cdleme41sn4aw

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(r) is for on and off P .\/ Q line. TODO: FIX COMMENT. (Contributed by NM, 19-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	cdleme41sn4aw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ R / s ⦌ N \neq ⦋ S / s ⦌ N$

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	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
14		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to P \neq Q$
15		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
16		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
17		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
18		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
19		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to R \neq S$
20	19	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to S \neq R$
21	1 2 3 4 5 6 7 8 9 10 11 12	cdleme41sn3aw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \neq R)) \to ⦋ S / s ⦌ N \neq ⦋ R / s ⦌ N$
22	13 14 15 16 17 18 20 21	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ S / s ⦌ N \neq ⦋ R / s ⦌ N$
23	22	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ R / s ⦌ N \neq ⦋ S / s ⦌ N$

Metamath Proof Explorer

Theorem cdleme41sn4aw

Proof