Nuprl Lemma : poset-functors-equal
∀C:SmallCategory. ∀I:Cname List. ∀F,G:Functor(poset-cat(I);C).
(F = G ∈ Functor(poset-cat(I);C)
⇐⇒ (∀f:name-morph(I;[]). ((ob(F) f) = (ob(G) f) ∈ cat-ob(C)))
∧ (∀x:nameset(I). ∀f:{f:name-morph(I;[])| (f x) = 0 ∈ ℕ2} .
((arrow(F) f flip(f;x) (λx.Ax))
= (arrow(G) f flip(f;x) (λx.Ax))
∈ (cat-arrow(C) (ob(F) f) (ob(F) flip(f;x))))))
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
nil: [],
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T,
axiom: Ax
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
int_seg: {i..j-},
squash: ↓T,
true: True,
rev_implies: P ⇐ Q,
guard: {T},
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
poset-functor-extends: poset-functor-extends(C;I;L;E;F)
Lemmas referenced :
cat-functor_wf,
poset-cat_wf,
list_wf,
coordinate_name_wf,
small-category_wf,
equal_wf,
and_wf,
functor-ob_wf,
cat-ob_wf,
subtype_rel_self,
nameset_wf,
extd-nameset_wf,
nil_wf,
all_wf,
assert_wf,
isname_wf,
name-morph_wf,
member-poset-cat-arrow,
subtype_rel_set,
equal-wf-T-base,
int_seg_wf,
name-morph-flip_wf,
poset-cat-arrow-flip,
set_wf,
extd-nameset-nil,
functor-arrow_wf,
squash_wf,
true_wf,
poset-extend-unique,
subtype_rel_dep_function,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
iff_weakening_equal,
cat-arrow_wf,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
because_Cache,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
productElimination,
equalityTransitivity,
functionEquality,
applyEquality,
sqequalRule,
setEquality,
lambdaEquality,
functionExtensionality,
natural_numberEquality,
baseClosed,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
addLevel,
levelHypothesis,
imageElimination,
imageMemberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
universeEquality,
productEquality,
instantiate
Latex:
\mforall{}C:SmallCategory. \mforall{}I:Cname List. \mforall{}F,G:Functor(poset-cat(I);C).
(F = G
\mLeftarrow{}{}\mRightarrow{} (\mforall{}f:name-morph(I;[]). ((ob(F) f) = (ob(G) f)))
\mwedge{} (\mforall{}x:nameset(I). \mforall{}f:\{f:name-morph(I;[])| (f x) = 0\} .
((arrow(F) f flip(f;x) (\mlambda{}x.Ax)) = (arrow(G) f flip(f;x) (\mlambda{}x.Ax)))))
Date html generated:
2017_10_05-PM-03_36_15
Last ObjectModification:
2017_07_28-AM-11_25_03
Theory : cubical!sets
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