Nuprl Lemma : monad-extend_wf
∀[C:SmallCategory]. ∀[M:Monad(C)]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x M(y)].
(monad-extend(C;M;x;y;f) ∈ cat-arrow(C) M(x) M(y))
Proof
Definitions occuring in Statement :
monad-extend: monad-extend(C;M;x;y;f),
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,
cat-monad: Monad(C),
spreadn: spread3,
monad-fun: M(x),
monad-functor: monad-functor(M),
pi1: fst(t),
monad-extend: monad-extend(C;M;x;y;f),
monad-op: monad-op(M;x),
pi2: snd(t),
and: P ∧ Q,
nat-trans: nat-trans(C;D;F;G),
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 :
cat-comp_wf,
functor-ob_wf,
functor-arrow_wf,
subtype_rel-equal,
cat-arrow_wf,
functor-comp_wf,
monad-fun_wf,
cat-ob_wf,
cat-monad_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[M:Monad(C)]. \mforall{}[x,y:cat-ob(C)]. \mforall{}[f:cat-arrow(C) x M(y)].
(monad-extend(C;M;x;y;f) \mmember{} cat-arrow(C) M(x) M(y))
Date html generated:
2020_05_20-AM-07_59_05
Last ObjectModification:
2017_01_17-PM-03_49_38
Theory : small!categories
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