Nuprl Lemma : cat-comp-assoc
∀[C:SmallCategory]
∀x,y,z,w:cat-ob(C). ∀f:cat-arrow(C) x y. ∀g:cat-arrow(C) y z. ∀h:cat-arrow(C) z w.
((cat-comp(C) x z w (cat-comp(C) x y z f g) h) = (cat-comp(C) x y w f (cat-comp(C) y z w g h)) ∈ (cat-arrow(C) x w))
Proof
Definitions occuring in Statement :
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
small-category: SmallCategory,
cat-arrow: cat-arrow(C),
pi2: snd(t),
pi1: fst(t),
cat-ob: cat-ob(C),
cat-comp: cat-comp(C),
spreadn: spread4,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
cat-arrow_wf,
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
functionExtensionality,
cumulativity,
dependent_functionElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[C:SmallCategory]
\mforall{}x,y,z,w:cat-ob(C). \mforall{}f:cat-arrow(C) x y. \mforall{}g:cat-arrow(C) y z. \mforall{}h:cat-arrow(C) z w.
((cat-comp(C) x z w (cat-comp(C) x y z f g) h) = (cat-comp(C) x y w f (cat-comp(C) y z w g h)))
Date html generated:
2020_05_20-AM-07_49_57
Last ObjectModification:
2017_07_28-AM-09_18_59
Theory : small!categories
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