Nuprl Lemma : omral_times_wf2
∀g:OCMon. ∀r:CDRng. ∀ps,qs:|omral(g;r)|. (ps ** qs ∈ |omral(g;r)|)
Proof
Definitions occuring in Statement :
omral_times: ps ** qs,
omralist: omral(g;r),
all: ∀x:A. B[x],
member: t ∈ T,
cdrng: CDRng,
ocmon: OCMon,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
omralist: omral(g;r),
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
and: P ∧ Q,
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|,
cand: A c∧ B,
ocmon: OCMon,
omon: OMon,
so_lambda: λ2x.t[x],
prop: ℙ,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
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],
cdrng: CDRng,
crng: CRng,
rng: Rng,
guard: {T},
abdmonoid: AbDMon,
dmon: DMon,
not: ¬A,
loset: LOSet,
poset: POSet{i},
qoset: QOSet
Lemmas referenced :
set_car_wf,
omralist_wf,
dset_wf,
cdrng_wf,
ocmon_wf,
cdrng_is_abdmonoid,
omral_times_wf,
assert_wf,
sd_ordered_wf,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
grp_car_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
equal_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
map_wf,
rng_car_wf,
oset_of_ocmon_wf0,
not_wf,
mem_wf,
dset_of_mon_wf,
abdmonoid_wf,
rng_zero_wf,
dset_of_mon_wf0,
omral_times_sd_ordered,
omral_times_non_zero_vals,
set_prod_wf,
loset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
because_Cache,
promote_hyp,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
instantiate,
cumulativity,
universeEquality,
functionEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}ps,qs:|omral(g;r)|. (ps ** qs \mmember{} |omral(g;r)|)
Date html generated:
2017_10_01-AM-10_06_08
Last ObjectModification:
2017_03_03-PM-01_14_14
Theory : polynom_3
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