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 :
pi2: snd(t),
top: Top,
all: ∀x:A. B[x],
cat-comp: cat-comp(C),
and: P ∧ Q,
spreadn: spread4,
small-category: SmallCategory,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat_arrow_triple_lemma,
cat_ob_pair_lemma
Rules used in proof :
hypothesisEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
sqequalRule,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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:
2017_01_11-AM-09_17_41
Last ObjectModification:
2017_01_10-PM-06_15_40
Theory : small!categories
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