Nuprl Lemma : csm-id-comp
∀[A,B:CubicalSet]. ∀[sigma:A ⟶ B]. ((sigma o 1(A) = sigma ∈ A ⟶ B) ∧ (1(B) o sigma = sigma ∈ A ⟶ B))
Proof
Definitions occuring in Statement :
csm-id: 1(X),
csm-comp: G o F,
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
csm-id: 1(X),
csm-comp: G o F,
cube-set-map: A ⟶ B,
ext-eq: A ≡ B,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-set-is-functor,
trans-id-property,
name-cat_wf,
type-cat_wf,
cube-set-map_wf,
cubical-set_wf,
nat-trans_wf,
equal_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
hypothesis,
instantiate,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
independent_pairEquality,
axiomEquality,
isectElimination,
isect_memberEquality,
because_Cache,
natural_numberEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[A,B:CubicalSet]. \mforall{}[sigma:A {}\mrightarrow{} B]. ((sigma o 1(A) = sigma) \mwedge{} (1(B) o sigma = sigma))
Date html generated:
2017_10_05-AM-10_11_19
Last ObjectModification:
2017_07_28-AM-11_17_53
Theory : cubical!sets
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