Nuprl Lemma : groupoid_inv
∀[G:Groupoid]
∀x,y:cat-ob(cat(G)). ∀f:cat-arrow(cat(G)) x y.
(((cat-comp(cat(G)) x y x f groupoid-inv(G;x;y;f)) = (cat-id(cat(G)) x) ∈ (cat-arrow(cat(G)) x x))
∧ ((cat-comp(cat(G)) y x y groupoid-inv(G;x;y;f) f) = (cat-id(cat(G)) y) ∈ (cat-arrow(cat(G)) y y)))
Proof
Definitions occuring in Statement :
groupoid-inv: groupoid-inv(G;x;y;x_y),
groupoid-cat: cat(G),
groupoid: Groupoid,
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
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},
pi2: snd(t),
groupoid-inv: groupoid-inv(G;x;y;x_y),
pi1: fst(t),
groupoid-cat: cat(G),
groupoid: Groupoid,
cand: A c∧ B,
and: P ∧ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
groupoid_wf,
cat-ob_wf,
groupoid-cat_wf,
cat-arrow_wf
Rules used in proof :
because_Cache,
axiomEquality,
independent_pairEquality,
dependent_functionElimination,
lambdaEquality,
hypothesisEquality,
isectElimination,
lemma_by_obid,
applyEquality,
independent_pairFormation,
hypothesis,
sqequalRule,
rename,
setElimination,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[G:Groupoid]
\mforall{}x,y:cat-ob(cat(G)). \mforall{}f:cat-arrow(cat(G)) x y.
(((cat-comp(cat(G)) x y x f groupoid-inv(G;x;y;f)) = (cat-id(cat(G)) x))
\mwedge{} ((cat-comp(cat(G)) y x y groupoid-inv(G;x;y;f) f) = (cat-id(cat(G)) y)))
Date html generated:
2020_05_20-AM-07_55_16
Last ObjectModification:
2015_12_28-PM-02_23_11
Theory : small!categories
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