Nuprl Lemma : imonomial-cons-ringeq
∀r:CRng. ∀v:ℤ List. ∀u,a:ℤ. ∀f:ℤ ⟶ |r|. (ring_term_value(f;imonomial-term(<a, [u / v]>)) = ((f u) * ring_term_value(f;\000Cimonomial-term(<a, v>))) ∈ |r|)
Proof
Definitions occuring in Statement :
ring_term_value: ring_term_value(f;t),
crng: CRng,
rng_times: *,
rng_car: |r|,
imonomial-term: imonomial-term(m),
cons: [a / b],
list: T List,
infix_ap: x f y,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
true: True,
squash: ↓T,
infix_ap: x f y,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ,
imonomial-term: imonomial-term(m),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
rng_car_wf,
list_wf,
crng_wf,
cons_wf,
rng_times_wf,
equal_wf,
int-to-ring_wf,
ring_term_value_wf,
imonomial-term_wf,
iff_weakening_equal,
crng_times_ac_1,
squash_wf,
true_wf,
imonomial-term-linear-ringeq,
subtype_rel_self,
list_accum_cons_lemma,
imonomial-ringeq-lemma,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
list_accum_wf,
int_term_wf,
ring_term_value_mul_lemma,
ring_term_value_const_lemma,
ring_term_value_var_lemma,
rng_one_wf,
int-to-ring-one,
rng_times_assoc,
rng_times_one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
functionEquality,
intEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
because_Cache,
applyEquality,
natural_numberEquality,
lambdaEquality,
imageElimination,
independent_pairEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
universeEquality,
dependent_functionElimination,
instantiate,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}r:CRng. \mforall{}v:\mBbbZ{} List. \mforall{}u,a:\mBbbZ{}. \mforall{}f:\mBbbZ{} {}\mrightarrow{} |r|.
(ring\_term\_value(f;imonomial-term(<a, [u / v]>)) = ((f u) * ring\_term\_value(f;imonomial-term(<a, v\000C>))))
Date html generated:
2018_05_21-PM-03_16_56
Last ObjectModification:
2018_05_19-AM-08_08_04
Theory : rings_1
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