Nuprl Lemma : csm-comp-assoc
∀[A,B,C,D:CubicalSet]. ∀[F:A ⟶ B]. ∀[G:B ⟶ C]. ∀[H:C ⟶ D].  (H o G o F = H o G o F ∈ A ⟶ D)
Proof
Definitions occuring in Statement : 
csm-comp: G o F
, 
cube-set-map: A ⟶ B
, 
cubical-set: CubicalSet
, 
uall: ∀[x:A]. B[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
csm-comp: G o F
, 
cube-set-map: A ⟶ B
, 
ext-eq: A ≡ B
, 
and: P ∧ Q
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
, 
all: ∀x:A. B[x]
, 
true: True
, 
uimplies: b supposing a
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
Lemmas referenced : 
cubical-set-is-functor, 
equal_wf, 
nat-trans_wf, 
name-cat_wf, 
type-cat_wf, 
trans-comp-assoc, 
trans-comp_wf, 
iff_weakening_equal, 
cube-set-map_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
productElimination, 
thin, 
applyEquality, 
instantiate, 
lambdaEquality, 
imageElimination, 
isectElimination, 
because_Cache, 
hypothesis, 
hypothesisEquality, 
sqequalRule, 
dependent_functionElimination, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
equalityTransitivity, 
equalitySymmetry, 
independent_isectElimination, 
independent_functionElimination, 
isect_memberEquality, 
axiomEquality
Latex:
\mforall{}[A,B,C,D:CubicalSet].  \mforall{}[F:A  {}\mrightarrow{}  B].  \mforall{}[G:B  {}\mrightarrow{}  C].  \mforall{}[H:C  {}\mrightarrow{}  D].    (H  o  G  o  F  =  H  o  G  o  F)
Date html generated:
2017_10_05-AM-10_11_16
Last ObjectModification:
2017_07_28-AM-11_17_50
Theory : cubical!sets
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