Metamath Proof Explorer

Theorem cdlemg17bq

Description: cdlemg17b with P and Q swapped. Antecedent F e. ( TW ) is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b also? (Contributed by NM, 13-May-2013)

		Ref	Expression
	Hypotheses	cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemg12.a	$⊢ A = Atoms (K)$
		cdlemg12.h	$⊢ H = LHyp (K)$
		cdlemg12.t	$⊢ T = LTrn (K) (W)$
		cdlemg12b.r	$⊢ R = trL (K) (W)$
	Assertion	cdlemg17bq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (Q) = P$

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemg12.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemg12.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemg12.a	$⊢ A = Atoms (K)$
5		cdlemg12.h	$⊢ H = LHyp (K)$
6		cdlemg12.t	$⊢ T = LTrn (K) (W)$
7		cdlemg12b.r	$⊢ R = trL (K) (W)$
8	1 2 3 4 5 6 7	cdlemg17pq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r)))$
9		simp11	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to (K \in HL \land W \in H)$
10		simp12	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
11		simp13	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
12		simp22	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to G \in T$
13		simp23	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to Q \neq P$
14		simp3	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))$
15	1 2 3 4 5 6 7	cdlemg17b	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to G (Q) = P$
16	9 10 11 12 13 14 15	syl321anc	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to G (Q) = P$
17	8 16	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to G (Q) = P$