### Nuprl Lemma : strong-continuous-depproduct

`∀[A:Type]. ∀[G:T:Type ⟶ A ⟶ Type].  Continuous+(T.x:A × G[T;x]) supposing ∀a:A. Continuous+(T.G[T;a])`

Proof

Definitions occuring in Statement :  strong-type-continuous: `Continuous+(T.F[T])` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` product: `x:A × B[x]` universe: `Type`
Definitions unfolded in proof :  so_apply: `x[s1;s2]` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` strong-type-continuous: `Continuous+(T.F[T])` ext-eq: `A ≡ B` and: `P ∧ Q` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` pi1: `fst(t)` pi2: `snd(t)` istype: `istype(T)`
Lemmas referenced :  nat_wf strong-type-continuous_wf false_wf le_wf member_wf pi2_wf subtype_rel_product
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  introduction cut independent_pairFormation Error :lambdaEquality_alt,  Error :isectIsType,  Error :universeIsType,  because_Cache Error :productIsType,  hypothesisEquality applyEquality Error :isect_memberEquality_alt,  productElimination thin Error :dependent_pairEquality_alt,  functionExtensionality universeEquality extract_by_obid hypothesis isectEquality sqequalHypSubstitution independent_pairEquality axiomEquality Error :functionIsType,  Error :inhabitedIsType,  isectElimination equalityTransitivity equalitySymmetry rename Error :dependent_set_memberEquality_alt,  natural_numberEquality Error :lambdaFormation_alt,  Error :equalityIsType1,  dependent_functionElimination independent_functionElimination cumulativity applyLambdaEquality hyp_replacement dependent_pairEquality lambdaEquality independent_isectElimination

Latex:
\mforall{}[A:Type].  \mforall{}[G:T:Type  {}\mrightarrow{}  A  {}\mrightarrow{}  Type].
Continuous+(T.x:A  \mtimes{}  G[T;x])  supposing  \mforall{}a:A.  Continuous+(T.G[T;a])

Date html generated: 2019_06_20-PM-00_27_45
Last ObjectModification: 2018_09_30-PM-00_43_44

Theory : subtype_1

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