cdlemg13a

	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	$⊢ R = trL (K) (W)$
Assertion	cdlemg13a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} F (G (P)) = G (P) \underset{˙}{\lor} F (G (P))$

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		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to K \in HL$
9		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \in A$
10		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (K \in HL \land W \in H)$
11		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G \in T$
12	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land P \in A) \to G (P) \in A$
13	10 11 9 12	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G (P) \in A$
14	1 2 4	hlatlej1	$⊢ (K \in HL \land P \in A \land G (P) \in A) \to P \underset{˙}{\leq} (P \underset{˙}{\lor} G (P))$
15	8 9 13 14	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\leq} (P \underset{˙}{\lor} G (P))$
16		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to R (F) = R (G)$
17		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F \in T$
18		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
19	1 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
20	10 11 18 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)$
21	1 2 3 4 5 6 7	trlval2	$⊢ ((K \in HL \land W \in H) \land F \in T \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W)) \to R (F) = (G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W$
22	10 17 20 21	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to R (F) = (G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W$
23	1 2 3 4 5 6 7	trlval2	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
24	10 11 18 23	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to R (G) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
25	16 22 24	3eqtr3d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
26	25	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G (P) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W) = G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W)$
27	1 4 5 6	ltrncoat	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land P \in A) \to F (G (P)) \in A$
28	10 17 11 9 27	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \in A$
29		eqid	$⊢ (G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W$
30	1 2 3 4 5 29	cdleme0cp	$⊢ ((K \in HL \land W \in H) \land ((G (P) \in A \land \neg G (P) \underset{˙}{\leq} W) \land F (G (P)) \in A)) \to G (P) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W) = G (P) \underset{˙}{\lor} F (G (P))$
31	10 20 28 30	syl12anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G (P) \underset{˙}{\lor} ((G (P) \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W) = G (P) \underset{˙}{\lor} F (G (P))$
32		eqid	$⊢ (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W$
33	1 2 3 4 5 32	cdleme0cq	$⊢ ((K \in HL \land W \in H) \land (P \in A \land (G (P) \in A \land \neg G (P) \underset{˙}{\leq} W))) \to G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W) = P \underset{˙}{\lor} G (P)$
34	10 9 20 33	syl12anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G (P) \underset{˙}{\lor} ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} W) = P \underset{˙}{\lor} G (P)$
35	26 31 34	3eqtr3rd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} G (P) = G (P) \underset{˙}{\lor} F (G (P))$
36	15 35	breqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))$
37	1 2 4	hlatlej2	$⊢ (K \in HL \land G (P) \in A \land F (G (P)) \in A) \to F (G (P)) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))$
38	8 13 28 37	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))$
39	8	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to K \in Lat$
40		eqid	$⊢ {Base}_{K} = {Base}_{K}$
41	40 4	atbase	$⊢ P \in A \to P \in {Base}_{K}$
42	9 41	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \in {Base}_{K}$
43	40 4	atbase	$⊢ F (G (P)) \in A \to F (G (P)) \in {Base}_{K}$
44	28 43	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \in {Base}_{K}$
45	40 2 4	hlatjcl	$⊢ (K \in HL \land G (P) \in A \land F (G (P)) \in A) \to G (P) \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
46	8 13 28 45	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to G (P) \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
47	40 1 2	latjle12	$⊢ (K \in Lat \land (P \in {Base}_{K} \land F (G (P)) \in {Base}_{K} \land G (P) \underset{˙}{\lor} F (G (P)) \in {Base}_{K})) \to ((P \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P))) \land F (G (P)) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))) \leftrightarrow (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P))))$
48	39 42 44 46 47	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P))) \land F (G (P)) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))) \leftrightarrow (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P))))$
49	36 38 48	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P)))$
50		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
51		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q$
52	1 2 3 4 5 6	cdlemg11a	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \neq P$
53	10 18 50 17 11 51 52	syl123anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to F (G (P)) \neq P$
54	53	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \neq F (G (P))$
55	1 2 4	ps-1	$⊢ (K \in HL \land (P \in A \land F (G (P)) \in A \land P \neq F (G (P))) \land (G (P) \in A \land F (G (P)) \in A)) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P))) \leftrightarrow P \underset{˙}{\lor} F (G (P)) = G (P) \underset{˙}{\lor} F (G (P)))$
56	8 9 28 54 13 28 55	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\leq} (G (P) \underset{˙}{\lor} F (G (P))) \leftrightarrow P \underset{˙}{\lor} F (G (P)) = G (P) \underset{˙}{\lor} F (G (P)))$
57	49 56	mpbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T) \land (F (P) \neq P \land R (F) = R (G) \land F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} F (G (P)) = G (P) \underset{˙}{\lor} F (G (P))$

Metamath Proof Explorer

Theorem cdlemg13a

Proof