Nuprl Lemma : retraction-epic
∀[C:SmallCategory]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x y]. epic(f) supposing retraction(f)
Proof
Definitions occuring in Statement :
cat-epic: epic(f),
cat-retraction: retraction(g),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
cat-epic: epic(f),
member: t ∈ T,
cat-retraction: retraction(g),
exists: ∃x:A. B[x],
cat-inverse: fg=1,
prop: ℙ,
true: True,
squash: ↓T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
cat-arrow_wf,
cat-comp_wf,
cat-retraction_wf,
cat-ob_wf,
small-category_wf,
squash_wf,
true_wf,
cat-comp-assoc,
cat-comp-ident,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
rename,
hypothesis,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
natural_numberEquality,
lambdaEquality,
imageElimination,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[x,y:cat-ob(C)]. \mforall{}[f:cat-arrow(C) x y]. epic(f) supposing retraction(f)
Date html generated:
2020_05_20-AM-07_50_31
Last ObjectModification:
2017_07_28-AM-09_19_10
Theory : small!categories
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