Nuprl Lemma : mk-groupoid_wf
∀[C:SmallCategory]. ∀[inv:x:cat-ob(C) ⟶ y:cat-ob(C) ⟶ (cat-arrow(C) x y) ⟶ (cat-arrow(C) y x)].
Groupoid(C;
inv(x,y,f) = inv[x;y;f]) ∈ Groupoid
supposing ∀x,y:cat-ob(C). ∀f:cat-arrow(C) x y.
(((cat-comp(C) x y x f inv[x;y;f]) = (cat-id(C) x) ∈ (cat-arrow(C) x x))
∧ ((cat-comp(C) y x y inv[x;y;f] f) = (cat-id(C) y) ∈ (cat-arrow(C) y y)))
Proof
Definitions occuring in Statement :
mk-groupoid: mk-groupoid,
groupoid: Groupoid,
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
mk-groupoid: mk-groupoid,
groupoid: Groupoid,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
cat-ob_wf,
cat-arrow_wf,
all_wf,
equal_wf,
cat-comp_wf,
cat-id_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
dependent_pairEquality,
hypothesisEquality,
dependent_set_memberEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
because_Cache,
productEquality,
setEquality,
functionEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[inv:x:cat-ob(C) {}\mrightarrow{} y:cat-ob(C) {}\mrightarrow{} (cat-arrow(C) x y) {}\mrightarrow{} (cat-arrow(C) y x)].
Groupoid(C;
inv(x,y,f) = inv[x;y;f]) \mmember{} Groupoid
supposing \mforall{}x,y:cat-ob(C). \mforall{}f:cat-arrow(C) x y.
(((cat-comp(C) x y x f inv[x;y;f]) = (cat-id(C) x))
\mwedge{} ((cat-comp(C) y x y inv[x;y;f] f) = (cat-id(C) y)))
Date html generated:
2020_05_20-AM-07_55_21
Last ObjectModification:
2017_07_28-AM-09_20_10
Theory : small!categories
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