cdlemk

Description: Lemma K of Crawley p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use F , N , and u to represent f, k, and tau. (Contributed by NM, 1-Aug-2013)

	Ref	Expression
Hypotheses	cdlemk7.h	$⊢ H = LHyp (K)$
	cdlemk7.t	$⊢ T = LTrn (K) (W)$
	cdlemk7.r	$⊢ R = trL (K) (W)$
	cdlemk7.e	$⊢ E = TEndo (K) (W)$
Assertion	cdlemk	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to \exists u \in E u (F) = N$

Proof

Step	Hyp	Ref	Expression
1		cdlemk7.h	$⊢ H = LHyp (K)$
2		cdlemk7.t	$⊢ T = LTrn (K) (W)$
3		cdlemk7.r	$⊢ R = trL (K) (W)$
4		cdlemk7.e	$⊢ E = TEndo (K) (W)$
5		eqid	$⊢ {Base}_{K} = {Base}_{K}$
6		eqid	$⊢ join (K) = join (K)$
7		eqid	$⊢ meet (K) = meet (K)$
8		eqid	$⊢ oc (K) = oc (K)$
9		eqid	$⊢ Atoms (K) = Atoms (K)$
10		eqid	$⊢ oc (K) (W) = oc (K) (W)$
11		eqid	$⊢ (oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1})) = (oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))$
12		eqid	$⊢ (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1}))$
13		eqid	$⊢ (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))) = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1}))))$
14		eqid	$⊢ (f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) = (f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1}))))))$
15	5 6 7 8 9 1 2 3 10 11 12 13 14 4	cdlemk56w	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to ((f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) \in E \land (f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) (F) = N)$
16		fveq1	$⊢ u = (f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) \to u (F) = (f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) (F)$
17	16	eqeq1d	$⊢ u = (f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) \to (u (F) = N \leftrightarrow (f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) (F) = N)$
18	17	rspcev	$⊢ ((f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) \in E \land (f \in T ⟼ if (F = N, f, (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{{Base}_{K}} \land R (b) \neq R (F) \land R (b) \neq R (f)) \to z (oc (K) (W)) = (oc (K) (W) join (K) R (f)) meet (K) (((oc (K) (W) join (K) R (b)) meet (K) (N (oc (K) (W)) join (K) R (b \circ F^{-1}))) join (K) R (f \circ b^{-1})))))) (F) = N) \to \exists u \in E u (F) = N$
19	15 18	syl	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to \exists u \in E u (F) = N$

Metamath Proof Explorer

Theorem cdlemk

Proof