cdlemg29

Description: Eliminate ( FP ) =/= P and ( GP ) =/= P from cdlemg28 . TODO: would it be better to do this later? (Contributed by NM, 29-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	cdlemg29	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

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		simpl11	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land F (P) = P) \to (K \in HL \land W \in H)$
11		simpl12	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land F (P) = P) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
12		simpl13	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land F (P) = P) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
13		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \to F \in T$
14	13	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land F (P) = P) \to F \in T$
15		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \to G \in T$
16	15	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land F (P) = P) \to G \in T$
17		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land F (P) = P) \to F (P) = P$
18	1 2 3 4 5 6 7	cdlemg14f	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land F (P) = P)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
19	10 11 12 14 16 17 18	syl123anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land F (P) = P) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
20		simpl11	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land G (P) = P) \to (K \in HL \land W \in H)$
21		simpl12	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land G (P) = P) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
22		simpl13	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land G (P) = P) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
23	13	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land G (P) = P) \to F \in T$
24	15	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land G (P) = P) \to G \in T$
25		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land G (P) = P) \to G (P) = P$
26	1 2 3 4 5 6 7	cdlemg14g	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (F \in T \land G \in T \land G (P) = P)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
27	20 21 22 23 24 25 26	syl123anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land G (P) = P) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
28		simpl1	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
29		simpl2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T))$
30		simp31l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \to z \neq N$
31	30	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to z \neq N$
32		simp31r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \to z \neq O$
33	32	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to z \neq O$
34		simpl32	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to z \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
35	31 33 34	3jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to (z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v))$
36		simpl33	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to (v \neq R (F) \land v \neq R (G))$
37		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to (F (P) \neq P \land G (P) \neq P)$
38	1 2 3 4 5 6 7 8 9	cdlemg28	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
39	28 29 35 36 37 38	syl113anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \land (F (P) \neq P \land G (P) \neq P)) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$
40	19 27 39	pm2.61da2ne	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (v \neq R (F) \land v \neq R (G)))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg29

Proof