Nuprl Lemma : strong-continuity3-half-squash-surject
∀[B:Type]. ((∃g:ℕ ⟶ B. Surj(ℕ;B;g)) ⇒ (∀F:(ℕ ⟶ B) ⟶ ℕ. ⇃(strong-continuity3(B;F))))
Proof
Definitions occuring in Statement :
strong-continuity3: strong-continuity3(T;F),
surject: Surj(A;B;f),
quotient: x,y:A//B[x; y],
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
strong-continuity3_functionality_surject,
implies-quotient-true2,
trivial-quotient-true,
strong-continuity3_wf,
surject_wf,
exists_wf,
compose_wf,
subtype_rel_self,
nat_wf,
strong-continuity3-half-squash
Rules used in proof :
universeEquality,
cumulativity,
functionEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
dependent_functionElimination,
baseClosed,
hypothesisEquality,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
independent_pairFormation,
because_Cache,
independent_isectElimination,
hypothesis,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[B:Type]. ((\mexists{}g:\mBbbN{} {}\mrightarrow{} B. Surj(\mBbbN{};B;g)) {}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} B) {}\mrightarrow{} \mBbbN{}. \00D9(strong-continuity3(B;F))))
Date html generated:
2017_09_29-PM-06_05_31
Last ObjectModification:
2017_09_04-AM-09_47_26
Theory : continuity
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