Nuprl Lemma : mk-nat-trans_wf
∀[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[trans:A:cat-ob(C) ⟶ (cat-arrow(D) (functor-ob(F) A) (functor-ob(G) A))].
x |→ trans[x] ∈ nat-trans(C;D;F;G)
supposing ∀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) trans[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) trans[B])
∈ (cat-arrow(D) (functor-ob(F) A) (functor-ob(G) B)))
Proof
Definitions occuring in Statement :
mk-nat-trans: x |→ T[x],
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
so_apply: x[s],
mk-nat-trans: x |→ T[x],
nat-trans: nat-trans(C;D;F;G),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-functor_wf,
functor-arrow_wf,
cat-comp_wf,
functor-ob_wf,
equal_wf,
all_wf,
cat-arrow_wf,
cat-ob_wf
Rules used in proof :
functionEquality,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
because_Cache,
dependent_functionElimination,
sqequalRule,
lambdaFormation,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesisEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
dependent_set_memberEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G:Functor(C;D)]. \mforall{}[trans:A:cat-ob(C) {}\mrightarrow{} (cat-arrow(D) (functor-ob(F) A)
(functor-ob(G) A))].
x |\mrightarrow{} trans[x] \mmember{} nat-trans(C;D;F;G)
supposing \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) trans[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)
trans[B]))
Date html generated:
2017_01_11-AM-09_18_05
Last ObjectModification:
2017_01_10-PM-04_23_33
Theory : small!categories
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