Nuprl Lemma : is-nat-trans
∀[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[trans:A:cat-ob(C) ⟶ (cat-arrow(D) (F A) (G A))].
trans ∈ nat-trans(C;D;F;G)
supposing ∀A,B:cat-ob(C). ∀g:cat-arrow(C) A B.
((cat-comp(D) (F A) (G A) (G B) (trans A) (G A B g))
= (cat-comp(D) (F A) (F B) (G B) (F A B g) (trans B))
∈ (cat-arrow(D) (F A) (G B)))
Proof
Definitions occuring in Statement :
nat-trans: nat-trans(C;D;F;G),
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat-trans: nat-trans(C;D;F;G),
all: ∀x:A. B[x]
Lemmas referenced :
cat-ob_wf,
cat-arrow_wf,
functor-ob_wf,
cat-comp_wf,
functor-arrow_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
dependent_set_memberEquality_alt,
hypothesisEquality,
sqequalRule,
functionIsType,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
applyEquality,
equalityIstype,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G:Functor(C;D)]. \mforall{}[trans:A:cat-ob(C) {}\mrightarrow{} (cat-arrow(D) (F A) (G A))].
trans \mmember{} nat-trans(C;D;F;G)
supposing \mforall{}A,B:cat-ob(C). \mforall{}g:cat-arrow(C) A B.
((cat-comp(D) (F A) (G A) (G B) (trans A) (G A B g))
= (cat-comp(D) (F A) (F B) (G B) (F A B g) (trans B)))
Date html generated:
2019_10_31-AM-07_24_02
Last ObjectModification:
2018_12_13-AM-09_45_27
Theory : small!categories
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