lor} (R \underset{˙}{\land} W)))$
11		breq1	$⊢ s = R \to (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
12	11	notbid	$⊢ s = R \to (\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
13		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to (K \in HL \land W \in H)$
14		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to P \in A$
15		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to Q \in A$
16		simp3l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to s \in A$
17		simp3rr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
18		simp2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to P \neq Q$
19	1 2 3 4 5 6 7 8 9	cdlemefr27cl	$⊢ (((K \in HL \land W \in H) \land P \in A \land Q \in A) \land (s \in A \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to N \in B$
20	13 14 15 16 17 18 19	syl33anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to N \in B$
21	1 2 3 4 5 6 7 8 9	cdlemefr32snb	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / s ⦌ N \in B$
22	1 2 3 4 5 6 12 20 21 10	cdlemefrs29clN	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to O \in B$

Metamath Proof Explorer

Theorem cdlemefr29clN

Proof