Metamath Proof Explorer

Theorem cdlemkuel

Description: Part of proof of Lemma K of Crawley p. 118. Conditions for the sigma_1 (p) function to be a translation. TODO: combine cdlemkj ? (Contributed by NM, 2-Jul-2013)

		Ref	Expression
	Hypotheses	cdlemk1.b	$⊢ B = {Base}_{K}$
		cdlemk1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemk1.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemk1.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemk1.a	$⊢ A = Atoms (K)$
		cdlemk1.h	$⊢ H = LHyp (K)$
		cdlemk1.t	$⊢ T = LTrn (K) (W)$
		cdlemk1.r	$⊢ R = trL (K) (W)$
		cdlemk1.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
		cdlemk1.o	$⊢ O = S (D)$
		cdlemk1.u	$⊢ U = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
	Assertion	cdlemkuel	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (G) \in T$

Proof

Step	Hyp	Ref	Expression
1		cdlemk1.b	$⊢ B = {Base}_{K}$
2		cdlemk1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk1.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk1.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk1.a	$⊢ A = Atoms (K)$
6		cdlemk1.h	$⊢ H = LHyp (K)$
7		cdlemk1.t	$⊢ T = LTrn (K) (W)$
8		cdlemk1.r	$⊢ R = trL (K) (W)$
9		cdlemk1.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
10		cdlemk1.o	$⊢ O = S (D)$
11		cdlemk1.u	$⊢ U = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
12		simp13	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \in T$
13	1 2 3 5 6 7 8 4 11	cdlemksv	$⊢ G \in T \to U (G) = (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})))$
14	12 13	syl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (G) = (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})))$
15		eqid	$⊢ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))) = (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})))$
16	1 2 3 4 5 6 7 8 9 10 15	cdlemkj	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))) \in T$
17	14 16	eqeltrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (G) \in T$