### Nuprl Lemma : strong-continuity2_biject

`∀[T,S:Type].`
`  ∀g:S ⟶ ℕ`
`    (Bij(S;ℕ;g)`
`    `` (∀F:(ℕ ⟶ T) ⟶ S`
`          (strong-continuity2(T;g o F)`
`          `` (∃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 :  strong-continuity2: `strong-continuity2(T;F)` biject: `Bij(A;B;f)` compose: `f o g` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` 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 :  prop: `ℙ` nat: `ℕ` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]`
Lemmas referenced :  le_wf zero-le-nat subtype_rel_self nat_wf strong-continuity2_biject_retract
Rules used in proof :  universeEquality rename setElimination sqequalRule dependent_set_memberEquality applyEquality lambdaFormation because_Cache independent_functionElimination lambdaEquality dependent_functionElimination hypothesisEquality thin isectElimination sqequalHypSubstitution hypothesis isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution extract_by_obid introduction cut

Latex:
\mforall{}[T,S:Type].
\mforall{}g:S  {}\mrightarrow{}  \mBbbN{}
(Bij(S;\mBbbN{};g)
{}\mRightarrow{}  (\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  T)  {}\mrightarrow{}  S
(strong-continuity2(T;g  o  F)
{}\mRightarrow{}  (\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: 2017_09_29-PM-06_05_19
Last ObjectModification: 2017_09_04-AM-10_36_13

Theory : continuity

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