Nuprl Lemma : trans-id-property
∀C1,C2:SmallCategory. ∀x,y:Functor(C1;C2). ∀f:nat-trans(C1;C2;x;y).
((trans-comp(C1;C2;x;x;y;identity-trans(C1;C2;x);f) = f ∈ nat-trans(C1;C2;x;y))
∧ (trans-comp(C1;C2;x;y;y;f;identity-trans(C1;C2;y)) = f ∈ nat-trans(C1;C2;x;y)))
Proof
Definitions occuring in Statement :
trans-comp: trans-comp(C;D;F;G;H;t1;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 :
prop: ℙ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
member: t ∈ T,
trans-comp: trans-comp(C;D;F;G;H;t1;t2),
identity-trans: identity-trans(C;D;F),
nat-trans: nat-trans(C;D;F;G),
cand: A c∧ B,
and: P ∧ Q,
all: ∀x:A. B[x]
Lemmas referenced :
small-category_wf,
cat-functor_wf,
nat-trans_wf,
functor-arrow_wf,
cat-comp_wf,
equal_wf,
cat-arrow_wf,
all_wf,
cat-ob_wf,
functor-ob_wf,
cat-comp-ident,
ap_mk_nat_trans_lemma
Rules used in proof :
independent_pairFormation,
because_Cache,
lambdaEquality,
productElimination,
applyEquality,
hypothesisEquality,
isectElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
sqequalRule,
functionExtensionality,
dependent_set_memberEquality,
equalitySymmetry,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}C1,C2:SmallCategory. \mforall{}x,y:Functor(C1;C2). \mforall{}f:nat-trans(C1;C2;x;y).
((trans-comp(C1;C2;x;x;y;identity-trans(C1;C2;x);f) = f)
\mwedge{} (trans-comp(C1;C2;x;y;y;f;identity-trans(C1;C2;y)) = f))
Date html generated:
2017_01_11-AM-09_18_21
Last ObjectModification:
2017_01_10-PM-04_46_00
Theory : small!categories
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