cdlemefr29exN

Description: Lemma for cdlemefs29bpre1N . (Compare cdleme25a .) TODO: FIX COMMENT. TODO: IS THIS NEEDED? (Contributed by NM, 28-Mar-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemefr29.b	$⊢ B = {Base}_{K}$
	cdlemefr29.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemefr29.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemefr29.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemefr29.a	$⊢ A = Atoms (K)$
	cdlemefr29.h	$⊢ H = LHyp (K)$
Assertion	cdlemefr29exN	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)$

Proof

Step	Hyp	Ref	Expression
1		cdlemefr29.b	$⊢ B = {Base}_{K}$
2		cdlemefr29.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemefr29.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemefr29.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemefr29.a	$⊢ A = Atoms (K)$
6		cdlemefr29.h	$⊢ H = LHyp (K)$
7		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to (K \in HL \land W \in H)$
8		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
9	1 2 3 4 5 6	lhpmcvr2	$⊢ ((K \in HL \land W \in H) \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to \exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
10	7 8 9	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to \exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
11		nfv	$⊢ Ⅎ s ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
12		nfv	$⊢ Ⅎ s (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W))$
13		nfra1	$⊢ Ⅎ s \forall s \in A C \in B$
14	11 12 13	nf3an	$⊢ Ⅎ s (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B)$
15		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to K \in HL$
16	15	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to K \in HL$
17	16	hllatd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to K \in Lat$
18		simpl3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to \forall s \in A C \in B$
19		simprl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to s \in A$
20		rsp	$⊢ \forall s \in A C \in B \to (s \in A \to C \in B)$
21	18 19 20	sylc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to C \in B$
22	15	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to K \in Lat$
23		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to X \in B$
24		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to W \in H$
25	1 6	lhpbase	$⊢ W \in H \to W \in B$
26	24 25	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to W \in B$
27	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
28	22 23 26 27	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to X \underset{˙}{\land} W \in B$
29	28	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to X \underset{˙}{\land} W \in B$
30	1 3	latjcl	$⊢ (K \in Lat \land C \in B \land X \underset{˙}{\land} W \in B) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
31	17 21 29 30	syl3anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B$
32	31	expr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land s \in A) \to (\neg s \underset{˙}{\leq} W \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)$
33	32	adantrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land s \in A) \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)$
34	33	ancld	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \land s \in A) \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B))$
35	34	ex	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to (s \in A \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)))$
36	14 35	reximdai	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to (\exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B))$
37	10 36	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land \forall s \in A C \in B) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land C \underset{˙}{\lor} (X \underset{˙}{\land} W) \in B)$

Metamath Proof Explorer

Theorem cdlemefr29exN

Proof