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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