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.
(h o g o f = h o g o f ∈ (cat-arrow(C) x w))
Proof
Definitions occuring in Statement :
cat_comp: g o f,
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 :
cat_comp: g o f,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
cat-arrow_wf,
cat-comp-assoc,
cat-comp_wf,
iff_weakening_equal,
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
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.
(h o g o f = h o g o f)
Date html generated:
2020_05_20-AM-07_49_58
Last ObjectModification:
2017_07_28-AM-09_19_01
Theory : small!categories
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