Nuprl Lemma : strong-continuity2_functionality_surject
∀[T,S:Type].
∀g:T ⟶ S. (Surj(T;S;g) ⇒ (∀F:(ℕ ⟶ S) ⟶ ℕ. (strong-continuity2(T;λf.(F (g o f))) ⇒ strong-continuity2(S;F))))
Proof
Definitions occuring in Statement :
strong-continuity2: strong-continuity2(T;F),
surject: Surj(A;B;f),
compose: f o g,
nat: ℕ,
uall: ∀[x:A]. B[x],
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,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
surject_wf,
strong-continuity2_wf,
strong-continuity3_functionality_surject,
compose_wf,
nat_wf,
strong-continuity2-iff-3
Rules used in proof :
universeEquality,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
productElimination,
cumulativity,
hypothesis,
functionEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T,S:Type].
\mforall{}g:T {}\mrightarrow{} S
(Surj(T;S;g)
{}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} S) {}\mrightarrow{} \mBbbN{}. (strong-continuity2(T;\mlambda{}f.(F (g o f))) {}\mRightarrow{} strong-continuity2(S;F))))
Date html generated:
2017_09_29-PM-06_05_15
Last ObjectModification:
2017_09_04-PM-00_14_08
Theory : continuity
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