Nuprl Lemma : omral_inj_mon_op
∀g:OCMon. ∀r:CDRng. ∀k,k':|g|. (inj(k * k',1) = (inj(k,1) ** inj(k',1)) ∈ |omral(g;r)|)
Proof
Definitions occuring in Statement :
omral_times: ps ** qs,
omral_inj: inj(k,v),
omralist: omral(g;r),
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
cdrng: CDRng,
rng_one: 1,
ocmon: OCMon,
grp_op: *,
grp_car: |g|,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
cdrng: CDRng,
crng: CRng,
rng: Rng,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
abgrp: AbGrp,
grp: Group{i},
iabmonoid: IAbMonoid,
imon: IMonoid,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
squash: ↓T,
omon: OMon,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
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|,
dset: DSet,
rng_car: |r|,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rng_when: rng_when,
rng_mssum: rng_mssum,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
finite_set: FiniteSet{s},
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
omral_lookups_same_a,
omral_inj_wf,
grp_op_wf,
rng_one_wf,
omral_times_wf2,
cdrng_is_abdmonoid,
add_grp_of_rng_wf_b,
subtype_rel_sets,
grp_sig_wf,
monoid_p_wf,
grp_car_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
cdrng_wf,
ocmon_wf,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
lookup_omral_inj,
lookup_omral_times,
mset_for_functionality,
oset_of_ocmon_wf,
ulinorder_wf,
assert_wf,
grp_le_wf,
bool_wf,
grp_eq_wf,
band_wf,
qoset_subtype_dset,
poset_subtype_qoset,
loset_subtype_poset,
subtype_rel_transitivity,
loset_wf,
poset_wf,
qoset_wf,
dset_wf,
mset_for_wf,
rng_when_wf,
rng_times_wf,
lookup_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
set_car_wf,
omralist_wf,
subtype_rel_self,
add_grp_of_rng_wf,
omral_dom_wf,
mon_when_wf,
add_grp_of_rng_wf_a,
rng_wf,
mset_mem_wf,
iff_weakening_equal,
rng_mssum_functionality_wrt_equal,
rng_mssum_wf,
ocmon_subtype_omon,
infix_ap_wf,
rng_eq_wf,
eqtt_to_assert,
assert_of_rng_eq,
cdrng_subtype_drng,
null_mset_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
mset_inj_wf,
ifthenelse_wf,
mset_wf,
omral_dom_inj,
finite_set_wf,
uiff_transitivity,
equal-wf-T-base,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
mset_for_null_lemma,
rng_when_of_zero,
mset_for_mset_inj,
imon_wf,
mon_when_true,
assert_of_mon_eq,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
abdmonoid_wf,
dmon_wf,
rng_times_one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
isectElimination,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
sqequalRule,
instantiate,
setEquality,
cumulativity,
lambdaEquality,
independent_isectElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_set_memberEquality,
productElimination,
productEquality,
functionEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
voidElimination,
independent_pairFormation,
impliesFunctionality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k,k':|g|. (inj(k * k',1) = (inj(k,1) ** inj(k',1)))
Date html generated:
2018_05_22-AM-07_47_15
Last ObjectModification:
2018_05_19-AM-08_29_01
Theory : polynom_3
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