cdleme36a

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 11-Mar-2013)

	Ref	Expression
Hypotheses	cdleme36.b	$⊢ B = {Base}_{K}$
	cdleme36.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme36.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme36.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme36.a	$⊢ A = Atoms (K)$
	cdleme36.h	$⊢ H = LHyp (K)$
	cdleme36.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme36.e	$⊢ E = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
Assertion	cdleme36a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg R \underset{˙}{\leq} (t \underset{˙}{\lor} E)$

Proof

Step	Hyp	Ref	Expression
1		cdleme36.b	$⊢ B = {Base}_{K}$
2		cdleme36.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme36.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme36.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme36.a	$⊢ A = Atoms (K)$
6		cdleme36.h	$⊢ H = LHyp (K)$
7		cdleme36.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme36.e	$⊢ E = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		simp3r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
10		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
11		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \in A$
12		simp3ll	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to t \in A$
13		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (K \in HL \land W \in H)$
14		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
15		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
16		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
17	2 3 4 5 6 7	cdleme0a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
18	13 14 15 16 17	syl112anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \in A$
19		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
20		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
21	2 3 4 5 6 7	cdleme0c	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \to U \neq R$
22	13 19 15 20 21	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \neq R$
23	22	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \neq U$
24	2 3 5	hlatexch2	$⊢ (K \in HL \land (R \in A \land t \in A \land U \in A) \land R \neq U) \to (R \underset{˙}{\leq} (t \underset{˙}{\lor} U) \to t \underset{˙}{\leq} (R \underset{˙}{\lor} U))$
25	10 11 12 18 23 24	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\leq} (t \underset{˙}{\lor} U) \to t \underset{˙}{\leq} (R \underset{˙}{\lor} U))$
26		simp3l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
27	2 3 4 5 6 7 8	cdleme1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (t \in A \land \neg t \underset{˙}{\leq} W))) \to t \underset{˙}{\lor} E = t \underset{˙}{\lor} U$
28	13 19 15 26 27	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to t \underset{˙}{\lor} E = t \underset{˙}{\lor} U$
29	28	breq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\leq} (t \underset{˙}{\lor} E) \leftrightarrow R \underset{˙}{\leq} (t \underset{˙}{\lor} U))$
30		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
31	2 3 4 5 6 7	cdleme4	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q = R \underset{˙}{\lor} U$
32	13 19 15 20 30 31	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q = R \underset{˙}{\lor} U$
33	32	breq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow t \underset{˙}{\leq} (R \underset{˙}{\lor} U))$
34	25 29 33	3imtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\leq} (t \underset{˙}{\lor} E) \to t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
35	9 34	mtod	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land ((t \in A \land \neg t \underset{˙}{\leq} W) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg R \underset{˙}{\leq} (t \underset{˙}{\lor} E)$

Metamath Proof Explorer

Theorem cdleme36a

Proof