### Nuprl Lemma : strong-continuity2-implies-uniform-continuity2

`∀F:(ℕ ⟶ 𝔹) ⟶ 𝔹. ∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹.  ((f = g ∈ (ℕn ⟶ 𝔹)) `` F f = F g)`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` implies: `P `` Q` uniform-continuity-pi: `ucA(T;F;n)` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` exists: `∃x:A. B[x]` uall: `∀[x:A]. B[x]` nat: `ℕ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` true: `True` so_lambda: `λ2x.t[x]` so_apply: `x[s]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cand: `A c∧ B` quotient: `x,y:A//B[x; y]` squash: `↓T` sq_type: `SQType(T)` guard: `{T}` uniform-continuity-pi-pi: `ucpB(T;F;n)`
Lemmas referenced :  istype-nat bool_wf strong-continuity2-implies-uniform-continuity uniform-continuity-pi-pi-prop2 decidable__equal_bool int_seg_wf subtype_rel_function nat_wf int_seg_subtype_nat istype-false subtype_rel_self true_wf quotient_wf exists_wf uniform-continuity-pi-pi_wf equiv_rel_true quotient-member-eq member_wf squash_wf istype-universe prop-truncation-implies uniform-continuity-pi-pi-prop subtype_base_sq set_subtype_base le_wf istype-int int_subtype_base equal_wf uniform-continuity-pi_wf le_witness
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  Error :functionIsType,  cut introduction extract_by_obid hypothesis Error :universeIsType,  sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality because_Cache independent_functionElimination Error :inhabitedIsType,  sqequalRule productElimination independent_pairFormation rename Error :productIsType,  Error :equalityIstype,  isectElimination natural_numberEquality setElimination applyEquality independent_isectElimination promote_hyp Error :lambdaEquality_alt,  productEquality pointwiseFunctionality pertypeElimination sqequalBase equalitySymmetry imageElimination equalityTransitivity instantiate universeEquality imageMemberEquality baseClosed cumulativity intEquality closedConclusion Error :dependent_pairEquality_alt,  independent_pairEquality Error :functionExtensionality_alt,  functionExtensionality functionEquality equalityElimination

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbB{}.  \mexists{}n:\mBbbN{}.  \mforall{}f,g:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.    ((f  =  g)  {}\mRightarrow{}  F  f  =  F  g)

Date html generated: 2019_06_20-PM-02_53_23
Last ObjectModification: 2018_11_28-AM-09_03_50

Theory : continuity

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