Nuprl Lemma : rmul-negative-iff
∀x,y:ℝ. ((x * y) < r0 ⇐⇒ ((r0 < x) ∧ (y < r0)) ∨ ((x < r0) ∧ (r0 < y)))
Proof
Definitions occuring in Statement :
rless: x < y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rmul-is-positive,
rminus_wf,
real_wf,
rless-implies-rless,
int-to-real_wf,
rmul_wf,
rless_wf,
rsub_wf,
itermSubtract_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
inhabitedIsType,
universeIsType,
productElimination,
independent_pairFormation,
natural_numberEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
unionElimination,
inlFormation_alt,
promote_hyp,
inrFormation_alt,
sqequalRule,
unionIsType,
productIsType,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}x,y:\mBbbR{}. ((x * y) < r0 \mLeftarrow{}{}\mRightarrow{} ((r0 < x) \mwedge{} (y < r0)) \mvee{} ((x < r0) \mwedge{} (r0 < y)))
Date html generated:
2019_10_29-AM-10_05_43
Last ObjectModification:
2019_01_13-PM-08_26_38
Theory : reals
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