Nuprl Lemma : equal-monads
∀[C:SmallCategory]. ∀[M1,M2:Monad(C)].
(M1 = M2 ∈ Monad(C)) supposing
((∀x:cat-ob(C). (monad-op(M1;x) = monad-op(M2;x) ∈ (cat-arrow(C) M1(M1(x)) M1(x)))) and
(∀x:cat-ob(C). (monad-unit(M1;x) = monad-unit(M2;x) ∈ (cat-arrow(C) x M1(x)))) and
(monad-functor(M1) = monad-functor(M2) ∈ Functor(C;C)))
Proof
Definitions occuring in Statement :
monad-op: monad-op(M;x),
monad-unit: monad-unit(M;x),
monad-fun: M(x),
monad-functor: monad-functor(M),
cat-monad: Monad(C),
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
monad-fun: M(x),
all: ∀x:A. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cat-monad: Monad(C),
spreadn: spread3,
nat-trans: nat-trans(C;D;F;G),
functor-comp: functor-comp(F;G),
top: Top,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
id_functor: 1,
monad-op: monad-op(M;x),
pi2: snd(t),
monad-functor: monad-functor(M),
pi1: fst(t),
monad-unit: monad-unit(M;x),
mk-functor: mk-functor,
functor-ob: ob(F)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
cat-ob_wf,
functor-ob_wf,
cat-functor_wf,
monad-functor_wf,
subtype_rel_self,
iff_weakening_equal,
ob_mk_functor_lemma,
istype-void,
arrow_mk_functor_lemma,
cat-arrow_wf,
cat-comp_wf,
cat-id_wf,
functor-arrow_wf,
monad-fun_wf,
monad-op_wf,
subtype_rel-equal,
monad-unit_wf,
cat-monad_wf,
small-category_wf,
nat-trans_wf,
id_functor_wf,
functor-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
lambdaFormation_alt,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
productIsType,
functionIsType,
equalityIsType1,
dependent_pairEquality_alt,
independent_pairEquality,
lambdaEquality,
functionExtensionality,
voidEquality,
isect_memberEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[M1,M2:Monad(C)].
(M1 = M2) supposing
((\mforall{}x:cat-ob(C). (monad-op(M1;x) = monad-op(M2;x))) and
(\mforall{}x:cat-ob(C). (monad-unit(M1;x) = monad-unit(M2;x))) and
(monad-functor(M1) = monad-functor(M2)))
Date html generated:
2020_05_20-AM-07_59_24
Last ObjectModification:
2018_11_13-AM-10_03_14
Theory : small!categories
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