cdlemg28b

Description: Part of proof of Lemma G of Crawley p. 116. Second equality of the equation of line 14 on p. 117. Note that -. z .<_ W is redundant here (but simplifies cdlemg28 .) (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	cdlemg28b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$

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		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
12		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (z \in A \land \neg z \underset{˙}{\leq} W)$
13		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to F \in T$
14		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to G \in T$
15		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
16		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to z \in A$
17		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (v \in A \land v \underset{˙}{\leq} W)$
18		simp311	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to z \neq N$
19	13 18	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (F \in T \land z \neq N)$
20		simp32l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to v \neq R (F)$
21		simp313	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to z \underset{˙}{\leq} (P \underset{˙}{\lor} v)$
22		simp33l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to F (P) \neq P$
23	1 2 3 4 5 6 7 8	cdlemg27b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (z \in A \land (v \in A \land v \underset{˙}{\leq} W) \land (F \in T \land z \neq N)) \land (v \neq R (F) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land F (P) \neq P)) \to \neg R (F) \underset{˙}{\leq} (Q \underset{˙}{\lor} z)$
24	15 16 17 19 20 21 22 23	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to \neg R (F) \underset{˙}{\leq} (Q \underset{˙}{\lor} z)$
25		simp312	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to z \neq O$
26	14 25	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (G \in T \land z \neq O)$
27		simp32r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to v \neq R (G)$
28		simp33r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to G (P) \neq P$
29	1 2 3 4 5 6 7 9	cdlemg27b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (z \in A \land (v \in A \land v \underset{˙}{\leq} W) \land (G \in T \land z \neq O)) \land (v \neq R (G) \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v) \land G (P) \neq P)) \to \neg R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} z)$
30	15 16 17 26 27 21 28 29	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to \neg R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} z)$
31	1 2 3 4 5 6 7	cdlemg26zz	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land F \in T) \land (G \in T \land \neg R (F) \underset{˙}{\leq} (Q \underset{˙}{\lor} z) \land \neg R (G) \underset{˙}{\leq} (Q \underset{˙}{\lor} z))) \to (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$
32	10 11 12 13 14 24 30 31	syl133anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((v \in A \land v \underset{˙}{\leq} W) \land (z \in A \land \neg z \underset{˙}{\leq} W) \land (F \in T \land G \in T)) \land ((z \neq N \land z \neq O \land z \underset{˙}{\leq} (P \underset{˙}{\lor} v)) \land (v \neq R (F) \land v \neq R (G)) \land (F (P) \neq P \land G (P) \neq P))) \to (Q \underset{˙}{\lor} F (G (Q))) \underset{˙}{\land} W = (z \underset{˙}{\lor} F (G (z))) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdlemg28b

Proof