Nuprl Lemma : omral_alg_umap_wf
∀g:OCMon. ∀a:CDRng. ∀n:algebra{i:l}(a). ∀f:|g| ⟶ n.car. (alg_umap(n,f) ∈ |omral(g;a)| ⟶ n.car)
Proof
Definitions occuring in Statement :
omral_alg_umap: alg_umap(n,f),
omralist: omral(g;r),
algebra: algebra{i:l}(A),
alg_car: a.car,
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
cdrng: CDRng,
ocmon: OCMon,
grp_car: |g|,
set_car: |p|
Definitions unfolded in proof :
omral_alg_umap: alg_umap(n,f),
all: ∀x:A. B[x],
member: t ∈ T,
tlambda: λx:T. b[x],
uall: ∀[x:A]. B[x],
ocmon: OCMon,
omon: OMon,
and: P ∧ Q,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
cdrng: CDRng,
so_lambda: λ2x.t[x],
crng: CRng,
rng: Rng,
algebra: algebra{i:l}(A),
module: A-Module,
omralist: omral(g;r),
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
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|,
alg_car: a.car,
rng_car: |r|,
rng_of_alg: a↓rg,
so_apply: x[s],
dset: DSet
Lemmas referenced :
rng_mssum_wf,
oset_of_ocmon_wf,
ulinorder_wf,
grp_car_wf,
assert_wf,
grp_le_wf,
equal_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,
rng_car_wf,
lookup_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
subtype_rel_self,
rng_of_alg_wf,
set_car_wf,
omral_dom_wf,
alg_car_wf,
omralist_wf,
algebra_wf,
cdrng_wf,
ocmon_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
lambdaEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
hypothesisEquality,
hypothesis,
productEquality,
applyEquality,
because_Cache,
functionEquality,
instantiate,
independent_isectElimination
Latex:
\mforall{}g:OCMon. \mforall{}a:CDRng. \mforall{}n:algebra\{i:l\}(a). \mforall{}f:|g| {}\mrightarrow{} n.car. (alg\_umap(n,f) \mmember{} |omral(g;a)| {}\mrightarrow{} n.car)
Date html generated:
2018_05_22-AM-07_47_17
Last ObjectModification:
2018_05_19-AM-08_28_48
Theory : polynom_3
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