cdlemg4c

	Ref	Expression
Hypotheses	cdlemg4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg4.a	$⊢ A = Atoms (K)$
	cdlemg4.h	$⊢ H = LHyp (K)$
	cdlemg4.t	$⊢ T = LTrn (K) (W)$
	cdlemg4.r	$⊢ R = trL (K) (W)$
	cdlemg4.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg4b.v	$⊢ V = R (G)$
Assertion	cdlemg4c	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T) \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to \neg G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)$

Proof

Step	Hyp	Ref	Expression
1		cdlemg4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg4.a	$⊢ A = Atoms (K)$
3		cdlemg4.h	$⊢ H = LHyp (K)$
4		cdlemg4.t	$⊢ T = LTrn (K) (W)$
5		cdlemg4.r	$⊢ R = trL (K) (W)$
6		cdlemg4.j	$⊢ \underset{˙}{\lor} = join (K)$
7		cdlemg4b.v	$⊢ V = R (G)$
8		simpll	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to (K \in HL \land W \in H)$
9		simplr2	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
10		simplr3	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to G \in T$
11	1 2 3 4 5 6 7	cdlemg4b2	$⊢ ((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T) \to G (Q) \underset{˙}{\lor} V = Q \underset{˙}{\lor} G (Q)$
12	8 9 10 11	syl3anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to G (Q) \underset{˙}{\lor} V = Q \underset{˙}{\lor} G (Q)$
13		simpr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
14		simpll	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to K \in HL$
15	14	hllatd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to K \in Lat$
16		simpr1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to P \in A$
17		eqid	$⊢ {Base}_{K} = {Base}_{K}$
18	17 2	atbase	$⊢ P \in A \to P \in {Base}_{K}$
19	16 18	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to P \in {Base}_{K}$
20		simpl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to (K \in HL \land W \in H)$
21		simpr3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to G \in T$
22	17 3 4 5	trlcl	$⊢ ((K \in HL \land W \in H) \land G \in T) \to R (G) \in {Base}_{K}$
23	20 21 22	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to R (G) \in {Base}_{K}$
24	7 23	eqeltrid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to V \in {Base}_{K}$
25	17 1 6	latlej2	$⊢ (K \in Lat \land P \in {Base}_{K} \land V \in {Base}_{K}) \to V \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
26	15 19 24 25	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to V \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
27	26	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to V \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
28		simpr2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to Q \in A$
29	17 2	atbase	$⊢ Q \in A \to Q \in {Base}_{K}$
30	28 29	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to Q \in {Base}_{K}$
31	17 3 4	ltrncl	$⊢ ((K \in HL \land W \in H) \land G \in T \land Q \in {Base}_{K}) \to G (Q) \in {Base}_{K}$
32	20 21 30 31	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to G (Q) \in {Base}_{K}$
33	17 6	latjcl	$⊢ (K \in Lat \land P \in {Base}_{K} \land V \in {Base}_{K}) \to P \underset{˙}{\lor} V \in {Base}_{K}$
34	15 19 24 33	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to P \underset{˙}{\lor} V \in {Base}_{K}$
35	17 1 6	latjle12	$⊢ (K \in Lat \land (G (Q) \in {Base}_{K} \land V \in {Base}_{K} \land P \underset{˙}{\lor} V \in {Base}_{K})) \to ((G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land V \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \leftrightarrow (G (Q) \underset{˙}{\lor} V) \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
36	15 32 24 34 35	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to ((G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land V \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \leftrightarrow (G (Q) \underset{˙}{\lor} V) \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
37	36	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to ((G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land V \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \leftrightarrow (G (Q) \underset{˙}{\lor} V) \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
38	13 27 37	mpbi2and	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to (G (Q) \underset{˙}{\lor} V) \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
39	12 38	eqbrtrrd	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to (Q \underset{˙}{\lor} G (Q)) \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
40	15	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to K \in Lat$
41	30	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to Q \in {Base}_{K}$
42	32	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to G (Q) \in {Base}_{K}$
43	19	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to P \in {Base}_{K}$
44	8 10 22	syl2anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to R (G) \in {Base}_{K}$
45	7 44	eqeltrid	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to V \in {Base}_{K}$
46	40 43 45 33	syl3anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to P \underset{˙}{\lor} V \in {Base}_{K}$
47	17 1 6	latjle12	$⊢ (K \in Lat \land (Q \in {Base}_{K} \land G (Q) \in {Base}_{K} \land P \underset{˙}{\lor} V \in {Base}_{K})) \to ((Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \leftrightarrow (Q \underset{˙}{\lor} G (Q)) \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
48	40 41 42 46 47	syl13anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to ((Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \leftrightarrow (Q \underset{˙}{\lor} G (Q)) \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
49	39 48	mpbird	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to (Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
50	49	simpld	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \land G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to Q \underset{˙}{\leq} (P \underset{˙}{\lor} V)$
51	50	ex	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to (G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V) \to Q \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
52	51	con3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T)) \to (\neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V) \to \neg G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V))$
53	52	3impia	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land G \in T) \land \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} V)) \to \neg G (Q) \underset{˙}{\leq} (P \underset{˙}{\lor} V)$

Metamath Proof Explorer

Theorem cdlemg4c

Proof