### Nuprl Lemma : weak-continuity-skolem_functionality

`∀[T,S:Type].`
`  ∀e:T ~ S. ∀F:(ℕ ⟶ S) ⟶ ℕ.  (weak-continuity-skolem(T;λf.(F ((fst(e)) o f))) `` weak-continuity-skolem(S;F))`

Proof

Definitions occuring in Statement :  weak-continuity-skolem: `weak-continuity-skolem(T;F)` equipollent: `A ~ B` compose: `f o g` nat: `ℕ` uall: `∀[x:A]. B[x]` pi1: `fst(t)` 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 :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` equipollent: `A ~ B` exists: `∃x:A. B[x]` pi1: `fst(t)` implies: `P `` Q` member: `t ∈ T` and: `P ∧ Q` weak-continuity-skolem: `weak-continuity-skolem(T;F)` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` compose: `f o g` nat: `ℕ` squash: `↓T` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  biject-inverse weak-continuity-skolem_wf nat_wf compose_wf equipollent_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf all_wf and_wf squash_wf true_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin sqequalRule cut introduction extract_by_obid isectElimination hypothesisEquality independent_functionElimination hypothesis rename cumulativity lambdaEquality applyEquality functionExtensionality functionEquality universeEquality dependent_pairFormation natural_numberEquality because_Cache independent_isectElimination independent_pairFormation dependent_functionElimination equalitySymmetry dependent_set_memberEquality equalityTransitivity setElimination applyLambdaEquality hyp_replacement imageElimination imageMemberEquality baseClosed

Latex:
\mforall{}[T,S:Type].
\mforall{}e:T  \msim{}  S.  \mforall{}F:(\mBbbN{}  {}\mrightarrow{}  S)  {}\mrightarrow{}  \mBbbN{}.
(weak-continuity-skolem(T;\mlambda{}f.(F  ((fst(e))  o  f)))  {}\mRightarrow{}  weak-continuity-skolem(S;F))

Date html generated: 2017_04_17-AM-09_54_00
Last ObjectModification: 2017_02_27-PM-05_49_00

Theory : continuity

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