Metamath Proof Explorer

Theorem cdlemksel

Description: Part of proof of Lemma K of Crawley p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki ? (Contributed by NM, 26-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.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	Assertion	cdlemksel	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to S (G) \in T$

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.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		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to G \in T$
11	1 2 3 4 5 6 7 8 9	cdlemksv	$⊢ G \in T \to S (G) = (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})))$
12	10 11	syl	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to S (G) = (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})))$
13		eqid	$⊢ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) = (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})))$
14	1 2 3 4 5 6 7 8 13	cdlemki	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \in T$
15	12 14	eqeltrd	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to S (G) \in T$