Nuprl Lemma : rng_times_over_minus
∀[r:Rng]. ∀[a,b:|r|]. ((((-r a) * b) = (-r (a * b)) ∈ |r|) ∧ ((a * (-r b)) = (-r (a * b)) ∈ |r|))
Proof
Definitions occuring in Statement :
rng: Rng,
rng_times: *,
rng_minus: -r,
rng_car: |r|,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
rng: Rng,
infix_ap: x f y,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
rng_car_wf,
rng_wf,
rng_plus_cancel_l,
rng_times_wf,
rng_minus_wf,
equal_wf,
squash_wf,
true_wf,
infix_ap_wf,
rng_plus_wf,
rng_plus_inv,
iff_weakening_equal,
rng_times_over_plus,
rng_zero_wf,
rng_times_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
because_Cache,
applyEquality,
independent_isectElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:|r|]. ((((-r a) * b) = (-r (a * b))) \mwedge{} ((a * (-r b)) = (-r (a * b))))
Date html generated:
2017_10_01-AM-08_17_27
Last ObjectModification:
2017_02_28-PM-02_02_43
Theory : rings_1
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