Nuprl Lemma : omral_times_dom
∀g:OCMon. ∀r:CDRng. ∀ps,qs:|omral(g;r)|. (↑(dom(ps ** qs) ⊆b (dom(ps) × dom(qs))))
Proof
Definitions occuring in Statement :
omral_times: ps ** qs,
omral_dom: dom(ps),
omralist: omral(g;r),
mset_prod: a × b,
bsubmset: a ⊆b b,
assert: ↑b,
all: ∀x:A. B[x],
cdrng: CDRng,
ocmon: OCMon,
dset_of_mon: g↓set,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
dset: DSet,
oset_of_ocmon: g↓oset,
cdrng: CDRng,
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,
dset_of_mon: g↓set,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
grp_car: |g|,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x.t[x],
so_apply: x[s],
omral_times: ps ** qs,
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
omral_dom: dom(ps),
null_mset: 0{s},
oal_dom: dom(ps),
mk_mset: mk_mset(as),
list_ind: list_ind,
nil: [],
it: ⋅,
crng: CRng,
rng: Rng,
mset_inj: mset_inj{s}(x),
mset_sum: a + b,
append: as @ bs,
omon: OMon,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
infix_ap: x f y,
abdmonoid: AbDMon,
bor_mon: <𝔹,∨b>,
assert: ↑b,
false: False,
or: P ∨ Q,
finite_set: FiniteSet{s},
fset_map: fs-map(f, a),
squash: ↓T,
exists: ∃x:A. B[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
mset_prod: a × b,
mset_union_mon: <MSet{s},⋃,0>,
grp_op: *,
mset_map: msmap{s,s'}(f;a),
monoid_hom: MonHom(M1,M2),
mset_mon: mset_mon{s},
true: True,
set_eq: =b
Lemmas referenced :
cdrng_is_abdmonoid,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
ocmon_wf,
abdmonoid_wf,
dmon_wf,
set_car_wf,
omralist_wf,
dset_wf,
cdrng_wf,
mem_bsubmset,
dset_of_mon_wf,
omral_dom_wf2,
omral_times_wf2,
mset_prod_wf,
omral_dom_wf,
assert_wf,
mset_mem_wf,
omral_times_wf,
dset_of_mon_wf0,
omralist_ind_a,
list_ind_nil_lemma,
map_nil_lemma,
nil_wf,
grp_car_wf,
rng_car_wf,
list_ind_cons_lemma,
map_cons_lemma,
cons_wf,
not_wf,
equal_wf,
rng_zero_wf,
before_wf,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
map_wf,
set_prod_wf,
oset_of_ocmon_wf0,
assert_functionality_wrt_uiff,
null_mset_wf,
mset_for_wf,
bor_mon_wf,
set_eq_wf,
mset_mem_char,
mset_for_null_lemma,
assert_functionality_wrt_bimplies,
omral_plus_wf,
omral_scale_wf,
mset_union_wf,
mset_mem_functionality_wrt_bsubmset,
omral_plus_wf2,
omral_scale_wf2,
omral_plus_dom,
bor_wf,
fset_mem_union,
assert_of_bor,
fset_map_wf,
finite_set_wf,
omral_dom_scale,
fset_of_mset_wf,
mset_map_wf,
fset_of_mset_mem,
abmonoid_subtype_iabmonoid,
bmsexists_char_a,
sq_stable_from_decidable,
mset_sum_wf,
mset_inj_wf_f,
loset_wf,
decidable__assert,
assert_of_dset_eq,
bmsexists_char,
equal_functionality_wrt_subtype_rel2,
mset_for_inj_lemma,
mset_union_mon_wf,
mset_inj_wf,
dist_hom_over_mset_for,
mset_union_bor_mon_hom,
monoid_hom_p_wf,
mset_mon_wf,
mset_sum_bor_mon_hom,
mset_for_functionality,
squash_wf,
true_wf,
mset_wf,
abmonoid_comm,
abdmonoid_abmonoid,
iabmonoid_wf,
mset_prod_mem,
bmsexists_char_a_rw,
mset_mem_inj_sum_lemma,
iff_transitivity,
or_wf,
iff_weakening_uiff,
assert_of_mon_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_pairFormation,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
lambdaEquality,
setElimination,
rename,
because_Cache,
independent_functionElimination,
functionEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
independent_pairEquality,
cumulativity,
universeEquality,
unionElimination,
equalityElimination,
setEquality,
imageElimination,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
inlFormation,
dependent_set_memberEquality,
functionExtensionality,
natural_numberEquality,
orFunctionality,
inrFormation
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}ps,qs:|omral(g;r)|. (\muparrow{}(dom(ps ** qs) \msubseteq{}\msubb{} (dom(ps) \mtimes{} dom(qs))))
Date html generated:
2017_10_01-AM-10_06_24
Last ObjectModification:
2017_03_03-PM-01_17_25
Theory : polynom_3
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