Nuprl Lemma : free_abmon_umap_properties
∀S:DSet. ∀M:FAbMon(S). ∀N:AbMon. ∀p:|S| ⟶ |N|.
((((M.umap N p) o M.inj) = p ∈ (|S| ⟶ |N|))
∧ (∀f:MonHom(M.mon,N). (((f o M.inj) = p ∈ (|S| ⟶ |N|)) ⇒ (f = (M.umap N p) ∈ (|M.mon| ⟶ |N|)))))
Proof
Definitions occuring in Statement :
free_abmon_umap: f.umap,
free_abmon_inj: f.inj,
free_abmon_mon: f.mon,
free_abmonoid: FAbMon(S),
compose: f o g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
monoid_hom: MonHom(M1,M2),
abmonoid: AbMon,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
abmonoid: AbMon,
mon: Mon,
free_abmonoid: FAbMon(S),
free_abmon_umap: f.umap,
pi2: snd(t),
free_abmon_inj: f.inj,
pi1: fst(t),
free_abmon_mon: f.mon,
unique_set: {!x:T | P[x]},
squash: ↓T,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
monoid_hom: MonHom(M1,M2),
subtype_rel: A ⊆r B
Lemmas referenced :
set_car_wf,
grp_car_wf,
abmonoid_wf,
free_abmonoid_wf,
dset_wf,
equal_wf,
compose_wf,
monoid_hom_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
productElimination,
sqequalRule,
applyEquality,
functionExtensionality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
because_Cache,
independent_functionElimination,
lambdaEquality
Latex:
\mforall{}S:DSet. \mforall{}M:FAbMon(S). \mforall{}N:AbMon. \mforall{}p:|S| {}\mrightarrow{} |N|.
((((M.umap N p) o M.inj) = p) \mwedge{} (\mforall{}f:MonHom(M.mon,N). (((f o M.inj) = p) {}\mRightarrow{} (f = (M.umap N p)))))
Date html generated:
2017_10_01-AM-10_01_05
Last ObjectModification:
2017_03_03-PM-01_03_21
Theory : polynom_1
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