Nuprl Lemma : mcopower_umap_unique
∀s:DSet. ∀g:AbMon. ∀c:MCopower(s;g). ∀h:AbMon. ∀f:|s| ⟶ MonHom(g,h). ∀a':|c.mon| ⟶ |h|.
(IsMonHom{c.mon,h}(a')
⇒ (∀j:|s|. ((f j) = (a' o (c.inj j)) ∈ (|g| ⟶ |h|)))
⇒ (a' = (c.umap h f) ∈ (|c.mon| ⟶ |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,
all: ∀x:A. B[x],
implies: 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],
implies: P ⇒ Q,
uni_sat: a = !x:T. Q[x],
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
dset: DSet,
prop: ℙ,
so_lambda: λ2x.t[x],
abmonoid: AbMon,
mon: Mon,
subtype_rel: A ⊆r B,
mcopower: MCopower(s;g),
so_apply: x[s]
Lemmas referenced :
mcopower_properties,
set_car_wf,
all_wf,
equal_wf,
grp_car_wf,
compose_wf,
mcopower_mon_wf,
abmonoid_wf,
mcopower_inj_wf,
monoid_hom_p_wf,
monoid_hom_wf,
mcopower_wf,
dset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
isectElimination,
setElimination,
rename,
lambdaEquality,
functionEquality,
applyEquality,
because_Cache
Latex:
\mforall{}s:DSet. \mforall{}g:AbMon. \mforall{}c:MCopower(s;g). \mforall{}h:AbMon. \mforall{}f:|s| {}\mrightarrow{} MonHom(g,h). \mforall{}a':|c.mon| {}\mrightarrow{} |h|.
(IsMonHom\{c.mon,h\}(a') {}\mRightarrow{} (\mforall{}j:|s|. ((f j) = (a' o (c.inj j)))) {}\mRightarrow{} (a' = (c.umap h f)))
Date html generated:
2016_05_16-AM-08_13_12
Last ObjectModification:
2015_12_28-PM-06_09_58
Theory : polynom_1
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