Metamath Proof Explorer

Theorem cdlemkuel-3

Description: Part of proof of Lemma K of Crawley p. 118. Conditions for the sigma_2 (p) function to be a translation. TODO: combine cdlemkj ? (Contributed by NM, 11-Jul-2013)

		Ref	Expression
	Hypotheses	cdlemk3.b	$⊢ B = {Base}_{K}$
		cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemk3.a	$⊢ A = Atoms (K)$
		cdlemk3.h	$⊢ H = LHyp (K)$
		cdlemk3.t	$⊢ T = LTrn (K) (W)$
		cdlemk3.r	$⊢ R = trL (K) (W)$
		cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
		cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
	Assertion	cdlemkuel-3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to D Y G \in T$

Proof

Step	Hyp	Ref	Expression
1		cdlemk3.b	$⊢ B = {Base}_{K}$
2		cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk3.a	$⊢ A = Atoms (K)$
6		cdlemk3.h	$⊢ H = LHyp (K)$
7		cdlemk3.t	$⊢ T = LTrn (K) (W)$
8		cdlemk3.r	$⊢ R = trL (K) (W)$
9		cdlemk3.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		cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
11		simp22	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to D \in T$
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		eqid	$⊢ S (D) = S (D)$
14		eqid	$⊢ (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
15	1 2 3 4 5 6 7 8 9 10 13 14	cdlemkuu	$⊢ (D \in T \land G \in T) \to D Y G = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) (G)$
16	11 12 15	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to D Y G = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) (G)$
17	1 2 3 4 5 6 7 8 9 13 14	cdlemkuel	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) (G) \in T$
18	16 17	eqeltrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to D Y G \in T$