Nuprl Lemma : omral_dom_one
∀g:OCMon. ∀r:CDRng. ((¬(0 = 1 ∈ |r|)) ⇒ (dom(11) = mset_inj{g↓oset}(e) ∈ FiniteSet{g↓oset}))
Proof
Definitions occuring in Statement :
omral_one: 11,
omral_dom: dom(ps),
mset_inj: mset_inj{s}(x),
finite_set: FiniteSet{s},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T,
cdrng: CDRng,
rng_one: 1,
rng_zero: 0,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_id: e
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
omral_one: 11,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
ocmon: OCMon,
omon: OMon,
and: P ∧ Q,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
cdrng: CDRng,
crng: CRng,
rng: Rng,
grp_car: |g|,
pi1: fst(t),
set_car: |p|,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
not: ¬A,
false: False,
bfalse: ff
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
finite_set_wf,
oset_of_ocmon_wf,
ulinorder_wf,
grp_car_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,
omral_dom_inj,
grp_id_wf,
rng_one_wf,
mset_inj_wf_f,
subtype_rel_self,
set_car_wf,
oset_of_ocmon_wf0,
iff_weakening_equal,
not_wf,
rng_car_wf,
rng_zero_wf,
cdrng_wf,
ocmon_wf,
rng_eq_wf,
uiff_transitivity,
equal-wf-T-base,
eqtt_to_assert,
assert_of_rng_eq,
cdrng_subtype_drng,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
productEquality,
because_Cache,
functionEquality,
instantiate,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination,
equalityElimination,
voidElimination,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. ((\mneg{}(0 = 1)) {}\mRightarrow{} (dom(11) = mset\_inj\{g\mdownarrow{}oset\}(e)))
Date html generated:
2018_05_22-AM-07_47_02
Last ObjectModification:
2018_05_19-AM-08_27_48
Theory : polynom_3
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