Nuprl Lemma : ringeq-iff-rsub-is-0
∀[r:Rng]. ∀[a,b:|r|]. uiff(a = b ∈ |r|;(a +r (-r b)) = 0 ∈ |r|)
Proof
Definitions occuring in Statement :
rng: Rng,
rng_minus: -r,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
rng: Rng,
infix_ap: x f y,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
equal_wf,
rng_car_wf,
rng_plus_wf,
rng_minus_wf,
rng_zero_wf,
iff_weakening_equal,
rng_plus_inv,
rng_properties,
rng_wf,
rng_plus_zero,
squash_wf,
true_wf,
subtype_rel_self,
rng_plus_assoc,
rng_plus_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
setElimination,
rename,
hypothesisEquality,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
equalityTransitivity,
independent_isectElimination,
productElimination,
independent_functionElimination,
equalityIstype,
inhabitedIsType,
independent_pairEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
applyLambdaEquality,
hyp_replacement,
lambdaEquality,
universeEquality,
instantiate
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:|r|]. uiff(a = b;(a +r (-r b)) = 0)
Date html generated:
2020_05_19-PM-10_08_12
Last ObjectModification:
2020_01_08-PM-06_00_28
Theory : rings_1
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