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Metamath Proof Explorer

Theorem cdlemg17irq

Description: cdlemg17ir with P and Q swapped. (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	cdlemg17irq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (Q)) = F (P)$

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	1 2 3 4 5 6 7	cdlemg17ir	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land Q \neq P) \land (G (Q) \neq Q \land R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} P) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land Q \underset{˙}{\lor} r = P \underset{˙}{\lor} r))) \to F (G (Q)) = F (P)$
10	8 9	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (G (P) \neq P \land R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to F (G (Q)) = F (P)$