Nuprl Lemma : free_abmon_endomorph_is_id
∀S:DSet. ∀M:FAbMon(S). ∀f:MonHom(M.mon,M.mon).
(((f o M.inj) = M.inj ∈ (|S| ⟶ |M.mon|)) ⇒ (f = Id{|M.mon|} ∈ (|M.mon| ⟶ |M.mon|)))
Proof
Definitions occuring in Statement :
free_abmon_inj: f.inj,
free_abmon_mon: f.mon,
free_abmonoid: FAbMon(S),
compose: f o g,
tidentity: Id{T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
monoid_hom: MonHom(M1,M2),
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
dset: DSet,
subtype_rel: A ⊆r B,
monoid_hom: MonHom(M1,M2),
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
squash: ↓T,
abmonoid: AbMon,
mon: Mon,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
set_car_wf,
grp_car_wf,
free_abmon_mon_wf,
compose_wf,
free_abmon_inj_wf,
monoid_hom_wf,
free_abmonoid_wf,
dset_wf,
free_abmon_umap_properties,
tidentity_wf_for_mon_hom,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
subtype_rel_transitivity,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
squash_wf,
true_wf,
comp_id_l,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
productElimination,
instantiate,
independent_isectElimination,
independent_functionElimination,
lambdaEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}S:DSet. \mforall{}M:FAbMon(S). \mforall{}f:MonHom(M.mon,M.mon). (((f o M.inj) = M.inj) {}\mRightarrow{} (f = Id\{|M.mon|\}))
Date html generated:
2017_10_01-AM-10_01_10
Last ObjectModification:
2017_03_03-PM-01_03_30
Theory : polynom_1
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