Nuprl Lemma : mcopower_properties
∀s:DSet. ∀g:AbMon. ∀c:MCopower(s;g).
((∀j:|s|. IsMonHom{g,c.mon}(c.inj j))
∧ (∀h:AbMon. ∀f:|s| ⟶ MonHom(g,h).
(c.umap h f) = !v:|c.mon| ⟶ |h|. (IsMonHom{c.mon,h}(v) ∧ (∀j:|s|. ((f j) = (v o (c.inj j)) ∈ (|g| ⟶ |h|))))))
Proof
Definitions occuring in Statement :
mcopower: MCopower(s;g),
mcopower_umap: m.umap,
mcopower_inj: m.inj,
mcopower_mon: m.mon,
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,
monoid_hom: MonHom(M1,M2),
monoid_hom_p: IsMonHom{M1,M2}(f),
abmonoid: AbMon,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
mcopower: MCopower(s;g),
uall: ∀[x:A]. B[x],
member: t ∈ T,
abmonoid: AbMon,
mon: Mon,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
dset: DSet,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
monoid_hom: MonHom(M1,M2),
prop: ℙ,
monoid_hom_p: IsMonHom{M1,M2}(f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f)
Lemmas referenced :
sq_stable__equal,
sq_stable__all,
sq_stable__and,
sq_stable__uni_sat,
squash_wf,
compose_wf,
equal_wf,
all_wf,
monoid_hom_p_wf,
subtype_rel_dep_function,
mcopower_umap_wf,
grp_car_wf,
dset_wf,
mcopower_wf,
abmonoid_wf,
monoid_hom_wf,
set_car_wf,
mcopower_inj_wf,
mcopower_mon_wf,
sq_stable__monoid_hom_p
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lemma_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
applyEquality,
lambdaEquality,
because_Cache,
sqequalRule,
independent_functionElimination,
introduction,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
functionEquality,
independent_isectElimination,
productEquality,
isect_memberEquality,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}s:DSet. \mforall{}g:AbMon. \mforall{}c:MCopower(s;g).
((\mforall{}j:|s|. IsMonHom\{g,c.mon\}(c.inj j))
\mwedge{} (\mforall{}h:AbMon. \mforall{}f:|s| {}\mrightarrow{} MonHom(g,h).
(c.umap h f) = !v:|c.mon| {}\mrightarrow{} |h|
(IsMonHom\{c.mon,h\}(v) \mwedge{} (\mforall{}j:|s|. ((f j) = (v o (c.inj j)))))))
Date html generated:
2016_05_16-AM-08_13_04
Last ObjectModification:
2016_01_16-PM-11_41_55
Theory : polynom_1
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