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

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

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` prop: `ℙ` exists: `∃x:A. B[x]` nat: `ℕ` and: `P ∧ Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` isl: `isl(x)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` sq_exists: `∃x:A [B[x]]`
Lemmas referenced :  uniform-continuity-from-fan-ext bool_wf istype-nat strong-continuity2-no-inner-squash-cantor4 implies-quotient-true2 nat_wf int_seg_wf unit_wf2 equal_wf subtype_rel_function int_seg_subtype_nat istype-false subtype_rel_self assert_wf btrue_wf bfalse_wf sq_exists_wf trivial-quotient-true istype-assert
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis dependent_functionElimination hypothesisEquality independent_functionElimination Error :functionIsType,  Error :universeIsType,  sqequalRule productEquality functionEquality natural_numberEquality setElimination rename unionEquality applyEquality because_Cache independent_isectElimination independent_pairFormation Error :inlEquality_alt,  isectEquality Error :inhabitedIsType,  unionElimination Error :equalityIstype,  equalityTransitivity equalitySymmetry Error :lambdaEquality_alt,  Error :unionIsType,  Error :productIsType,  Error :isectIsType,  productElimination Error :dependent_set_memberEquality_alt

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbB{}.  \00D9(\mexists{}n:\mBbbN{}.  \mforall{}f,g:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.    ((f  =  g)  {}\mRightarrow{}  F  f  =  F  g))

Date html generated: 2019_06_20-PM-02_52_43
Last ObjectModification: 2019_01_26-PM-06_15_00

Theory : continuity

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