Metamath Proof Explorer

Theorem cdleme32snaw

Description: Show that [_ R / s ]_ N is an atom not under W . (Contributed by NM, 6-Mar-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	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
	Assertion	cdleme32snaw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (⦋ R / s ⦌ N \in A \land \neg ⦋ R / s ⦌ N \underset{˙}{\leq} W)$

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		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W)) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W))$
14		eqid	$⊢ (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W)))) = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W))))$
15	1 2 3 4 5 6 7 9 10 11 12 13 14	cdlemefs32sn1aw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (⦋ R / s ⦌ N \in A \land \neg ⦋ R / s ⦌ N \underset{˙}{\leq} W)$
16	15	3expa	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W))) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (⦋ R / s ⦌ N \in A \land \neg ⦋ R / s ⦌ N \underset{˙}{\leq} W)$
17	1 2 3 4 5 6 7 8 12	cdlemefr32sn2aw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (⦋ R / s ⦌ N \in A \land \neg ⦋ R / s ⦌ N \underset{˙}{\leq} W)$
18	17	3expa	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W))) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (⦋ R / s ⦌ N \in A \land \neg ⦋ R / s ⦌ N \underset{˙}{\leq} W)$
19	16 18	pm2.61dan	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (⦋ R / s ⦌ N \in A \land \neg ⦋ R / s ⦌ N \underset{˙}{\leq} W)$