Nuprl Lemma : monad-extend-unit
∀[C:SmallCategory]. ∀[M:Monad(C)]. ∀[y:cat-ob(C)].
(monad-extend(C;M;y;y;monad-unit(M;y)) = (cat-id(C) M(y)) ∈ (cat-arrow(C) M(y) M(y)))
Proof
Definitions occuring in Statement :
monad-extend: monad-extend(C;M;x;y;f),
monad-unit: monad-unit(M;x),
monad-fun: M(x),
cat-monad: Monad(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cat-monad: Monad(C),
nat-trans: nat-trans(C;D;F;G),
functor-comp: functor-comp(F;G),
all: ∀x:A. B[x],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
id_functor: 1,
monad-fun: M(x),
monad-extend: monad-extend(C;M;x;y;f),
monad-functor: monad-functor(M),
monad-op: monad-op(M;x),
monad-unit: monad-unit(M;x),
pi1: fst(t),
pi2: snd(t),
spreadn: spread3,
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 :
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
equal_wf,
squash_wf,
true_wf,
cat-arrow_wf,
functor-ob_wf,
cat-id_wf,
iff_weakening_equal,
cat-ob_wf,
cat-monad_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
lambdaEquality,
imageElimination,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[M:Monad(C)]. \mforall{}[y:cat-ob(C)].
(monad-extend(C;M;y;y;monad-unit(M;y)) = (cat-id(C) M(y)))
Date html generated:
2017_10_05-AM-00_52_15
Last ObjectModification:
2017_07_28-AM-09_20_50
Theory : small!categories
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