cdlemg28a

Description: Part of proof of Lemma G of Crawley p. 116. First equality of the equation of line 14 on p. 117. (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	$⊢ R = trL (K) (W)$
Assertion	cdlemg28a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to (K \in HL \land W \in H)$
9		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
10		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to (z \in A \land \neg z \underset{˙}{\leq} W)$
11		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to F \in T$
12		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to G \in T$
13		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W))$
14		simp21l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to z \in A$
15		simp31l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to v \neq R (F)$
16		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to z \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
17		simp33l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to F (P) \neq P$
18	1 2 3 4 5 6 7	cdlemg27a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (z \in A \land F \in T) \land (v \neq R (F) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land F (P) \neq P)) \to \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} z)$
19	13 14 11 15 16 17 18	syl123anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} z)$
20		simp31r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to v \neq R (G)$
21		simp33r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to G (P) \neq P$
22	1 2 3 4 5 6 7	cdlemg27a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land (z \in A \land G \in T) \land (v \neq R (G) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land G (P) \neq P)) \to \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} z)$
23	13 14 12 20 16 21 22	syl123anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} z)$
24	1 2 3 4 5 6 7	cdlemg25zz	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (P \underset{˙}{\lor} z) \land \neg R (G) \underset{˙}{\leq} (P \underset{˙}{\lor} z))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$
25	8 9 10 11 12 19 23 24	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (v \in A \land v \underset{˙}{\leq} W)) \land ((z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T \land G \in T) \land ((v \neq R (F) \land v \neq R (G)) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land (F (P) \neq P \land G (P) \neq P))) \to (P \underset{˙}{\lor} F (G (P))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg28a

Proof