Nuprl Lemma : pscm-comp-context-map
∀[C:SmallCategory]. ∀[Gamma,Delta:ps_context{j:l}(C)]. ∀[sigma:psc_map{j:l}(C; Delta; Gamma)]. ∀[I:cat-ob(C)].
∀[rho:Delta(I)].
(sigma o <rho> = <(sigma)rho> ∈ psc_map{[i | j]:l}(C; Yoneda(I); Gamma))
Proof
Definitions occuring in Statement :
pscm-comp: G o F,
pscm-ap: (s)x,
ps-context-map: <rho>,
psc_map: A ⟶ B,
Yoneda: Yoneda(I),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
cat-comp: cat-comp(C),
compose: f o g,
uimplies: b supposing a,
I_set: A(I),
ps-context-map: <rho>,
pscm-comp: G o F,
functor-arrow: arrow(F),
psc-restriction: f(s),
Yoneda: Yoneda(I),
squash: ↓T,
prop: ℙ,
functor-ob: ob(F),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
pscm-ap: (s)x
Lemmas referenced :
pscm-equal,
Yoneda_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
pscm-comp_wf,
ps-context-map_wf,
subtype_rel_self,
psc_map_wf,
I_set_wf,
cat-ob_wf,
ps_context_wf,
small-category_wf,
pscm-ap_wf,
subtype_rel-equal,
op-cat_wf,
cat_ob_op_lemma,
ob_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
psc-restriction_wf,
pscm-ap-restriction,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
dependent_functionElimination,
functionExtensionality,
independent_isectElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
Error :memTop,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma,Delta:ps\_context\{j:l\}(C)]. \mforall{}[sigma:psc\_map\{j:l\}(C; Delta; Gamma)].
\mforall{}[I:cat-ob(C)]. \mforall{}[rho:Delta(I)].
(sigma o <rho> = <(sigma)rho>)
Date html generated:
2020_05_20-PM-01_27_08
Last ObjectModification:
2020_04_20-AM-10_34_58
Theory : presheaf!models!of!type!theory
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