cdlemg16zz

Description: Eliminate P =/= Q from cdlemg16z . TODO: Use this only if needed. (Contributed by NM, 26-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	cdlemg16zz	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

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		id	$⊢ P = Q \to P = Q$
9		2fveq3	$⊢ P = Q \to F (G (P)) = F (G (Q))$
10	8 9	oveq12d	$⊢ P = Q \to P \underset{˙}{\lor} F (G (P)) = Q \underset{˙}{\lor} F (G (Q))$
11	10	oveq1d	$⊢ P = Q \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
12	11	adantl	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P = Q) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
13		simpl1	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to (K \in HL \land W \in H)$
14		simpl21	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
15		simpl22	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
16		simpl23	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to F \in T$
17		simpl31	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to G \in T$
18		simpr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to P \neq Q$
19		simpl32	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
20		simpl33	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
21	1 2 3 4 5 6 7	cdlemg16z	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land P \neq Q) \land (\neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
22	13 14 15 16 17 18 19 20 21	syl332anc	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land P \neq Q) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
23	12 22	pm2.61dane	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg16zz

Proof