Nuprl Lemma : groupoid-left-cancellation
∀[G:Groupoid]. ∀[x,y,z:cat-ob(cat(G))]. ∀[a,b:cat-arrow(cat(G)) y z]. ∀[c:cat-arrow(cat(G)) x y].
uiff((cat-comp(cat(G)) x y z c a) = (cat-comp(cat(G)) x y z c b) ∈ (cat-arrow(cat(G)) x z);a
= b
∈ (cat-arrow(cat(G)) y z))
Proof
Definitions occuring in Statement :
groupoid-cat: cat(G),
groupoid: Groupoid,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
true: True,
squash: ↓T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
cat-arrow_wf,
groupoid-cat_wf,
cat-comp_wf,
and_wf,
cat-ob_wf,
groupoid_wf,
groupoid-inv_wf,
squash_wf,
true_wf,
cat-comp-assoc,
groupoid_inv,
cat-comp-ident,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
productElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
natural_numberEquality,
lambdaEquality,
imageElimination,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[G:Groupoid]. \mforall{}[x,y,z:cat-ob(cat(G))]. \mforall{}[a,b:cat-arrow(cat(G)) y z]. \mforall{}[c:cat-arrow(cat(G)) x y].
uiff((cat-comp(cat(G)) x y z c a) = (cat-comp(cat(G)) x y z c b);a = b)
Date html generated:
2020_05_20-AM-07_55_52
Last ObjectModification:
2017_07_28-AM-09_20_14
Theory : small!categories
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