cdlemg10

	Ref	Expression
Hypotheses	cdlemg8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg8.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg8.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg8.a	$⊢ A = Atoms (K)$
	cdlemg8.h	$⊢ H = LHyp (K)$
	cdlemg8.t	$⊢ T = LTrn (K) (W)$
	cdlemg10.r	$⊢ R = trL (K) (W)$
Assertion	cdlemg10	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} W$

Proof

Step	Hyp	Ref	Expression
1		cdlemg8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg8.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg8.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg8.a	$⊢ A = Atoms (K)$
5		cdlemg8.h	$⊢ H = LHyp (K)$
6		cdlemg8.t	$⊢ T = LTrn (K) (W)$
7		cdlemg10.r	$⊢ R = trL (K) (W)$
8		eqid	$⊢ {Base}_{K} = {Base}_{K}$
9		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
10	9	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in Lat$
11		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
12		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (K \in HL \land W \in H)$
13		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in T$
14		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G \in T$
15	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land P \in A) \to G (P) \in A$
16	12 14 11 15	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (P) \in A$
17	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land F \in T \land G (P) \in A) \to F (G (P)) \in A$
18	12 13 16 17	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F (G (P)) \in A$
19	8 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land F (G (P)) \in A) \to P \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
20	9 11 18 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} F (G (P)) \in {Base}_{K}$
21		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
22	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land Q \in A) \to G (Q) \in A$
23	12 14 21 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (Q) \in A$
24	1 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land F \in T \land G (Q) \in A) \to F (G (Q)) \in A$
25	12 13 23 24	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F (G (Q)) \in A$
26	8 2 4	hlatjcl	$⊢ (K \in HL \land Q \in A \land F (G (Q)) \in A) \to Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}$
27	9 21 25 26	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}$
28	8 3	latmcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} F (G (P)) \in {Base}_{K} \land Q \underset{˙}{\lor} F (G (Q)) \in {Base}_{K}) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \in {Base}_{K}$
29	10 20 27 28	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q))) \in {Base}_{K}$
30	8 5 6 7	trlcl	$⊢ ((K \in HL \land W \in H) \land F \in T) \to R (F) \in {Base}_{K}$
31	12 13 30	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (F) \in {Base}_{K}$
32	8 5 6 7	trlcl	$⊢ ((K \in HL \land W \in H) \land G \in T) \to R (G) \in {Base}_{K}$
33	12 14 32	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (G) \in {Base}_{K}$
34	8 2	latjcl	$⊢ (K \in Lat \land R (F) \in {Base}_{K} \land R (G) \in {Base}_{K}) \to R (F) \underset{˙}{\lor} R (G) \in {Base}_{K}$
35	10 31 33 34	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (F) \underset{˙}{\lor} R (G) \in {Base}_{K}$
36		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
37	8 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
38	36 37	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in {Base}_{K}$
39	1 2 3 4 5 6 7	cdlemg10a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} (R (F) \underset{˙}{\lor} R (G))$
40	1 5 6 7	trlle	$⊢ ((K \in HL \land W \in H) \land F \in T) \to R (F) \underset{˙}{\leq} W$
41	12 13 40	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (F) \underset{˙}{\leq} W$
42	1 5 6 7	trlle	$⊢ ((K \in HL \land W \in H) \land G \in T) \to R (G) \underset{˙}{\leq} W$
43	12 14 42	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R (G) \underset{˙}{\leq} W$
44	8 1 2	latjle12	$⊢ (K \in Lat \land (R (F) \in {Base}_{K} \land R (G) \in {Base}_{K} \land W \in {Base}_{K})) \to ((R (F) \underset{˙}{\leq} W \land R (G) \underset{˙}{\leq} W) \leftrightarrow (R (F) \underset{˙}{\lor} R (G)) \underset{˙}{\leq} W)$
45	10 31 33 38 44	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((R (F) \underset{˙}{\leq} W \land R (G) \underset{˙}{\leq} W) \leftrightarrow (R (F) \underset{˙}{\lor} R (G)) \underset{˙}{\leq} W)$
46	41 43 45	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R (F) \underset{˙}{\lor} R (G)) \underset{˙}{\leq} W$
47	8 1 10 29 35 38 39 46	lattrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (F (G (P)) \underset{˙}{\lor} F (G (Q)) \neq P \underset{˙}{\lor} Q \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} (Q \underset{˙}{\lor} F (G (Q)))) \underset{˙}{\leq} W$

Metamath Proof Explorer

Theorem cdlemg10

Proof