cdlemki

Description: Part of proof of Lemma K of Crawley p. 118. TODO: Eliminate and put into cdlemksel . (Contributed by NM, 25-Jun-2013)

	Ref	Expression
Hypotheses	cdlemk.b	$⊢ B = {Base}_{K}$
	cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk.a	$⊢ A = Atoms (K)$
	cdlemk.h	$⊢ H = LHyp (K)$
	cdlemk.t	$⊢ T = LTrn (K) (W)$
	cdlemk.r	$⊢ R = trL (K) (W)$
	cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk.i	$⊢ I = (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})))$
Assertion	cdlemki	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to I \in T$

Proof

Step	Hyp	Ref	Expression
1		cdlemk.b	$⊢ B = {Base}_{K}$
2		cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk.a	$⊢ A = Atoms (K)$
5		cdlemk.h	$⊢ H = LHyp (K)$
6		cdlemk.t	$⊢ T = LTrn (K) (W)$
7		cdlemk.r	$⊢ R = trL (K) (W)$
8		cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
9		cdlemk.i	$⊢ I = (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})))$
10		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (K \in HL \land W \in H)$
11		simp22	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
12		simp1	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to ((K \in HL \land W \in H) \land F \in T \land G \in T)$
13		simp21	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N \in T$
14	2 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (N (P) \in A \land \neg N (P) \underset{˙}{\leq} W)$
15	10 13 11 14	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (N (P) \in A \land \neg N (P) \underset{˙}{\leq} W)$
16		simp11l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to K \in HL$
17		simp22l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \in A$
18	14	simpld	$⊢ ((K \in HL \land W \in H) \land N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to N (P) \in A$
19	10 13 11 18	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N (P) \in A$
20	2 3 4	hlatlej2	$⊢ (K \in HL \land P \in A \land N (P) \in A) \to N (P) \underset{˙}{\leq} (P \underset{˙}{\lor} N (P))$
21	16 17 19 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N (P) \underset{˙}{\leq} (P \underset{˙}{\lor} N (P))$
22		simp23	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to R (F) = R (N)$
23	22	oveq2d	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \underset{˙}{\lor} R (F) = P \underset{˙}{\lor} R (N)$
24	2 3 4 5 6 7	trljat1	$⊢ ((K \in HL \land W \in H) \land N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to P \underset{˙}{\lor} R (N) = P \underset{˙}{\lor} N (P)$
25	10 13 11 24	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \underset{˙}{\lor} R (N) = P \underset{˙}{\lor} N (P)$
26	23 25	eqtr2d	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \underset{˙}{\lor} N (P) = P \underset{˙}{\lor} R (F)$
27	21 26	breqtrd	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (F))$
28		simp31	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to F \neq {I ↾}_{B}$
29		simp32	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to G \neq {I ↾}_{B}$
30		simp33	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to R (G) \neq R (F)$
31	30	necomd	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to R (F) \neq R (G)$
32		eqid	$⊢ (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))$
33	1 2 3 8 4 5 6 7 32	cdlemh	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (N (P) \in A \land \neg N (P) \underset{˙}{\leq} W) \land N (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (F))) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (F) \neq R (G))) \to ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} W)$
34	12 11 15 27 28 29 31 33	syl133anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} W)$
35	2 4 5 6 9	ltrniotacl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} W)) \to I \in T$
36	10 11 34 35	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to I \in T$

Metamath Proof Explorer

Theorem cdlemki

Proof