Nuprl Lemma : rdiv-is-positive
∀x,y:ℝ. (y ≠ r0 ⇒ (r0 < (x/y) ⇐⇒ ((r0 < x) ∧ (r0 < y)) ∨ ((x < r0) ∧ (y < r0))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
cand: A c∧ B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
rdiv: (x/y)
Lemmas referenced :
rless_wf,
int-to-real_wf,
rdiv_wf,
or_wf,
rneq_wf,
real_wf,
rmul_preserves_rless,
rminus_wf,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rsub_wf,
rmul_wf,
rinv_wf2,
rless_functionality,
itermMultiply_wf,
real_term_value_mul_lemma,
req_transitivity,
rminus_functionality,
rmul_functionality,
rmul-rinv,
req_weakening,
rmul-identity1,
rmul-zero-both,
rmul-rdiv-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_isectElimination,
productEquality,
unionElimination,
sqequalRule,
inrFormation,
inlFormation,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
promote_hyp,
addLevel
Latex:
\mforall{}x,y:\mBbbR{}. (y \mneq{} r0 {}\mRightarrow{} (r0 < (x/y) \mLeftarrow{}{}\mRightarrow{} ((r0 < x) \mwedge{} (r0 < y)) \mvee{} ((x < r0) \mwedge{} (y < r0))))
Date html generated:
2017_10_03-AM-08_47_47
Last ObjectModification:
2017_07_28-AM-07_33_04
Theory : reals
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