Nuprl Lemma : ps_context-ext
∀[C:SmallCategory]
{FM:F:cat-ob(C) ⟶ 𝕌{j'} × (I:cat-ob(C) ⟶ J:cat-ob(C) ⟶ (cat-arrow(C) J I) ⟶ (F I) ⟶ (F J))|
let F,M = FM
in (∀I:cat-ob(C). ∀s:FM(I). (cat-id(C) I(s) = s ∈ FM(I)))
∧ (∀I,J,K:cat-ob(C). ∀f:cat-arrow(C) J I. ∀g:cat-arrow(C) K J. ∀s:FM(I).
(cat-comp(C) K J I g f(s) = g(f(s)) ∈ FM(K)))} ≡ ps_context{j:l}(C)
Proof
Definitions occuring in Statement :
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T,
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
small-category: SmallCategory,
spreadn: spread4,
ps_context: __⊢,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
op-cat: op-cat(C),
cat-functor: Functor(C1;C2),
compose: f o g,
psc-restriction: f(s),
I_set: A(I),
functor-ob: ob(F),
ext-eq: A ≡ B,
cand: A c∧ B,
guard: {T},
true: True,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
sq_stable__ext-eq,
small-category_wf,
cat-ob_wf,
cat-arrow_wf,
I_set_pair_redex_lemma,
psc_restriction_pair_lemma,
equal_wf,
cat-id_wf,
cat-comp_wf,
ps_context_wf,
small-category-cumulativity-2,
cat_id_tuple_lemma,
cat_comp_tuple_lemma,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
istype-universe,
iff_weakening_equal,
squash_wf,
true_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
imageElimination,
universeIsType,
setEquality,
productEquality,
functionEquality,
cumulativity,
universeEquality,
applyEquality,
productElimination,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
independent_pairFormation,
lambdaEquality_alt,
setIsType,
productIsType,
functionIsType,
because_Cache,
equalityIstype,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
inhabitedIsType,
lambdaFormation_alt,
functionExtensionality_alt,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination
Latex:
\mforall{}[C:SmallCategory]
\{FM:F:cat-ob(C) {}\mrightarrow{} \mBbbU{}\{j'\} \mtimes{} (I:cat-ob(C) {}\mrightarrow{} J:cat-ob(C) {}\mrightarrow{} (cat-arrow(C) J I) {}\mrightarrow{} (F I) {}\mrightarrow{} (F J))|
let F,M = FM
in (\mforall{}I:cat-ob(C). \mforall{}s:FM(I). (cat-id(C) I(s) = s))
\mwedge{} (\mforall{}I,J,K:cat-ob(C). \mforall{}f:cat-arrow(C) J I. \mforall{}g:cat-arrow(C) K J. \mforall{}s:FM(I).
(cat-comp(C) K J I g f(s) = g(f(s))))\} \mequiv{} ps\_context\{j:l\}(C)
Date html generated:
2020_05_20-PM-01_23_22
Last ObjectModification:
2020_03_31-PM-07_30_09
Theory : presheaf!models!of!type!theory
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