Nuprl Lemma : cube-set-restriction-comp2
∀X:CubicalSet. ∀I,J,J2,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) supposing J = J2 ∈ (Cname List)
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,
uimplies: b supposing a,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
and_wf,
equal_wf,
list_wf,
coordinate_name_wf,
cube-set-restriction_wf,
subtype_base_sq,
list_subtype_base,
set_subtype_base,
le_wf,
int_subtype_base,
subtype_rel_self,
name-morph_wf,
I-cube_wf,
squash_wf,
true_wf,
length_wf_nat,
nat_wf,
subtype_rel_wf,
member_wf,
cube-set-restriction-comp,
iff_weakening_equal,
all_wf,
cubical-set_wf,
isect_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
dependent_set_memberEquality,
hypothesis,
independent_pairFormation,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyLambdaEquality,
setElimination,
rename,
productElimination,
applyEquality,
instantiate,
cumulativity,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
dependent_functionElimination,
independent_functionElimination,
equalitySymmetry,
because_Cache,
addLevel,
allFunctionality,
imageElimination,
equalityTransitivity,
universeEquality,
hyp_replacement,
imageMemberEquality,
baseClosed
Latex:
\mforall{}X:CubicalSet. \mforall{}I,J,J2,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) supposing J = J2
Date html generated:
2017_10_05-AM-10_12_01
Last ObjectModification:
2017_03_03-AM-01_52_42
Theory : cubical!sets
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