Nuprl Lemma : strong-continuity2-half-squash-surject-biject
∀[T,S,U:Type].
((U ⊆r ℕ)
⇒ (∃r:ℕ ⟶ U. ∀x:U. ((r x) = x ∈ U))
⇒ (∃g:ℕ ⟶ T. Surj(ℕ;T;g))
⇒ (∃h:S ⟶ U. Bij(S;U;h))
⇒ (∀F:(ℕ ⟶ T) ⟶ S
⇃(∃M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (S?)
∀f:ℕ ⟶ T
((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (S?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (S?) supposing ↑isl(M n f))))))
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
surject: Surj(A;B;f),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
squash: ↓T,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T}
Lemmas referenced :
strong-continuity2_biject_retract-ext,
strong-continuity2-half-squash,
nat_wf,
subtype_rel_self,
compose_wf,
exists_wf,
biject_wf,
surject_wf,
all_wf,
equal_wf,
subtype_rel_wf,
strong-continuity2_wf,
trivial-quotient-true,
strong-continuity2_functionality_surject,
int_seg_wf,
unit_wf2,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
isect_wf,
assert_wf,
isl_wf,
implies-quotient-true2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
independent_isectElimination,
because_Cache,
independent_pairFormation,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
dependent_functionElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
cumulativity,
universeEquality,
natural_numberEquality,
setElimination,
rename,
unionEquality,
productEquality,
inlEquality
Latex:
\mforall{}[T,S,U:Type].
((U \msubseteq{}r \mBbbN{})
{}\mRightarrow{} (\mexists{}r:\mBbbN{} {}\mrightarrow{} U. \mforall{}x:U. ((r x) = x))
{}\mRightarrow{} (\mexists{}g:\mBbbN{} {}\mrightarrow{} T. Surj(\mBbbN{};T;g))
{}\mRightarrow{} (\mexists{}h:S {}\mrightarrow{} U. Bij(S;U;h))
{}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} S
\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} (S?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} T
((\mexists{}n:\mBbbN{}. ((M n f) = (inl (F f))))
\mwedge{} (\mforall{}n:\mBbbN{}. (M n f) = (inl (F f)) supposing \muparrow{}isl(M n f))))))
Date html generated:
2019_06_20-PM-03_17_37
Last ObjectModification:
2019_06_20-PM-03_13_04
Theory : continuity
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