Nuprl Lemma : rmul-is-positive
∀x,y:ℝ. (r0 < (x * y) ⇐⇒ ((r0 < x) ∧ (r0 < y)) ∨ ((x < r0) ∧ (y < r0)))
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
or: P ∨ Q,
cand: A c∧ B,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rneq: x ≠ y
Lemmas referenced :
rless_wf,
int-to-real_wf,
rmul_wf,
or_wf,
real_wf,
rmul-is-negative,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rsub_wf,
rmul-one-both,
rdiv-zero,
rmul-rdiv-cancel,
rmul-ac,
rmul_comm,
rmul_functionality,
rmul-assoc,
req_inversion,
req_functionality,
uiff_transitivity,
rmul-int-rdiv,
rmul-rdiv-cancel2,
rmul-zero-both,
rless_functionality,
req_weakening,
req_wf,
rless-int,
rdiv_wf,
rmul_preserves_rless,
rminus-rminus,
req_transitivity,
rminus_functionality,
rmul_over_rminus,
rmul-minus,
rmul-int,
rmul_reverses_rless_iff,
rminus_wf,
rmul_reverses_rless,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
productEquality,
dependent_functionElimination,
minusEquality,
independent_functionElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
unionElimination,
inlFormation,
inrFormation,
promote_hyp,
addLevel,
multiplyEquality,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}x,y:\mBbbR{}. (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_17
Last ObjectModification:
2017_07_28-AM-07_32_57
Theory : reals
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