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 :
mk-functor: mk-functor,
functor-comp: functor-comp(F;G),
functor-ob: ob(F),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat-trans: nat-trans(C;D;F;G),
and: P ∧ Q,
pi2: snd(t),
monad-op: monad-op(M;x),
monad-extend: monad-extend(C;M;x;y;f),
pi1: fst(t),
monad-functor: monad-functor(M),
monad-fun: M(x),
spreadn: spread3,
cat-monad: Monad(C),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-monad_wf,
cat-ob_wf,
monad-fun_wf,
functor-comp_wf,
cat-arrow_wf,
subtype_rel-equal,
functor-arrow_wf,
functor-ob_wf,
cat-comp_wf
Rules used in proof :
independent_isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
applyEquality,
sqequalRule,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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:
2017_01_19-PM-02_58_34
Last ObjectModification:
2017_01_17-PM-03_49_38
Theory : small!categories
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