Nuprl Lemma : omral_alg_umap_is_hom
∀g:OCMon. ∀a:CDRng. ∀n:algebra{i:l}(a). ∀f:MonHom(g,n↓rg↓xmn). alg_hom_p(a; omral_alg(g;a); n; alg_umap(n,f))
Proof
Definitions occuring in Statement :
omral_alg_umap: alg_umap(n,f),
omral_alg: omral_alg(g;r),
alg_hom_p: alg_hom_p(a; m; n; f),
algebra: algebra{i:l}(A),
rng_of_alg: a↓rg,
all: ∀x:A. B[x],
mul_mon_of_rng: r↓xmn,
cdrng: CDRng,
ocmon: OCMon,
monoid_hom: MonHom(M1,M2)
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
alg_hom_p: alg_hom_p(a; m; n; f),
and: P ∧ Q,
module_hom_p: module_hom_p(a; m; n; f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
uall: ∀[x:A]. B[x],
cdrng: CDRng,
crng: CRng,
rng: Rng,
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
algebra: algebra{i:l}(A),
module: A-Module,
omral_alg: omral_alg(g;r),
alg_car: a.car,
pi1: fst(t),
alg_plus: a.plus,
pi2: snd(t),
omral_alg_umap: alg_umap(n,f),
tlambda: λx:T. b[x],
infix_ap: x f y,
squash: ↓T,
prop: ℙ,
omon: OMon,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
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,
grp_car: |g|,
monoid_hom: MonHom(M1,M2),
mul_mon_of_rng: r↓xmn,
rng_car: |r|,
rng_of_alg: a↓rg,
so_apply: x[s],
implies: P ⇒ Q,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
rng_zero: 0,
rev_uimplies: rev_uimplies(P;Q),
rng_plus: +r,
alg_act: a.act,
rng_mssum: rng_mssum,
abgrp: AbGrp,
grp: Group{i},
iabmonoid: IAbMonoid,
imon: IMonoid,
dset: DSet,
grp_of_module: m↓grp,
label: ...$L... t,
mset_for: mset_for,
mon_for: For{g} x ∈ as. f[x],
for: For{T,op,id} x ∈ as. f[x],
reduce: reduce(f;k;as),
list_ind: list_ind,
map: map(f;as),
omral_dom: dom(ps),
oal_dom: dom(ps),
mk_mset: mk_mset(as),
alg_times: a.times,
mod_mssum: mod_mssum,
grp_op: *,
set_eq: =b,
rng_times: *,
alg_one: a.one,
omral_one: 11,
finite_set: FiniteSet{s},
not: ¬A,
false: False,
top: Top,
rng_one: 1
Lemmas referenced :
omon_inc,
ocmon_subtype_omon,
abdmonoid_dmon,
ocmon_wf,
alg_car_wf,
rng_car_wf,
omral_alg_wf,
monoid_hom_wf,
mul_mon_of_rng_wf,
rng_of_alg_wf,
algebra_wf,
cdrng_wf,
equal_wf,
squash_wf,
true_wf,
rng_mssum_dom_shift,
oset_of_ocmon_wf,
ulinorder_wf,
grp_car_wf,
assert_wf,
grp_le_wf,
bool_wf,
grp_eq_wf,
band_wf,
qoset_subtype_dset,
poset_subtype_qoset,
loset_subtype_poset,
subtype_rel_transitivity,
loset_wf,
poset_wf,
qoset_wf,
dset_wf,
rng_of_alg_wf2,
alg_act_wf,
lookup_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
omral_plus_wf,
subtype_rel_self,
list_wf,
set_car_wf,
omral_dom_wf,
mset_union_wf,
omral_plus_dom,
mset_mem_wf,
mset_diff_wf,
alg_plus_wf,
iff_weakening_equal,
lookup_omral_eq_zero,
omral_plus_wf2,
assert_functionality_wrt_uiff,
bor_wf,
bnot_wf,
mset_mem_diff,
mset_union_wf_f,
omral_dom_wf2,
fset_mem_union,
iff_transitivity,
eqtt_to_assert,
assert_of_bor,
or_wf,
not_wf,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
module_act_zero_l,
alg_zero_wf,
mem_bsubmset,
all_wf,
rng_mssum_functionality_wrt_equal,
rng_plus_wf,
lookup_omral_plus,
infix_ap_wf,
rng_mssum_wf,
dset_of_mon_wf0,
add_grp_of_rng_wf,
mon_subtype_grp_sig,
dmon_subtype_mon,
ocmon_subtype_abdmonoid,
abdmonoid_wf,
dmon_wf,
mon_wf,
grp_sig_wf,
module_act_plus,
rng_mssum_of_plus,
mset_for_functionality_wrt_bsubmset,
add_grp_of_rng_wf_b,
subtype_rel_sets,
monoid_p_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
omral_action_wf,
omralist_wf,
omral_dom_action,
mset_for_wf,
rng_times_wf,
lookup_omral_action,
dist_hom_over_mset_for,
module_act_is_grp_hom,
monoid_hom_p_wf,
module_action_p,
grp_of_module_wf2,
omral_times_wf,
grp_of_module_wf,
mset_prod_wf,
omral_times_dom,
alg_times_wf,
omral_times_wf2,
mset_prod_wf2,
mod_mssum_functionality,
rng_when_wf,
lookup_omral_times,
mod_mssum_wf,
mon_when_wf,
mod_action_mssum_r,
mod_action_when_r,
mod_mssum_swap,
imon_wf,
grp_eq_sym,
dset_of_mon_wf,
set_eq_wf,
fset_for_when_eq,
prod_in_mset_prod,
mod_times_mssum_r,
mod_times_mssum_l,
monoid_hom_op,
algebra_act_times_lr,
omral_inj_wf,
rng_one_wf,
ifthenelse_wf,
rng_eq_wf,
mset_wf,
null_mset_wf,
mset_inj_wf,
omral_dom_inj,
finite_set_wf,
alg_one_wf,
uiff_transitivity,
equal-wf-T-base,
assert_of_rng_eq,
cdrng_subtype_drng,
eqff_to_assert,
mset_for_null_lemma,
module_over_triv_rng,
mset_for_mset_inj,
lookup_omral_inj,
mon_when_true,
assert_of_mon_eq,
monoid_hom_id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
independent_pairFormation,
isect_memberFormation,
isectElimination,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
because_Cache,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_set_memberEquality,
productElimination,
productEquality,
functionEquality,
instantiate,
independent_isectElimination,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
inlFormation,
addLevel,
allFunctionality,
impliesFunctionality,
levelHypothesis,
allLevelFunctionality,
impliesLevelFunctionality,
inrFormation,
setEquality,
cumulativity,
voidElimination,
voidEquality
Latex:
\mforall{}g:OCMon. \mforall{}a:CDRng. \mforall{}n:algebra\{i:l\}(a). \mforall{}f:MonHom(g,n\mdownarrow{}rg\mdownarrow{}xmn).
alg\_hom\_p(a; omral\_alg(g;a); n; alg\_umap(n,f))
Date html generated:
2018_05_22-AM-07_47_33
Last ObjectModification:
2018_05_19-AM-08_33_14
Theory : polynom_3
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