Nuprl Lemma : mk_fabmon
∀s:DSet. ∀g:AbMon. ∀i:|s| ⟶ |g|. ∀U:g':AbMon ⟶ (|s| ⟶ |g'|) ⟶ |g| ⟶ |g'|.
((∀g':AbMon. ∀f:|s| ⟶ |g'|.
(IsMonHom{g,g'}(U g' f)
∧ (((U g' f) o i) = f ∈ (|s| ⟶ |g'|))
∧ (∀u:|g| ⟶ |g'|. (IsMonHom{g,g'}(u) ⇒ ((u o i) = f ∈ (|s| ⟶ |g'|)) ⇒ (u = (U g' f) ∈ (|g| ⟶ |g'|))))))
⇒ (<g, i, U> ∈ FAbMon(s)))
Proof
Definitions occuring in Statement :
free_abmonoid: FAbMon(S),
compose: f o g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
equal: s = t ∈ T,
monoid_hom_p: IsMonHom{M1,M2}(f),
abmonoid: AbMon,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
free_abmonoid: FAbMon(S),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
subtype_rel: A ⊆r B,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x.t[x],
monoid_hom: MonHom(M1,M2),
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
unique_set: {!x:T | P[x]},
cand: A c∧ B,
guard: {T}
Lemmas referenced :
set_car_wf,
abmonoid_wf,
grp_car_wf,
unique_set_wf,
monoid_hom_wf,
equal_wf,
compose_wf,
all_wf,
monoid_hom_p_wf,
dset_wf,
monoid_hom_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
dependent_pairEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
functionEquality,
lambdaEquality,
cumulativity,
universeEquality,
productEquality,
instantiate,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}s:DSet. \mforall{}g:AbMon. \mforall{}i:|s| {}\mrightarrow{} |g|. \mforall{}U:g':AbMon {}\mrightarrow{} (|s| {}\mrightarrow{} |g'|) {}\mrightarrow{} |g| {}\mrightarrow{} |g'|.
((\mforall{}g':AbMon. \mforall{}f:|s| {}\mrightarrow{} |g'|.
(IsMonHom\{g,g'\}(U g' f)
\mwedge{} (((U g' f) o i) = f)
\mwedge{} (\mforall{}u:|g| {}\mrightarrow{} |g'|. (IsMonHom\{g,g'\}(u) {}\mRightarrow{} ((u o i) = f) {}\mRightarrow{} (u = (U g' f))))))
{}\mRightarrow{} (<g, i, U> \mmember{} FAbMon(s)))
Date html generated:
2017_10_01-AM-10_01_14
Last ObjectModification:
2017_03_03-PM-01_03_32
Theory : polynom_1
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