Nuprl Lemma : drng_properties
∀[r:DRng]. (IsRing(|r|;+r;0;-r;*;1) ∧ IsEqFun(|r|;=b))
Proof
Definitions occuring in Statement :
drng: DRng,
ring_p: IsRing(T;plus;zero;neg;times;one),
rng_one: 1,
rng_times: *,
rng_minus: -r,
rng_zero: 0,
rng_plus: +r,
rng_eq: =b,
rng_car: |r|,
eqfun_p: IsEqFun(T;eq),
uall: ∀[x:A]. B[x],
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
drng: DRng,
ring_p: IsRing(T;plus;zero;neg;times;one),
prop: ℙ,
implies: P ⇒ Q,
sq_stable: SqStable(P),
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
ident: Ident(T;op;id),
bilinear: BiLinear(T;pl;tm),
squash: ↓T,
group_p: IsGroup(T;op;id;inv),
inverse: Inverse(T;op;id;inv),
eqfun_p: IsEqFun(T;eq),
uiff: uiff(P;Q),
uimplies: b supposing a,
infix_ap: x f y
Lemmas referenced :
drng_wf,
equal_wf,
assert_witness,
assert_wf,
rng_eq_wf,
sq_stable__eqfun_p,
squash_wf,
sq_stable__bilinear,
sq_stable__monoid_p,
sq_stable__group_p,
bilinear_wf,
rng_one_wf,
rng_times_wf,
monoid_p_wf,
and_wf,
rng_minus_wf,
rng_zero_wf,
rng_plus_wf,
rng_car_wf,
group_p_wf,
sq_stable__and
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
isect_memberEquality,
independent_functionElimination,
lambdaFormation,
because_Cache,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
applyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[r:DRng]. (IsRing(|r|;+r;0;-r;*;1) \mwedge{} IsEqFun(|r|;=\msubb{}))
Date html generated:
2016_05_15-PM-00_20_33
Last ObjectModification:
2016_01_15-AM-08_51_43
Theory : rings_1
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