cdleme28a

Description: Lemma for cdleme25b . TODO: FIX COMMENT. (Contributed by NM, 4-Feb-2013)

	Ref	Expression
Hypotheses	cdleme26.b	$⊢ B = {Base}_{K}$
	cdleme26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme26.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme26.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme26.a	$⊢ A = Atoms (K)$
	cdleme26.h	$⊢ H = LHyp (K)$
	cdleme27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme27.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme27.z	$⊢ Z = (z \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((s \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.d	$⊢ D = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N))$
	cdleme27.c	$⊢ C = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F)$
	cdleme27.g	$⊢ G = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme27.o	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((t \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.e	$⊢ E = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = O))$
	cdleme27.y	$⊢ Y = if (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q), E, G)$
	cdleme28a.v	$⊢ V = (s \underset{˙}{\lor} t) \underset{˙}{\land} (X \underset{˙}{\land} W)$
Assertion	cdleme28a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (C \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	$⊢ B = {Base}_{K}$
2		cdleme26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme26.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme26.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme26.a	$⊢ A = Atoms (K)$
6		cdleme26.h	$⊢ H = LHyp (K)$
7		cdleme27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme27.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme27.z	$⊢ Z = (z \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} z) \underset{˙}{\land} W))$
10		cdleme27.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((s \underset{˙}{\lor} z) \underset{˙}{\land} W))$
11		cdleme27.d	$⊢ D = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N))$
12		cdleme27.c	$⊢ C = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F)$
13		cdleme27.g	$⊢ G = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
14		cdleme27.o	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((t \underset{˙}{\lor} z) \underset{˙}{\land} W))$
15		cdleme27.e	$⊢ E = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = O))$
16		cdleme27.y	$⊢ Y = if (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q), E, G)$
17		cdleme28a.v	$⊢ V = (s \underset{˙}{\lor} t) \underset{˙}{\land} (X \underset{˙}{\land} W)$
18		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to K \in HL$
19	18	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to K \in Lat$
20		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to W \in H$
21		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
22		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
23		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
24		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to P \neq Q$
25	1 2 3 4 5 6 7 8 9 10 11 12	cdleme27cl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land P \neq Q)) \to C \in B$
26	18 20 21 22 23 24 25	syl222anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \in B$
27		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
28	1 2 3 4 5 6 7 13 9 14 15 16	cdleme27cl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land P \neq Q)) \to Y \in B$
29	18 20 21 22 27 24 28	syl222anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to Y \in B$
30		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
31	30 23 27	3jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to ((K \in HL \land W \in H) \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W))$
32		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
33		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to s \neq t$
34		simp32l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
35		simp32r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
36	33 34 35	3jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (s \neq t \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
37	1 2 3 4 5 6 17	cdleme23b	$⊢ (((K \in HL \land W \in H) \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (s \neq t \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to V \in A$
38	31 32 36 37	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to V \in A$
39	1 5	atbase	$⊢ V \in A \to V \in B$
40	38 39	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to V \in B$
41	1 3	latjcl	$⊢ (K \in Lat \land Y \in B \land V \in B) \to Y \underset{˙}{\lor} V \in B$
42	19 29 40 41	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to Y \underset{˙}{\lor} V \in B$
43		simp33l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to X \in B$
44	1 6	lhpbase	$⊢ W \in H \to W \in B$
45	20 44	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to W \in B$
46	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
47	19 43 45 46	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to X \underset{˙}{\land} W \in B$
48	1 3	latjcl	$⊢ (K \in Lat \land Y \in B \land X \underset{˙}{\land} W \in B) \to Y \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
49	19 29 47 48	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to Y \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
50	1 2 3 4 5 6 17	cdleme23c	$⊢ (((K \in HL \land W \in H) \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (s \neq t \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to s \underset{˙}{\leq} (t \underset{˙}{\lor} V)$
51	31 32 36 50	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to s \underset{˙}{\leq} (t \underset{˙}{\lor} V)$
52	33 51	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V))$
53	1 2 3 4 5 6 17	cdleme23a	$⊢ (((K \in HL \land W \in H) \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land (s \neq t \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to V \underset{˙}{\leq} W$
54	31 32 36 53	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to V \underset{˙}{\leq} W$
55	38 54	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (V \in A \land V \underset{˙}{\leq} W)$
56	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16	cdleme27a	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V)$
57	30 24 23 21 22 27 52 55 56	syl332anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V)$
58		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to s \in A$
59		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to t \in A$
60	1 3 5	hlatjcl	$⊢ (K \in HL \land s \in A \land t \in A) \to s \underset{˙}{\lor} t \in B$
61	18 58 59 60	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to s \underset{˙}{\lor} t \in B$
62	1 2 4	latmle2	$⊢ (K \in Lat \land s \underset{˙}{\lor} t \in B \land X \underset{˙}{\land} W \in B) \to ((s \underset{˙}{\lor} t) \underset{˙}{\land} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (X \underset{˙}{\land} W)$
63	19 61 47 62	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to ((s \underset{˙}{\lor} t) \underset{˙}{\land} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (X \underset{˙}{\land} W)$
64	17 63	eqbrtrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to V \underset{˙}{\leq} (X \underset{˙}{\land} W)$
65	1 2 3	latjlej2	$⊢ (K \in Lat \land (V \in B \land X \underset{˙}{\land} W \in B \land Y \in B)) \to (V \underset{˙}{\leq} (X \underset{˙}{\land} W) \to (Y \underset{˙}{\lor} V) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W)))$
66	19 40 47 29 65	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (V \underset{˙}{\leq} (X \underset{˙}{\land} W) \to (Y \underset{˙}{\lor} V) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W)))$
67	64 66	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (Y \underset{˙}{\lor} V) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$
68	1 2 19 26 42 49 57 67	lattrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$
69	1 2 3	latlej2	$⊢ (K \in Lat \land Y \in B \land X \underset{˙}{\land} W \in B) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$
70	19 29 47 69	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$
71	1 2 3	latjle12	$⊢ (K \in Lat \land (C \in B \land X \underset{˙}{\land} W \in B \land Y \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)) \to ((C \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W)) \land (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))) \leftrightarrow (C \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W)))$
72	19 26 47 49 71	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to ((C \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W)) \land (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))) \leftrightarrow (C \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W)))$
73	68 70 72	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to (C \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$

Metamath Proof Explorer

Theorem cdleme28a

Proof