Nuprl Lemma : trans-comp_wf
∀[C,D:SmallCategory]. ∀[F,G,H:Functor(C;D)]. ∀[t1:nat-trans(C;D;F;G)]. ∀[t2:nat-trans(C;D;G;H)].
(t1 o t2 ∈ nat-trans(C;D;F;H))
Proof
Definitions occuring in Statement :
trans-comp: t1 o t2,
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,
trans-comp: t1 o t2,
so_lambda: λ2x.t[x],
nat-trans: nat-trans(C;D;F;G),
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
true: True,
squash: ↓T,
prop: ℙ,
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-comp_wf,
functor-ob_wf,
cat-ob_wf,
cat-arrow_wf,
nat-trans_wf,
cat-functor_wf,
small-category_wf,
functor-arrow_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cat-comp-assoc,
nat-trans-equation,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
applyEquality,
because_Cache,
hypothesis,
setElimination,
rename,
universeIsType,
independent_isectElimination,
lambdaFormation_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
natural_numberEquality,
imageElimination,
instantiate,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G,H:Functor(C;D)]. \mforall{}[t1:nat-trans(C;D;F;G)]. \mforall{}[t2:nat-trans(C;D;G;H)].
(t1 o t2 \mmember{} nat-trans(C;D;F;H))
Date html generated:
2020_05_20-AM-07_51_40
Last ObjectModification:
2019_12_30-PM-05_48_14
Theory : small!categories
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