Nuprl Lemma : pscm-ap-presheaf-app-fun
∀[C:SmallCategory]. ∀[X,Delta:ps_context{j:l}(C)]. ∀[A,B:{X ⊢ _}]. ∀[w:{X ⊢ _:(A ⟶ B)}]. ∀[u:{X ⊢ _:A}].
∀[s:psc_map{j:l}(C; Delta; X)].
((app(w; u))s = app((w)s; (u)s) ∈ {Delta ⊢ _:(B)s})
Proof
Definitions occuring in Statement :
presheaf-app: app(w; u),
presheaf-fun: (A ⟶ B),
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
pscm-presheaf-app,
presheaf-app_wf_fun,
pscm-ap-type_wf,
presheaf-term_wf,
squash_wf,
true_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
equal_wf,
istype-universe,
ps_context_cumulativity2,
pscm-presheaf-fun,
presheaf-type-cumulativity2,
presheaf-fun_wf,
subtype_rel_self,
iff_weakening_equal,
pscm-ap-term_wf,
psc_map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Delta:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}].
\mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[s:psc\_map\{j:l\}(C; Delta; X)].
((app(w; u))s = app((w)s; (u)s))
Date html generated:
2020_05_20-PM-01_33_19
Last ObjectModification:
2020_04_02-PM-06_31_09
Theory : presheaf!models!of!type!theory
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