Nuprl Lemma : prod-image-is-image
∀[A:Type]. ∀[f:Base]. ∀[B:Image(A,f) ⟶ Type]. ∀[g:Base].
y:Image(A,f) × Image(B[y],g) ≡ Image((z:A × B[f z]),(λp.let a,b = p
in <f a, g b>))
Proof
Definitions occuring in Statement :
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
image-type: Image(T,f),
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
spread: spread def,
pair: <a, b>,
product: x:A × B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s]
Lemmas referenced :
base_wf,
image-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaEquality,
productElimination,
thin,
productEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
cumulativity,
imageMemberEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality,
imageElimination,
rename,
dependent_pairEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:Base]. \mforall{}[B:Image(A,f) {}\mrightarrow{} Type]. \mforall{}[g:Base].
y:Image(A,f) \mtimes{} Image(B[y],g) \mequiv{} Image((z:A \mtimes{} B[f z]),(\mlambda{}p.let a,b = p
in <f a, g b>))
Date html generated:
2016_05_13-PM-04_14_08
Last ObjectModification:
2016_01_14-PM-07_28_47
Theory : subtype_1
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