cdleme25a

	Ref	Expression
Hypotheses	cdleme24.b	$⊢ B = {Base}_{K}$
	cdleme24.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme24.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme24.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme24.a	$⊢ A = Atoms (K)$
	cdleme24.h	$⊢ H = LHyp (K)$
	cdleme24.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme24.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme24.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} s) \underset{˙}{\land} W))$
Assertion	cdleme25a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land N \in B)$

Proof

Step	Hyp	Ref	Expression
1		cdleme24.b	$⊢ B = {Base}_{K}$
2		cdleme24.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme24.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme24.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme24.a	$⊢ A = Atoms (K)$
6		cdleme24.h	$⊢ H = LHyp (K)$
7		cdleme24.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme24.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme24.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} s) \underset{˙}{\land} W))$
10		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
11		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
12		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
13		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
14		simp3l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
15	2 3 5 6	cdlemb2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to \exists s \in A (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
16	10 11 12 13 14 15	syl221anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \exists s \in A (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
17	10	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to K \in HL$
18	11	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to W \in H$
19		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
20	19	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to P \in A$
21		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
22	21	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to Q \in A$
23		simpl2l	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to R \in A$
24		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to s \in A$
25	2 3 4 5 6 7 8 9 1	cdleme22gb	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land s \in A)) \to N \in B$
26	17 18 20 22 23 24 25	syl222anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to N \in B$
27	26	a1d	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to N \in B)$
28	27	ancld	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land s \in A) \to ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land N \in B))$
29	28	reximdva	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (\exists s \in A (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land N \in B))$
30	16 29	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \exists s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land N \in B)$

Metamath Proof Explorer

Theorem cdleme25a

Proof