Nuprl Lemma : identity-trans_wf
∀[C,D:SmallCategory]. ∀[F:Functor(C;D)]. (identity-trans(C;D;F) ∈ nat-trans(C;D;F;F))
Proof
Definitions occuring in Statement :
identity-trans: identity-trans(C;D;F),
nat-trans: nat-trans(C;D;F;G),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
identity-trans: identity-trans(C;D;F),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
mk-nat-trans_wf,
cat-id_wf,
functor-ob_wf,
cat-ob_wf,
equal_wf,
squash_wf,
true_wf,
cat-arrow_wf,
cat-comp-ident1,
functor-arrow_wf,
cat-comp_wf,
iff_weakening_equal,
cat-comp-ident2,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
lambdaEquality,
applyEquality,
hypothesis,
independent_isectElimination,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F:Functor(C;D)]. (identity-trans(C;D;F) \mmember{} nat-trans(C;D;F;F))
Date html generated:
2020_05_20-AM-07_51_35
Last ObjectModification:
2017_07_28-AM-09_19_19
Theory : small!categories
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