Nuprl Lemma : uniform-continuity-from-fan3
∀[T:Type]
((∃P:ℕ ⟶ ℙ. ∃a:ℕ. (P[a] ∧ (∀n:ℕ. Dec(P[n])) ∧ (∃h:T ⟶ {n:ℕ| P[n]} . Bij(T;{n:ℕ| P[n]} ;h))))
⇒ (∀F:(ℕ ⟶ 𝔹) ⟶ T. ⇃(∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕn ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ T)))))
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False
Lemmas referenced :
uniform-continuity-from-fan2,
nat_wf,
biject_wf,
subtype_rel_wf,
exists_wf,
all_wf,
equal_wf,
bool_wf,
subtype_rel_self,
decidable_wf,
not_wf,
set_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
productElimination,
dependent_pairFormation,
setEquality,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
setElimination,
rename,
independent_pairFormation,
productEquality,
functionEquality,
dependent_functionElimination,
instantiate,
cumulativity,
universeEquality,
unionEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
dependent_set_memberEquality,
voidElimination
Latex:
\mforall{}[T:Type]
((\mexists{}P:\mBbbN{} {}\mrightarrow{} \mBbbP{}. \mexists{}a:\mBbbN{}. (P[a] \mwedge{} (\mforall{}n:\mBbbN{}. Dec(P[n])) \mwedge{} (\mexists{}h:T {}\mrightarrow{} \{n:\mBbbN{}| P[n]\} . Bij(T;\{n:\mBbbN{}| P[n]\} ;h))))
{}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T. \00D9(\mexists{}n:\mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((f = g) {}\mRightarrow{} ((F f) = (F g))))))
Date html generated:
2019_06_20-PM-02_52_30
Last ObjectModification:
2018_08_21-PM-01_56_54
Theory : continuity
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