Nuprl Lemma : cat-comp_wf
∀[C:SmallCategory]
(cat-comp(C) ∈ x:cat-ob(C)
⟶ y:cat-ob(C)
⟶ z:cat-ob(C)
⟶ (cat-arrow(C) x y)
⟶ (cat-arrow(C) y z)
⟶ (cat-arrow(C) x z))
Proof
Definitions occuring in Statement :
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory,
spreadn: spread4,
and: P ∧ Q,
cat-comp: cat-comp(C),
all: ∀x:A. B[x],
top: Top,
pi2: snd(t)
Lemmas referenced :
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
hypothesisEquality
Latex:
\mforall{}[C:SmallCategory]
(cat-comp(C) \mmember{} x:cat-ob(C)
{}\mrightarrow{} y:cat-ob(C)
{}\mrightarrow{} z:cat-ob(C)
{}\mrightarrow{} (cat-arrow(C) x y)
{}\mrightarrow{} (cat-arrow(C) y z)
{}\mrightarrow{} (cat-arrow(C) x z))
Date html generated:
2020_05_20-AM-07_49_37
Last ObjectModification:
2017_01_10-PM-06_15_40
Theory : small!categories
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