Nuprl Lemma : trans-id-property
∀C1,C2:SmallCategory. ∀x,y:Functor(C1;C2). ∀f:nat-trans(C1;C2;x;y).
((identity-trans(C1;C2;x) o f = f ∈ nat-trans(C1;C2;x;y)) ∧ (f o identity-trans(C1;C2;y) = f ∈ nat-trans(C1;C2;x;y)))
Proof
Definitions occuring in Statement :
trans-comp: t1 o t2,
identity-trans: identity-trans(C;D;F),
nat-trans: nat-trans(C;D;F;G),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
all: ∀x:A. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
nat-trans: nat-trans(C;D;F;G),
identity-trans: identity-trans(C;D;F),
trans-comp: t1 o t2,
member: t ∈ T,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
uall: ∀[x:A]. B[x],
prop: ℙ
Lemmas referenced :
ap_mk_nat_trans_lemma,
cat-comp-ident,
functor-ob_wf,
cat-ob_wf,
all_wf,
cat-arrow_wf,
equal_wf,
cat-comp_wf,
functor-arrow_wf,
nat-trans_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
equalitySymmetry,
dependent_set_memberEquality,
functionExtensionality,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
productElimination,
lambdaEquality,
because_Cache,
independent_pairFormation
Latex:
\mforall{}C1,C2:SmallCategory. \mforall{}x,y:Functor(C1;C2). \mforall{}f:nat-trans(C1;C2;x;y).
((identity-trans(C1;C2;x) o f = f) \mwedge{} (f o identity-trans(C1;C2;y) = f))
Date html generated:
2020_05_20-AM-07_51_44
Last ObjectModification:
2017_01_10-PM-04_46_00
Theory : small!categories
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