cdleme42mN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT . f preserves join: f(r \/ s) = f(r) \/ s, p. 115 10th line from bottom. (Contributed by NM, 20-Mar-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme41.b	$⊢ B = {Base}_{K}$
	cdleme41.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme41.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme41.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme41.a	$⊢ A = Atoms (K)$
	cdleme41.h	$⊢ H = LHyp (K)$
	cdleme41.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme41.d	$⊢ D = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme41.e	$⊢ E = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme41.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (E \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme41.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = G))$
	cdleme41.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, D)$
	cdleme41.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
	cdleme41.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
Assertion	cdleme42mN	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to F (R \underset{˙}{\lor} S) = F (R) \underset{˙}{\lor} F (S)$

Proof

Step	Hyp	Ref	Expression
1		cdleme41.b	$⊢ B = {Base}_{K}$
2		cdleme41.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme41.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme41.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme41.a	$⊢ A = Atoms (K)$
6		cdleme41.h	$⊢ H = LHyp (K)$
7		cdleme41.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme41.d	$⊢ D = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme41.e	$⊢ E = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdleme41.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (E \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
11		cdleme41.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = G))$
12		cdleme41.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, D)$
13		cdleme41.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
14		cdleme41.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
15		simpl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
16		simpr2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
17		simpr3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
18		simpr1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to P \neq Q$
19		eqid	$⊢ (R \underset{˙}{\lor} S) \underset{˙}{\land} W = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
20	1 2 3 4 5 6 7 8 9 10 11 12 13 14 19	cdleme42g	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (R \underset{˙}{\lor} S) = F (R) \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)$
21	15 16 17 18 20	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to F (R \underset{˙}{\lor} S) = F (R) \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)$
22	1 2 3 4 5 6 7 8 9 10 11 12 13 14 19	cdleme42ke	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to F (R) \underset{˙}{\lor} F (S) = F (R) \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)$
23	21 22	eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to F (R \underset{˙}{\lor} S) = F (R) \underset{˙}{\lor} F (S)$

Metamath Proof Explorer

Theorem cdleme42mN

Proof