Nuprl Lemma : fmonalg_properties
∀g:AbMon. ∀a:CRng. ∀m:FMonAlg(g;a).
(IsMonHom{g,m.alg↓rg↓xmn}(m.inj)
∧ (∀n:algebra{i:l}(a). ∀f:MonHom(g,n↓rg↓xmn).
(m.umap n f) = !f':m.alg.car ⟶ n.car. (alg_hom_p(a; m.alg; n; f') ∧ ((f' o m.inj) = f ∈ (|g| ⟶ n.car)))))
Proof
Definitions occuring in Statement :
fmonalg: FMonAlg(g;a),
fma_umap: f.umap,
fma_inj: f.inj,
fma_alg: f.alg,
alg_hom_p: alg_hom_p(a; m; n; f),
algebra: algebra{i:l}(A),
rng_of_alg: a↓rg,
alg_car: a.car,
compose: f o g,
uni_sat: a = !x:T. Q[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
mul_mon_of_rng: r↓xmn,
crng: CRng,
monoid_hom: MonHom(M1,M2),
monoid_hom_p: IsMonHom{M1,M2}(f),
abmonoid: AbMon,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
fmonalg: FMonAlg(g;a),
and: P ∧ Q,
cand: A c∧ B,
uni_sat: a = !x:T. Q[x],
alg_hom_p: alg_hom_p(a; m; n; f),
module_hom_p: module_hom_p(a; m; n; f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
abmonoid: AbMon,
mon: Mon,
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
implies: P ⇒ Q,
algebra: algebra{i:l}(A),
module: A-Module,
prop: ℙ,
monoid_hom: MonHom(M1,M2),
subtype_rel: A ⊆r B,
mul_mon_of_rng: r↓xmn,
grp_car: |g|,
pi1: fst(t),
rng_of_alg: a↓rg,
rng_car: |r|,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
fmonalg_wf,
crng_wf,
abmonoid_wf,
alg_car_wf,
rng_car_wf,
fma_alg_wf,
alg_hom_p_wf,
equal_wf,
grp_car_wf,
compose_wf,
fma_inj_wf,
monoid_hom_wf,
mul_mon_of_rng_wf,
rng_of_alg_wf,
algebra_wf,
sq_stable__monoid_hom_p
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
comment,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
independent_pairFormation,
productElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
isectElimination,
axiomEquality,
because_Cache,
lambdaEquality,
productEquality,
functionExtensionality,
applyEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}g:AbMon. \mforall{}a:CRng. \mforall{}m:FMonAlg(g;a).
(IsMonHom\{g,m.alg\mdownarrow{}rg\mdownarrow{}xmn\}(m.inj)
\mwedge{} (\mforall{}n:algebra\{i:l\}(a). \mforall{}f:MonHom(g,n\mdownarrow{}rg\mdownarrow{}xmn).
(m.umap n f) = !f':m.alg.car {}\mrightarrow{} n.car. (alg\_hom\_p(a; m.alg; n; f') \mwedge{} ((f' o m.inj) = f))))
Date html generated:
2017_10_01-AM-10_01_25
Last ObjectModification:
2017_03_03-PM-01_03_55
Theory : polynom_1
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