cdlemg28

Description: Part of proof of Lemma G of Crawley p. 116. Chain the equalities of line 14 on p. 117. TODO: rearrange hypotheses in the order of cdlemg29 (and maybe leading up to this too)? (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	cdlemg28	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (K \in HL \land W \in H)$
11		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
12		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (v \in A \land v \underset{˙}{\leq} W)$
13		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (z \in A \land \neg z \underset{˙}{\leq} W)$
14		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to F \in T$
15		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to G \in T$
16		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (v \neq R (F) \land v \neq R (G))$
17		simp313	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to z \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
18		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (F (P) \neq P \land G (P) \neq P)$
19	1 2 3 4 5 6 7	cdlemg28a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$
20	10 11 12 13 14 15 16 17 18 19	syl333anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$
21	1 2 3 4 5 6 7 8 9	cdlemg28b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$
22	20 21	eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg28

Proof