Nuprl Lemma : gcopower_properties
∀s:DSet. ∀g:AbGrp. ∀c:gcopower{i}(s;g).
(IsEqFun(|c.grp|;=b)
∧ grp_p(c.grp)
∧ Comm(|c.grp|;*)
∧ (∀j:|s|. IsMonHom{g,c.grp}(c.inj j))
∧ (∀h:AbGrp. ∀f:|s| ⟶ MonHom(g,h).
(c.umap h f) = !v:|c.grp| ⟶ |h|. (IsMonHom{c.grp,h}(v) ∧ (∀j:|s|. ((f j) = (v o (c.inj j)) ∈ (|g| ⟶ |h|))))))
Proof
Definitions occuring in Statement :
gcopower: gcopower{i}(s;g),
grp_p: grp_p(g),
gcopower_umap: g1.umap,
gcopower_inj: g1.inj,
gcopower_grp: g1.grp,
eqfun_p: IsEqFun(T;eq),
comm: Comm(T;op),
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),
abgrp: AbGrp,
grp_op: *,
grp_eq: =b,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
gcopower: gcopower{i}(s;g),
uall: ∀[x:A]. B[x],
member: t ∈ T,
dset: DSet,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
guard: {T},
subtype_rel: A ⊆r B,
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),
grp_p: grp_p(g)
Lemmas referenced :
grp_inv_wf,
grp_id_wf,
sq_stable__group_p,
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,
gcopower_umap_wf,
dset_wf,
gcopower_wf,
abgrp_wf,
monoid_hom_wf,
set_car_wf,
gcopower_inj_wf,
sq_stable__monoid_hom_p,
grp_op_wf,
sq_stable__comm,
grp_eq_wf,
gcopower_grp_wf,
grp_car_wf,
sq_stable__eqfun_p
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lemma_by_obid,
isectElimination,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
introduction,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
because_Cache,
applyEquality,
functionEquality,
lambdaEquality,
independent_isectElimination,
productEquality,
isect_memberEquality,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}s:DSet. \mforall{}g:AbGrp. \mforall{}c:gcopower\{i\}(s;g).
(IsEqFun(|c.grp|;=\msubb{})
\mwedge{} grp\_p(c.grp)
\mwedge{} Comm(|c.grp|;*)
\mwedge{} (\mforall{}j:|s|. IsMonHom\{g,c.grp\}(c.inj j))
\mwedge{} (\mforall{}h:AbGrp. \mforall{}f:|s| {}\mrightarrow{} MonHom(g,h).
(c.umap h f) = !v:|c.grp| {}\mrightarrow{} |h|
(IsMonHom\{c.grp,h\}(v) \mwedge{} (\mforall{}j:|s|. ((f j) = (v o (c.inj j)))))))
Date html generated:
2016_05_16-AM-08_14_07
Last ObjectModification:
2016_01_16-PM-11_41_58
Theory : polynom_1
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