Nuprl Lemma : counit-unit-equations_wf
∀[A,B:SmallCategory]. ∀[F:Functor(A;B)]. ∀[G:Functor(B;A)]. ∀[eps:b:cat-ob(B) ⟶ (cat-arrow(B) (F (G b)) b)].
∀[eta:a:cat-ob(A) ⟶ (cat-arrow(A) a (G (F a)))].
(counit-unit-equations(A;B;F;G;eps;eta) ∈ ℙ)
Proof
Definitions occuring in Statement :
counit-unit-equations: counit-unit-equations(D;C;F;G;eps;eta),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
counit-unit-equations: counit-unit-equations(D;C;F;G;eps;eta),
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
all_wf,
cat-ob_wf,
equal_wf,
cat-arrow_wf,
functor-ob_wf,
cat-comp_wf,
functor-arrow_wf,
cat-id_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
productEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
because_Cache,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality
Latex:
\mforall{}[A,B:SmallCategory]. \mforall{}[F:Functor(A;B)]. \mforall{}[G:Functor(B;A)]. \mforall{}[eps:b:cat-ob(B) {}\mrightarrow{} (cat-arrow(B)
(F (G b))
b)].
\mforall{}[eta:a:cat-ob(A) {}\mrightarrow{} (cat-arrow(A) a (G (F a)))].
(counit-unit-equations(A;B;F;G;eps;eta) \mmember{} \mBbbP{})
Date html generated:
2020_05_20-AM-07_58_08
Last ObjectModification:
2017_01_16-PM-00_19_09
Theory : small!categories
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