cdlemg34

Description: Use cdlemg33 to eliminate z from cdlemg29 . TODO: Fix comment. (Contributed by NM, 31-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	$⊢ R = trL (K) (W)$
	cdlemg31.n	$⊢ N = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
	cdlemg33.o	$⊢ O = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (G))$
Assertion	cdlemg34	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

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		cdlemg31.n	$⊢ N = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (F))$
9		cdlemg33.o	$⊢ O = (P \underset{˙}{\lor} v) \underset{˙}{\land} (Q \underset{˙}{\lor} R (G))$
10	1 2 3 4 5 6 7 8 9	cdlemg33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to \exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))$
11		simp11	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
12		simp121	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to (v \in A \land v \underset{˙}{\leq} W)$
13		simp2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to z \in A$
14		simp3l	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to \neg z \underset{˙}{\leq} W$
15	13 14	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to (z \in A \land \neg z \underset{˙}{\leq} W)$
16		simp122	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to (F \in T \land G \in T)$
17		simp3r1	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to z \neq N$
18		simp3r2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to z \neq O$
19	17 18	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to (z \neq N \land z \neq O)$
20		simp3r3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to z \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
21		simp131	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to v \neq R (F)$
22		simp132	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to v \neq R (G)$
23	21 22	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to (v \neq R (F) \land v \neq R (G))$
24	1 2 3 4 5 6 7 8 9	cdlemg29	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
25	11 12 15 16 19 20 23 24	syl133anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \land z \in A \land (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
26	25	rexlimdv3a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (\exists z \in A (\neg z \underset{˙}{\leq} W \land (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W)$
27	10 26	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land G \in T) \land P \neq Q) \land (v \neq R (F) \land v \neq R (G) \land \exists r \in A (\neg r \underset{˙}{\leq} W \land P \underset{˙}{\lor} r = Q \underset{˙}{\lor} r))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg34

Proof