Nuprl Lemma : cat-comp-ident
∀[C:SmallCategory]
∀x,y:cat-ob(C). ∀f:cat-arrow(C) x y.
(((cat-comp(C) x x y (cat-id(C) x) f) = f ∈ (cat-arrow(C) x y))
∧ ((cat-comp(C) x y y f (cat-id(C) y)) = f ∈ (cat-arrow(C) x y)))
Proof
Definitions occuring in Statement :
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
cand: A c∧ B,
and: P ∧ Q,
spreadn: spread4,
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
cat-arrow: cat-arrow(C),
small-category: SmallCategory,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-ob_wf,
cat-arrow_wf
Rules used in proof :
because_Cache,
axiomEquality,
independent_pairEquality,
dependent_functionElimination,
lambdaEquality,
hypothesisEquality,
isectElimination,
lemma_by_obid,
applyEquality,
hypothesis,
independent_pairFormation,
sqequalRule,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C:SmallCategory]
\mforall{}x,y:cat-ob(C). \mforall{}f:cat-arrow(C) x y.
(((cat-comp(C) x x y (cat-id(C) x) f) = f) \mwedge{} ((cat-comp(C) x y y f (cat-id(C) y)) = f))
Date html generated:
2020_05_20-AM-07_50_00
Last ObjectModification:
2015_12_28-PM-02_23_55
Theory : small!categories
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