cdleme12

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, first part of 3rd sentence. F and G represent f(s) and f(t) respectively. (Contributed by NM, 16-Jun-2012)

	Ref	Expression
Hypotheses	cdleme12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme12.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme12.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme12.a	$⊢ A = Atoms (K)$
	cdleme12.h	$⊢ H = LHyp (K)$
	cdleme12.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme12.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme12.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
Assertion	cdleme12	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G) = U$

Proof

Step	Hyp	Ref	Expression
1		cdleme12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme12.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme12.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme12.a	$⊢ A = Atoms (K)$
5		cdleme12.h	$⊢ H = LHyp (K)$
6		cdleme12.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme12.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme12.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
9		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (K \in HL \land W \in H)$
10		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to P \in A$
11		simp22	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to Q \in A$
12		simp31	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
13	1 2 3 4 5 6 7	cdleme1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to S \underset{˙}{\lor} F = S \underset{˙}{\lor} U$
14	9 10 11 12 13	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to S \underset{˙}{\lor} F = S \underset{˙}{\lor} U$
15		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to K \in HL$
16		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
17		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to P \neq Q$
18	1 2 3 4 5 6	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
19	9 16 11 17 18	syl112anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to U \in A$
20		simp31l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to S \in A$
21	2 4	hlatjcom	$⊢ (K \in HL \land U \in A \land S \in A) \to U \underset{˙}{\lor} S = S \underset{˙}{\lor} U$
22	15 19 20 21	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to U \underset{˙}{\lor} S = S \underset{˙}{\lor} U$
23	14 22	eqtr4d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to S \underset{˙}{\lor} F = U \underset{˙}{\lor} S$
24		simp32	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (T \in A \land \neg T \underset{˙}{\leq} W)$
25	1 2 3 4 5 6 8	cdleme1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (T \in A \land \neg T \underset{˙}{\leq} W))) \to T \underset{˙}{\lor} G = T \underset{˙}{\lor} U$
26	9 10 11 24 25	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to T \underset{˙}{\lor} G = T \underset{˙}{\lor} U$
27		simp32l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to T \in A$
28	2 4	hlatjcom	$⊢ (K \in HL \land U \in A \land T \in A) \to U \underset{˙}{\lor} T = T \underset{˙}{\lor} U$
29	15 19 27 28	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to U \underset{˙}{\lor} T = T \underset{˙}{\lor} U$
30	26 29	eqtr4d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to T \underset{˙}{\lor} G = U \underset{˙}{\lor} T$
31	23 30	oveq12d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G) = (U \underset{˙}{\lor} S) \underset{˙}{\land} (U \underset{˙}{\lor} T)$
32		simp33	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))$
33	1 2 3 4	2llnma2	$⊢ (K \in HL \land (S \in A \land T \in A \land U \in A) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (U \underset{˙}{\lor} S) \underset{˙}{\land} (U \underset{˙}{\lor} T) = U$
34	15 20 27 19 32 33	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (U \underset{˙}{\lor} S) \underset{˙}{\land} (U \underset{˙}{\lor} T) = U$
35	31 34	eqtrd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G) = U$

Metamath Proof Explorer

Theorem cdleme12

Proof