cdlemg42

Description: Part of proof of Lemma G of Crawley p. 116, first line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013)

	Ref	Expression
Hypotheses	cdlemg42.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg42.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg42.a	$⊢ A = Atoms (K)$
	cdlemg42.h	$⊢ H = LHyp (K)$
	cdlemg42.t	$⊢ T = LTrn (K) (W)$
	cdlemg42.r	$⊢ R = trL (K) (W)$
Assertion	cdlemg42	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to \neg G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))$

Proof

Step	Hyp	Ref	Expression
1		cdlemg42.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg42.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg42.a	$⊢ A = Atoms (K)$
4		cdlemg42.h	$⊢ H = LHyp (K)$
5		cdlemg42.t	$⊢ T = LTrn (K) (W)$
6		cdlemg42.r	$⊢ R = trL (K) (W)$
7		simp33	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to R (F) \neq R (G)$
8		simpl1l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to K \in HL$
9		simp31l	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to P \in A$
10	9	adantr	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to P \in A$
11		simp1	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to (K \in HL \land W \in H)$
12		simp2l	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to F \in T$
13	1 3 4 5	ltrnat	$⊢ ((K \in HL \land W \in H) \land F \in T \land P \in A) \to F (P) \in A$
14	11 12 9 13	syl3anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to F (P) \in A$
15	14	adantr	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to F (P) \in A$
16	1 2 3	hlatlej1	$⊢ (K \in HL \land P \in A \land F (P) \in A) \to P \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))$
17	8 10 15 16	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to P \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))$
18		simpr	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))$
19	8	hllatd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to K \in Lat$
20		eqid	$⊢ {Base}_{K} = {Base}_{K}$
21	20 3	atbase	$⊢ P \in A \to P \in {Base}_{K}$
22	10 21	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to P \in {Base}_{K}$
23		simp2r	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to G \in T$
24	1 3 4 5	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land P \in A) \to G (P) \in A$
25	11 23 9 24	syl3anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to G (P) \in A$
26	25	adantr	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to G (P) \in A$
27	20 3	atbase	$⊢ G (P) \in A \to G (P) \in {Base}_{K}$
28	26 27	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to G (P) \in {Base}_{K}$
29	20 2 3	hlatjcl	$⊢ (K \in HL \land P \in A \land F (P) \in A) \to P \underset{˙}{\lor} F (P) \in {Base}_{K}$
30	8 10 15 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to P \underset{˙}{\lor} F (P) \in {Base}_{K}$
31	20 1 2	latjle12	$⊢ (K \in Lat \land (P \in {Base}_{K} \land G (P) \in {Base}_{K} \land P \underset{˙}{\lor} F (P) \in {Base}_{K})) \to ((P \underset{˙}{\leq} (P \underset{˙}{\lor} F (P)) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \leftrightarrow (P \underset{˙}{\lor} G (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P)))$
32	19 22 28 30 31	syl13anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to ((P \underset{˙}{\leq} (P \underset{˙}{\lor} F (P)) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \leftrightarrow (P \underset{˙}{\lor} G (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P)))$
33	17 18 32	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to (P \underset{˙}{\lor} G (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))$
34		simpl32	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to G (P) \neq P$
35	34	necomd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to P \neq G (P)$
36	1 2 3	ps-1	$⊢ (K \in HL \land (P \in A \land G (P) \in A \land P \neq G (P)) \land (P \in A \land F (P) \in A)) \to ((P \underset{˙}{\lor} G (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P)) \leftrightarrow P \underset{˙}{\lor} G (P) = P \underset{˙}{\lor} F (P))$
37	8 10 26 35 10 15 36	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to ((P \underset{˙}{\lor} G (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P)) \leftrightarrow P \underset{˙}{\lor} G (P) = P \underset{˙}{\lor} F (P))$
38	33 37	mpbid	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to P \underset{˙}{\lor} G (P) = P \underset{˙}{\lor} F (P)$
39	38	oveq1d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to (P \underset{˙}{\lor} G (P)) meet (K) W = (P \underset{˙}{\lor} F (P)) meet (K) W$
40		simpl1	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to (K \in HL \land W \in H)$
41		simpl2r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to G \in T$
42		simpl31	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
43		eqid	$⊢ meet (K) = meet (K)$
44	1 2 43 3 4 5 6	trlval2	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (G) = (P \underset{˙}{\lor} G (P)) meet (K) W$
45	40 41 42 44	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to R (G) = (P \underset{˙}{\lor} G (P)) meet (K) W$
46		simpl2l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to F \in T$
47	1 2 43 3 4 5 6	trlval2	$⊢ ((K \in HL \land W \in H) \land F \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (F) = (P \underset{˙}{\lor} F (P)) meet (K) W$
48	40 46 42 47	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to R (F) = (P \underset{˙}{\lor} F (P)) meet (K) W$
49	39 45 48	3eqtr4rd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \land G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))) \to R (F) = R (G)$
50	49	ex	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to (G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P)) \to R (F) = R (G))$
51	50	necon3ad	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to (R (F) \neq R (G) \to \neg G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P)))$
52	7 51	mpd	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G (P) \neq P \land R (F) \neq R (G))) \to \neg G (P) \underset{˙}{\leq} (P \underset{˙}{\lor} F (P))$

Metamath Proof Explorer

Theorem cdlemg42

Proof