Nuprl Lemma : ring_polynomial_null
∀r:CRng. ∀t:int_term(). t ≡ "0" supposing inl Ax ≤ null(int_term_to_ipoly(t))
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
crng: CRng,
int_term_to_ipoly: int_term_to_ipoly(t),
itermConstant: "const",
int_term: int_term(),
null: null(as),
uimplies: b supposing a,
all: ∀x:A. B[x],
inl: inl x,
natural_number: $n,
sqle: s ≤ t,
axiom: Ax
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iPolynomial: iPolynomial(),
or: P ∨ Q,
uimplies: b supposing a,
ringeq_int_terms: t1 ≡ t2,
crng: CRng,
rng: Rng,
prop: ℙ,
cons: [a / b],
top: Top,
ipolynomial-term: ipolynomial-term(p),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
not: ¬A,
false: False
Lemmas referenced :
ring_term_polynomial,
int_term_to_ipoly_wf,
iPolynomial_wf,
iMonomial_wf,
list-cases,
null_nil_lemma,
rng_car_wf,
sqle_wf_base,
ringeq_int_terms_wf,
ipolynomial-term_wf,
nil_wf,
product_subtype_list,
null_cons_lemma,
cons_wf,
equal_wf,
int_term_wf,
crng_wf,
not-btrue-sqle-bfalse
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
unionElimination,
sqequalRule,
isect_memberFormation,
lambdaEquality,
axiomEquality,
functionEquality,
intEquality,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination
Latex:
\mforall{}r:CRng. \mforall{}t:int\_term(). t \mequiv{} "0" supposing inl Ax \mleq{} null(int\_term\_to\_ipoly(t))
Date html generated:
2018_05_21-PM-03_17_28
Last ObjectModification:
2018_05_19-AM-08_08_22
Theory : rings_1
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