Nuprl Lemma : unit-cube-map-comp
∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K).
(unit-cube-map((f o g)) = unit-cube-map(f) o unit-cube-map(g) ∈ unit-cube(K) ⟶ unit-cube(I))
Proof
Definitions occuring in Statement :
unit-cube-map: unit-cube-map(f),
unit-cube: unit-cube(I),
csm-comp: G o F,
cube-set-map: A ⟶ B,
name-comp: (f o g),
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
cube-set-map: A ⟶ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
nat-trans: nat-trans(C;D;F;G),
unit-cube-map: unit-cube-map(f),
unit-cube: unit-cube(I),
csm-comp: G o F,
functor-ob: ob(F),
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
name-cat: NameCat,
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
compose: f o g,
trans-comp: t1 o t2,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-set-is-functor,
nat-trans-equal,
name-cat_wf,
type-cat_wf,
unit-cube_wf,
subtype_rel_weakening,
cubical-set_wf,
cat-functor_wf,
csm-comp_wf,
unit-cube-map_wf,
name-comp_wf,
cube-set-map_wf,
name-morph_wf,
list_wf,
coordinate_name_wf,
cat_comp_tuple_lemma,
ob_pair_lemma,
ap_mk_nat_trans_lemma,
equal_wf,
squash_wf,
true_wf,
name-comp-assoc,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
equalitySymmetry,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
hypothesis,
hypothesisEquality,
applyEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionExtensionality,
imageElimination,
equalityTransitivity,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}I,J,K:Cname List. \mforall{}f:name-morph(I;J). \mforall{}g:name-morph(J;K).
(unit-cube-map((f o g)) = unit-cube-map(f) o unit-cube-map(g))
Date html generated:
2017_10_05-AM-10_12_18
Last ObjectModification:
2017_07_28-AM-11_18_12
Theory : cubical!sets
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