Nuprl Lemma : cat-inverse-unique
∀[C:SmallCategory]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x y]. ∀[g2,g1:cat-arrow(C) y x].
(g1 = g2 ∈ (cat-arrow(C) y x)) supposing ((∃h:cat-arrow(C) y x. hf=1) and fg2=1 and fg1=1)
Proof
Definitions occuring in Statement :
cat-inverse: fg=1,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
exists: ∃x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ
Lemmas referenced :
left-right-inverse-unique,
exists_wf,
cat-arrow_wf,
cat-inverse_wf,
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
applyEquality,
sqequalRule,
lambdaEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[x,y:cat-ob(C)]. \mforall{}[f:cat-arrow(C) x y]. \mforall{}[g2,g1:cat-arrow(C) y x].
(g1 = g2) supposing ((\mexists{}h:cat-arrow(C) y x. hf=1) and fg2=1 and fg1=1)
Date html generated:
2017_01_09-AM-09_11_15
Last ObjectModification:
2017_01_08-PM-01_00_35
Theory : small!categories
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