Nuprl Lemma : monad-op_wf
∀[C:SmallCategory]. ∀[M:Monad(C)]. ∀[x:cat-ob(C)]. (monad-op(M;x) ∈ cat-arrow(C) M(M(x)) M(x))
Proof
Definitions occuring in Statement :
monad-op: monad-op(M;x),
monad-fun: M(x),
cat-monad: Monad(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
monad-op: monad-op(M;x),
cat-monad: Monad(C),
nat-trans: nat-trans(C;D;F;G),
monad-fun: M(x),
pi2: snd(t),
monad-functor: monad-functor(M),
pi1: fst(t),
id_functor: 1,
all: ∀x:A. B[x],
top: Top,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
functor-ob: ob(F),
functor-comp: functor-comp(F;G),
mk-functor: mk-functor
Lemmas referenced :
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
subtype_rel-equal,
cat-arrow_wf,
functor-ob_wf,
functor-comp_wf,
cat-ob_wf,
cat-monad_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
independent_isectElimination,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[M:Monad(C)]. \mforall{}[x:cat-ob(C)]. (monad-op(M;x) \mmember{} cat-arrow(C) M(M(x)) M(x))
Date html generated:
2020_05_20-AM-07_58_55
Last ObjectModification:
2017_01_17-PM-03_46_27
Theory : small!categories
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