cdleme35sn2aw

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one outside of P .\/ Q line case; compare cdleme32sn2awN . TODO: FIX COMMENT. (Contributed by NM, 18-Mar-2013)

	Ref	Expression
Hypotheses	cdleme32s.b	$⊢ B = {Base}_{K}$
	cdleme32s.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme32s.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme32s.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme32s.a	$⊢ A = Atoms (K)$
	cdleme32s.h	$⊢ H = LHyp (K)$
	cdleme32s.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme32s.d	$⊢ D = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme32s.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, D)$
Assertion	cdleme35sn2aw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ R / s ⦌ N \neq ⦋ S / s ⦌ N$

Proof

Step	Hyp	Ref	Expression
1		cdleme32s.b	$⊢ B = {Base}_{K}$
2		cdleme32s.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme32s.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme32s.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme32s.a	$⊢ A = Atoms (K)$
6		cdleme32s.h	$⊢ H = LHyp (K)$
7		cdleme32s.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme32s.d	$⊢ D = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme32s.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, D)$
10		eqid	$⊢ (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
11		eqid	$⊢ (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W)) = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
12	2 3 4 5 6 7 10 11	cdleme35h2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \neq (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
13		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to R \in A$
14		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
15	8 9 10	cdleme31sn2	$⊢ (R \in A \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / s ⦌ N = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
16	13 14 15	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ R / s ⦌ N = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
17		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to S \in A$
18		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
19	8 9 11	cdleme31sn2	$⊢ (S \in A \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ S / s ⦌ N = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
20	17 18 19	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ S / s ⦌ N = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
21	12 16 20	3netr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (\neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \neq S)) \to ⦋ R / s ⦌ N \neq ⦋ S / s ⦌ N$

Metamath Proof Explorer

Theorem cdleme35sn2aw

Proof