Nuprl Lemma : strong-continuity3_functionality
∀[T,S:Type]. ∀e:T ~ S. ∀F:(ℕ ⟶ S) ⟶ ℕ. (strong-continuity3(T;λf.(F ((fst(e)) o f))) ⇒ strong-continuity3(S;F))
Proof
Definitions occuring in Statement :
strong-continuity3: strong-continuity3(T;F),
equipollent: A ~ B,
compose: f o g,
nat: ℕ,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
prop: ℙ,
pi1: fst(t),
member: t ∈ T,
and: P ∧ Q,
biject: Bij(A;B;f),
exists: ∃x:A. B[x],
equipollent: A ~ B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
equipollent_wf,
compose_wf,
nat_wf,
strong-continuity3_wf,
strong-continuity3_functionality_surject
Rules used in proof :
universeEquality,
because_Cache,
functionEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
cumulativity,
sqequalRule,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T,S:Type].
\mforall{}e:T \msim{} S. \mforall{}F:(\mBbbN{} {}\mrightarrow{} S) {}\mrightarrow{} \mBbbN{}.
(strong-continuity3(T;\mlambda{}f.(F ((fst(e)) o f))) {}\mRightarrow{} strong-continuity3(S;F))
Date html generated:
2017_09_29-PM-06_05_12
Last ObjectModification:
2017_09_04-AM-09_02_31
Theory : continuity
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