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) (functor-ob(F) A) (functor-ob(G) A) (functor-ob(G) B) (T A) (functor-arrow(G) A B g))
= (cat-comp(D) (functor-ob(F) A) (functor-ob(F) B) (functor-ob(G) B) (functor-arrow(F) A B g) (T B))
∈ (cat-arrow(D) (functor-ob(F) A) (functor-ob(G) B)))
Proof
Definitions occuring in Statement :
nat-trans: nat-trans(C;D;F;G),
functor-arrow: functor-arrow(F),
functor-ob: functor-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 :
all: ∀x:A. B[x],
nat-trans: nat-trans(C;D;F;G),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-functor_wf,
nat-trans_wf,
cat-ob_wf,
cat-arrow_wf
Rules used in proof :
because_Cache,
axiomEquality,
isect_memberEquality,
sqequalRule,
isectElimination,
extract_by_obid,
applyEquality,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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) (functor-ob(F) A) (functor-ob(G) A) (functor-ob(G) B) (T A)
(functor-arrow(G) A B g))
= (cat-comp(D) (functor-ob(F) A) (functor-ob(F) B) (functor-ob(G) B) (functor-arrow(F) A B g)
(T B)))
Date html generated:
2017_01_11-AM-09_18_01
Last ObjectModification:
2017_01_10-PM-00_07_17
Theory : small!categories
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