cdleme41sn3aw

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(r) is different on and off the P .\/ Q line. TODO: FIX COMMENT. (Contributed by NM, 18-Mar-2013)

	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	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, D)$
Assertion	cdleme41sn3aw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ R / s ⦌ N \neq ⦋ S / s ⦌ N$

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		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
14		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to P \neq Q$
15		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
16		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
17		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} (E \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W)) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (E \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W))$
18		eqid	$⊢ (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (E \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W)))) = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (E \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W))))$
19	1 2 3 4 5 6 7 8 9 10 11 12 17 18	cdleme41sn3a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / s ⦌ N \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
20	13 14 15 16 19	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ R / s ⦌ N \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
21		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
22		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
23	1 2 3 4 5 6 7 8 12	cdleme35sn3a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg ⦋ S / s ⦌ N \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
24	13 14 21 22 23	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to \neg ⦋ S / s ⦌ N \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
25		nbrne2	$⊢ (⦋ R / s ⦌ N \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg ⦋ S / s ⦌ N \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / s ⦌ N \neq ⦋ S / s ⦌ N$
26	20 24 25	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ R / s ⦌ N \neq ⦋ S / s ⦌ N$

Metamath Proof Explorer

Theorem cdleme41sn3aw

Proof