Nuprl Lemma : pscm-ap-presheaf-pair
∀[C:SmallCategory]. ∀[X,Delta:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}].
∀[s:psc_map{j:l}(C; Delta; X)].
((presheaf-pair(u;v))s = presheaf-pair((u)s;(v)s) ∈ {Delta ⊢ _:(Σ A B)s})
Proof
Definitions occuring in Statement :
presheaf-pair: presheaf-pair(u;v),
presheaf-sigma: Σ A B,
pscm-id-adjoin: [u],
psc-adjoin: X.A,
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,
subtype_rel: A ⊆r B
Lemmas referenced :
pscm-presheaf-pair,
pscm-ap-term_wf,
presheaf-sigma_wf,
presheaf-pair_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
psc-adjoin_wf,
psc_map_wf,
presheaf-term_wf,
pscm-ap-type_wf,
pscm-id-adjoin_wf,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
universeIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Delta:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}].
\mforall{}[v:\{X \mvdash{} \_:(B)[u]\}]. \mforall{}[s:psc\_map\{j:l\}(C; Delta; X)].
((presheaf-pair(u;v))s = presheaf-pair((u)s;(v)s))
Date html generated:
2020_05_20-PM-01_33_13
Last ObjectModification:
2020_04_02-PM-06_30_48
Theory : presheaf!models!of!type!theory
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