Nuprl Lemma : nat-trans-equation
∀[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[T:nat-trans(C;D;F;G)]. ∀[A,B:cat-ob(C)]. ∀[g:cat-arrow(C) A B].
((cat-comp(D) (F A) (G A) (G B) (T A) (G A B g))
= (cat-comp(D) (F A) (F B) (G B) (F A B g) (T 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,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
nat-trans: nat-trans(C;D;F;G),
all: ∀x:A. B[x],
member: t ∈ T
Lemmas referenced :
cat-arrow_wf,
cat-ob_wf,
nat-trans_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
universeIsType,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
inhabitedIsType
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G:Functor(C;D)]. \mforall{}[T:nat-trans(C;D;F;G)]. \mforall{}[A,B:cat-ob(C)].
\mforall{}[g:cat-arrow(C) A B].
((cat-comp(D) (F A) (G A) (G B) (T A) (G A B g))
= (cat-comp(D) (F A) (F B) (G B) (F A B g) (T B)))
Date html generated:
2020_05_20-AM-07_51_18
Last ObjectModification:
2019_12_30-PM-02_11_00
Theory : small!categories
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