Nuprl Lemma : cube-set-restriction-comp
∀X:CubicalSet. ∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀a:X(I). (g(f(a)) = (f o g)(a) ∈ X(K))
Proof
Definitions occuring in Statement :
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
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],
member: t ∈ T,
cubical-set: CubicalSet,
cube-set-restriction: f(s),
pi2: snd(t),
and: P ∧ Q,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
I-cube: X(I),
top: Top,
subtype_rel: A ⊆r B,
functor-ob: ob(F),
pi1: fst(t),
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
compose: f o g
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
ob_pair_lemma,
subtype_rel_self,
iff_weakening_equal,
compose_wf,
name-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
functionExtensionality,
because_Cache,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}X:CubicalSet. \mforall{}I,J,K:Cname List. \mforall{}f:name-morph(I;J). \mforall{}g:name-morph(J;K). \mforall{}a:X(I).
(g(f(a)) = (f o g)(a))
Date html generated:
2017_10_05-AM-10_11_58
Last ObjectModification:
2017_07_28-AM-11_18_01
Theory : cubical!sets
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