cdleme29ex

Description: Lemma for cdleme29b . (Compare cdleme25a .) TODO: FIX COMMENT. (Contributed by NM, 7-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	$⊢ C = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F)$
Assertion	cdleme29ex	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
14		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
15	1 2 3 4 5 6	lhpmcvr2	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to \exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
16	13 14 15	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to \exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
17		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to K \in HL$
18	17	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to K \in HL$
19	18	hllatd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to K \in Lat$
20		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to W \in H$
21	20	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to W \in H$
22		simpl12	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
23		simpl13	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
24		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
25		simpl2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to P \neq Q$
26	1 2 3 4 5 6 7 8 9 10 11 12	cdleme27cl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land P \neq Q)) \to C \in B$
27	18 21 22 23 24 25 26	syl222anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to C \in B$
28		simpl3l	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to X \in B$
29	1 6	lhpbase	$⊢ W \in H \to W \in B$
30	21 29	syl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to W \in B$
31	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
32	19 28 30 31	syl3anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to X \underset{˙}{\land} W \in B$
33	1 3	latjcl	$⊢ (K \in Lat \land C \in B \land X \underset{˙}{\land} W \in B) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
34	19 27 32 33	syl3anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
35	34	expr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land s \in A) \to (\neg s \underset{˙}{\leq} W \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)$
36	35	adantrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land s \in A) \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)$
37	36	ancld	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land s \in A) \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B))$
38	37	reximdva	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to (\exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B))$
39	16 38	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)$

Metamath Proof Explorer

Theorem cdleme29ex

Proof