Nuprl Lemma : omral_action_times_r1
∀g:OCMon. ∀r:CDRng. ∀v:|r|. ∀ps,qs:|omral(g;r)|. ((v ⋅⋅ (ps ** qs)) = ((v ⋅⋅ ps) ** qs) ∈ |omral(g;r)|)
Proof
Definitions occuring in Statement :
omral_action: v ⋅⋅ ps,
omral_times: ps ** qs,
omralist: omral(g;r),
all: ∀x:A. B[x],
equal: s = t ∈ T,
cdrng: CDRng,
rng_car: |r|,
ocmon: OCMon,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
subtype_rel: A ⊆r B,
dset: DSet,
cdrng: CDRng,
crng: CRng,
rng: Rng,
squash: ↓T,
prop: ℙ,
infix_ap: x f y,
omon: OMon,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
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,
so_apply: x[s],
cand: A c∧ B,
abgrp: AbGrp,
grp: Group{i},
iabmonoid: IAbMonoid,
imon: IMonoid,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
add_grp_of_rng: r↓+gp,
grp_car: |g|,
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,
grp_id: e,
pi2: snd(t),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rng_mssum: rng_mssum,
loset: LOSet,
poset: POSet{i},
qoset: QOSet
Lemmas referenced :
omral_lookups_same_a,
omral_action_wf,
omral_times_wf2,
grp_car_wf,
set_car_wf,
omralist_wf,
dset_wf,
rng_car_wf,
cdrng_wf,
ocmon_wf,
equal_wf,
squash_wf,
true_wf,
lookup_omral_action,
rng_times_wf,
lookup_omral_times,
mset_for_functionality,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
add_grp_of_rng_wf_b,
grp_sig_wf,
monoid_p_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
mset_for_wf,
rng_when_wf,
oset_of_ocmon_wf0,
dset_of_mon_wf0,
add_grp_of_rng_wf,
lookup_wf,
rng_zero_wf,
omral_dom_wf,
rng_wf,
mset_mem_wf,
iff_weakening_equal,
loset_wf,
mset_for_functionality_wrt_bsubmset,
mset_diff_wf,
omral_dom_action,
assert_functionality_wrt_uiff,
bnot_wf,
mset_mem_diff,
omral_dom_wf2,
iff_transitivity,
not_wf,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
lookup_omral_eq_zero,
rng_times_zero,
rng_when_of_zero,
mset_for_of_id,
rng_times_mssum_l,
rng_mssum_wf,
rng_mssum_functionality_wrt_equal,
rng_times_when_l,
rng_times_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
isectElimination,
setElimination,
rename,
applyEquality,
lambdaEquality,
sqequalRule,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
productEquality,
cumulativity,
functionEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
setEquality,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}v:|r|. \mforall{}ps,qs:|omral(g;r)|. ((v \mcdot{}\mcdot{} (ps ** qs)) = ((v \mcdot{}\mcdot{} ps) ** qs))
Date html generated:
2017_10_01-AM-10_07_00
Last ObjectModification:
2017_03_03-PM-01_16_24
Theory : polynom_3
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