Nuprl Lemma : ps-ps-context-map-comp
∀[C:SmallCategory]. ∀[I,J,K:cat-ob(C)]. ∀[f:cat-arrow(C) J I]. ∀[g:cat-arrow(C) K J].
(<cat-comp(C) K J I g f> = <f> o <g> ∈ Yoneda(K) ⟶ Yoneda(I))
Proof
Definitions occuring in Statement :
pscm-comp: G o F,
ps-context-map: <rho>,
psc_map: A ⟶ B,
Yoneda: Yoneda(I),
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
cat-comp: cat-comp(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,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
I_set: A(I),
functor-ob: ob(F),
Yoneda: Yoneda(I),
uimplies: b supposing a,
ps-context-map: <rho>,
pscm-comp: G o F,
compose: f o g
Lemmas referenced :
pscm-equal2,
Yoneda_wf,
ps-context-map_wf,
cat-comp_wf,
subtype_rel_self,
I_set_wf,
pscm-comp_wf,
I_set_pair_redex_lemma,
arrow_pair_lemma,
cat-comp-assoc,
cat-arrow_wf,
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
applyEquality,
sqequalRule,
independent_isectElimination,
lambdaFormation_alt,
Error :memTop,
equalitySymmetry,
universeIsType,
because_Cache
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[I,J,K:cat-ob(C)]. \mforall{}[f:cat-arrow(C) J I]. \mforall{}[g:cat-arrow(C) K J].
(<cat-comp(C) K J I g f> = <f> o <g>)
Date html generated:
2020_05_20-PM-01_24_22
Last ObjectModification:
2020_04_03-PM-01_04_26
Theory : presheaf!models!of!type!theory
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