### Nuprl Lemma : strong-continuity2_biject_retract-ext

`∀[T,S,U:Type].`
`  ∀r:ℕ ⟶ U`
`    ((U ⊆r ℕ)`
`    `` (∀x:U. ((r x) = x ∈ U))`
`    `` (∀g:S ⟶ U`
`          (Bij(S;U;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` subtype_rel: `A ⊆r B` 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 :  member: `t ∈ T` bij_inv: `bij_inv(bi)` pi1: `fst(t)` pi2: `snd(t)` strong-continuity2_biject_retract biject-inverse uall: `∀[x:A]. B[x]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` so_apply: `x[s1;s2;s3;s4]` top: `Top` uimplies: `b supposing a` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]`
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution equalityTransitivity equalitySymmetry isectElimination baseClosed Error :isect_memberEquality_alt,  voidElimination independent_isectElimination

Latex:
\mforall{}[T,S,U:Type].
\mforall{}r:\mBbbN{}  {}\mrightarrow{}  U
((U  \msubseteq{}r  \mBbbN{})
{}\mRightarrow{}  (\mforall{}x:U.  ((r  x)  =  x))
{}\mRightarrow{}  (\mforall{}g:S  {}\mrightarrow{}  U
(Bij(S;U;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: 2019_06_20-PM-02_50_34
Last ObjectModification: 2019_03_26-AM-07_44_57

Theory : continuity

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