Nuprl Lemma : omral_alg_umap_unique
∀g:OCMon. ∀a:CDRng. ∀n:algebra{i:l}(a). ∀f:|g| ⟶ n.car. ∀f':algebra_hom(a; omral_alg(g;a); n).
(((f' o (λk:|g|. inj(k,1))) = f ∈ (|g| ⟶ n.car)) ⇒ (f' = alg_umap(n,f) ∈ (|omral(g;a)| ⟶ n.car)))
Proof
Definitions occuring in Statement :
omral_alg_umap: alg_umap(n,f),
omral_alg: omral_alg(g;r),
omral_inj: inj(k,v),
omralist: omral(g;r),
algebra_hom: algebra_hom(A; M; N),
algebra: algebra{i:l}(A),
alg_car: a.car,
compose: f o g,
tlambda: λx:T. b[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
cdrng: CDRng,
rng_one: 1,
ocmon: OCMon,
grp_car: |g|,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
cdrng: CDRng,
crng: CRng,
rng: Rng,
algebra: algebra{i:l}(A),
module: A-Module,
subtype_rel: A ⊆r B,
algebra_hom: algebra_hom(A; M; N),
module_hom: module_hom(A; M; N),
and: P ∧ Q,
omral_alg: omral_alg(g;r),
alg_car: a.car,
pi1: fst(t),
omralist: omral(g;r),
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
dset_list: s List,
set_prod: s × t,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
grp_car: |g|,
tlambda: λx:T. b[x],
omral_alg_umap: alg_umap(n,f),
squash: ↓T,
omon: OMon,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
infix_ap: x f y,
so_apply: x[s],
cand: A c∧ B,
rng_of_alg: a↓rg,
rng_car: |r|,
dset: DSet,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
compose: f o g,
rng_mssum: rng_mssum,
grp_of_module: m↓grp,
abgrp: AbGrp,
grp: Group{i},
iabmonoid: IAbMonoid,
imon: IMonoid,
calgebra: CAlg(A),
monoid_hom: MonHom(M1,M2),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
module_hom_p: module_hom_p(a; m; n; f),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
alg_act: a.act
Lemmas referenced :
equal_wf,
grp_car_wf,
alg_car_wf,
rng_car_wf,
compose_wf,
set_car_wf,
omralist_wf,
omral_inj_wf,
rng_one_wf,
algebra_hom_wf,
omral_alg_wf,
algebra_wf,
cdrng_wf,
ocmon_wf,
squash_wf,
true_wf,
rng_mssum_functionality_wrt_equal,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
rng_of_alg_wf2,
dset_of_mon_wf0,
add_grp_of_rng_wf,
rng_of_alg_wf,
alg_act_wf,
lookup_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
dset_wf,
omral_dom_wf,
mset_mem_wf,
iff_weakening_equal,
module_hom_action,
grp_of_module_wf2,
grp_sig_wf,
monoid_p_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
mset_for_functionality,
grp_of_module_wf,
dist_hom_over_mset_for,
omral_alg_wf2,
calgebra_wf,
algebra_hom_properties,
module_hom_properties,
module_hom_is_grp_hom,
monoid_hom_p_wf,
omral_action_wf,
rng_times_wf,
omral_action_inj,
rng_times_one,
omral_fact_a
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
functionExtensionality,
because_Cache,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
setElimination,
rename,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
productElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
productEquality,
cumulativity,
unionElimination,
equalityElimination,
independent_isectElimination,
independent_functionElimination,
setEquality,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
dependent_set_memberEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}g:OCMon. \mforall{}a:CDRng. \mforall{}n:algebra\{i:l\}(a). \mforall{}f:|g| {}\mrightarrow{} n.car. \mforall{}f':algebra\_hom(a; omral\_alg(g;a); n).
(((f' o (\mlambda{}k:|g|. inj(k,1))) = f) {}\mRightarrow{} (f' = alg\_umap(n,f)))
Date html generated:
2017_10_01-AM-10_07_41
Last ObjectModification:
2017_03_03-PM-01_17_35
Theory : polynom_3
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