cdlemefs27cl

Description: Part of proof of Lemma E in Crawley p. 113. Closure of N . TODO FIX COMMENT This is the start of a re-proof of cdleme27cl etc. with the s .<_ ( P .\/ Q ) condition (so as to not have the C hypothesis). (Contributed by NM, 24-Mar-2013)

	Ref	Expression
Hypotheses	cdlemefs26.b	$⊢ B = {Base}_{K}$
	cdlemefs26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemefs26.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemefs26.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemefs26.a	$⊢ A = Atoms (K)$
	cdlemefs26.h	$⊢ H = LHyp (K)$
	cdlemefs27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdlemefs27.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemefs27.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemefs27.i	$⊢ I = (ι u \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = E))$
	cdlemefs27.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
Assertion	cdlemefs27cl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to N \in B$

Proof

Step	Hyp	Ref	Expression
1		cdlemefs26.b	$⊢ B = {Base}_{K}$
2		cdlemefs26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemefs26.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemefs26.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemefs26.a	$⊢ A = Atoms (K)$
6		cdlemefs26.h	$⊢ H = LHyp (K)$
7		cdlemefs27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemefs27.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemefs27.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemefs27.i	$⊢ I = (ι u \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = E))$
11		cdlemefs27.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
12		simpr2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
13	12	iftrued	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C) = I$
14		simpl1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to (K \in HL \land W \in H)$
15		simpl2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
16		simpl3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
17		simpr1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
18		simpr3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to P \neq Q$
19	1 2 3 4 5 6 7 8 9 10	cdleme25cl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (P \neq Q \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to I \in B$
20	14 15 16 17 18 12 19	syl312anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to I \in B$
21	13 20	eqeltrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C) \in B$
22	11 21	eqeltrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to N \in B$

Metamath Proof Explorer

Theorem cdlemefs27cl

Proof