cdlemg31c

Description: Show that when N is an atom, it is not under W . TODO: Is there a shorter direct proof? TODO: should we eliminate ( FP ) =/= P here? (Contributed by NM, 29-May-2013)

	Ref	Expression
Hypotheses	cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg12.a	$⊢ A = Atoms (K)$
	cdlemg12.h	$⊢ H = LHyp (K)$
	cdlemg12.t	$⊢ T = LTrn (K) (W)$
	cdlemg12b.r	$⊢ R = trL (K) (W)$
	cdlemg31.n	$⊢ N = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
Assertion	cdlemg31c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to \neg N \underset{˙}{\leq} W$

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg12.a	$⊢ A = Atoms (K)$
5		cdlemg12.h	$⊢ H = LHyp (K)$
6		cdlemg12.t	$⊢ T = LTrn (K) (W)$
7		cdlemg12b.r	$⊢ R = trL (K) (W)$
8		cdlemg31.n	$⊢ N = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
9		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to K \in HL$
10		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to W \in H$
11	9 10	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to (K \in HL \land W \in H)$
12		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
13		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to v \neq R (F)$
14	13	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to R (F) \neq v$
15		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
16		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to F \in T$
17		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to F (P) \neq P$
18	1 4 5 6 7	trlat	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (F \in T \land F (P) \neq P)) \to R (F) \in A$
19	11 15 16 17 18	syl112anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to R (F) \in A$
20	1 5 6 7	trlle	$⊢ ((K \in HL \land W \in H) \land F \in T) \to R (F) \underset{˙}{\leq} W$
21	11 16 20	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to R (F) \underset{˙}{\leq} W$
22		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to (v \in A \land v \underset{˙}{\leq} W)$
23	1 2 4 5	lhp2atnle	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land R (F) \neq v) \land (R (F) \in A \land R (F) \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \to \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} R (F))$
24	11 12 14 19 21 22 23	syl321anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} R (F))$
25		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to P \in A$
26		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to Q \in A$
27		simp2ll	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to v \in A$
28	1 2 3 4 5 6 7 8	cdlemg31a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (v \in A \land F \in T)) \to N \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
29	9 10 25 26 27 16 28	syl222anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to N \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
30	29	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W) \to N \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
31		simp111	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W \land N \neq v) \to (K \in HL \land W \in H)$
32		simp112	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W \land N \neq v) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
33		simp3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W \land N \neq v) \to N \neq v$
34	33	necomd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W \land N \neq v) \to v \neq N$
35		simp12l	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W \land N \neq v) \to (v \in A \land v \underset{˙}{\leq} W)$
36		simp133	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W \land N \neq v) \to N \in A$
37		simp2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W \land N \neq v) \to N \underset{˙}{\leq} W$
38	1 2 4 5	lhp2atnle	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land v \neq N) \land (v \in A \land v \underset{˙}{\leq} W) \land (N \in A \land N \underset{˙}{\leq} W)) \to \neg N \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
39	31 32 34 35 36 37 38	syl312anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W \land N \neq v) \to \neg N \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
40	39	3expia	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W) \to (N \neq v \to \neg N \underset{˙}{\leq} (P \underset{˙}{\lor} v))$
41	40	necon4ad	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W) \to (N \underset{˙}{\leq} (P \underset{˙}{\lor} v) \to N = v)$
42	30 41	mpd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W) \to N = v$
43	1 2 3 4 5 6 7 8	cdlemg31b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (v \in A \land F \in T)) \to N \underset{˙}{\leq} (Q \underset{˙}{\lor} R (F))$
44	9 10 25 26 27 16 43	syl222anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to N \underset{˙}{\leq} (Q \underset{˙}{\lor} R (F))$
45	44	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W) \to N \underset{˙}{\leq} (Q \underset{˙}{\lor} R (F))$
46	42 45	eqbrtrrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \land N \underset{˙}{\leq} W) \to v \underset{˙}{\leq} (Q \underset{˙}{\lor} R (F))$
47	24 46	mtand	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land F \in T) \land (v \neq R (F) \land F (P) \neq P \land N \in A)) \to \neg N \underset{˙}{\leq} W$

Metamath Proof Explorer

Theorem cdlemg31c

Proof