Nuprl Lemma : strong-continuity3_functionality_surject
∀[T,S:Type].
∀g:T ⟶ S. (Surj(T;S;g) ⇒ (∀F:(ℕ ⟶ S) ⟶ ℕ. (strong-continuity3(T;λf.(F (g o f))) ⇒ strong-continuity3(S;F))))
Proof
Definitions occuring in Statement :
strong-continuity3: strong-continuity3(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 :
compose: f o g,
rev_implies: P ⇐ Q,
guard: {T},
true: True,
squash: ↓T,
cand: A c∧ B,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
nat: ℕ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
strong-continuity3: strong-continuity3(T;F),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
iff_weakening_equal,
true_wf,
squash_wf,
surject_wf,
strong-continuity3_wf,
isl_wf,
assert_wf,
false_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
unit_wf2,
equal_wf,
exists_wf,
all_wf,
int_seg_wf,
compose_wf,
nat_wf,
surject-inverse
Rules used in proof :
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
universeEquality,
inlEquality,
independent_pairFormation,
independent_isectElimination,
unionEquality,
productEquality,
functionEquality,
because_Cache,
cumulativity,
setElimination,
natural_numberEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
dependent_pairFormation,
rename,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
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-continuity3(T;\mlambda{}f.(F (g o f))) {}\mRightarrow{} strong-continuity3(S;F))))
Date html generated:
2017_09_29-PM-06_05_10
Last ObjectModification:
2017_09_04-PM-00_14_58
Theory : continuity
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