Nuprl Lemma : int-to-ring-minus
∀[r:Rng]. ∀[y:ℤ]. (int-to-ring(r;-y) = (-r int-to-ring(r;y)) ∈ |r|)
Proof
Definitions occuring in Statement :
int-to-ring: int-to-ring(r;n),
rng: Rng,
rng_minus: -r,
rng_car: |r|,
uall: ∀[x:A]. B[x],
apply: f a,
minus: -n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
squash: ↓T,
prop: ℙ,
rng: Rng,
infix_ap: x f y,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
int-to-ring-add,
rng_wf,
int-to-ring-zero,
minus-one-mul,
add-mul-special,
zero-mul,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_plus_wf,
rng_minus_wf,
int-to-ring_wf,
infix_ap_wf,
iff_weakening_equal,
rng_plus_ac_1,
rng_plus_comm,
rng_plus_inv,
rng_plus_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
minusEquality,
hypothesis,
intEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
voidElimination,
voidEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
multiplyEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[y:\mBbbZ{}]. (int-to-ring(r;-y) = (-r int-to-ring(r;y)))
Date html generated:
2017_10_01-AM-08_19_14
Last ObjectModification:
2017_02_28-PM-02_03_48
Theory : rings_1
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