Nuprl Lemma : lookup_omral_action
∀g:OCMon. ∀r:CDRng. ∀k:|g|. ∀v:|r|. ∀ps:|omral(g;r)|. (((v ⋅⋅ ps)[k]) = (v * (ps[k])) ∈ |r|)
Proof
Definitions occuring in Statement :
omral_action: v ⋅⋅ ps,
omralist: omral(g;r),
lookup: as[k],
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
cdrng: CDRng,
rng_times: *,
rng_zero: 0,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_car: |g|,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
omral_action: v ⋅⋅ ps,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
cdrng: CDRng,
crng: CRng,
rng: Rng,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
squash: ↓T,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
omralist: omral(g;r),
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
dset_list: s List,
set_prod: s × t,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
grp_car: |g|,
true: True,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
set_car_wf,
omralist_wf,
dset_wf,
rng_car_wf,
grp_car_wf,
cdrng_wf,
ocmon_wf,
equal_wf,
squash_wf,
true_wf,
lookup_wf,
list_wf,
poset_sig_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
mon_ident,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
abdmonoid_abmonoid,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
abdmonoid_wf,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
omral_scale_wf,
grp_id_wf,
iff_weakening_equal,
lookup_omral_scale_a,
infix_ap_wf,
dset_of_mon_wf0,
add_grp_of_rng_wf,
rng_times_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productEquality,
cumulativity,
because_Cache,
instantiate,
independent_isectElimination,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
functionEquality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k:|g|. \mforall{}v:|r|. \mforall{}ps:|omral(g;r)|. (((v \mcdot{}\mcdot{} ps)[k]) = (v * (ps[k])))
Date html generated:
2017_10_01-AM-10_06_44
Last ObjectModification:
2017_03_03-PM-01_14_19
Theory : polynom_3
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