Nuprl Lemma : rmul_reverses_rless_iff
∀x,y,z:ℝ. ((y < r0) ⇒ (x < z ⇐⇒ (z * y) < (x * 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,
implies: 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],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rless_wf,
rmul_wf,
int-to-real_wf,
real_wf,
rmul_reverses_rless,
rdiv_wf,
rinv-negative,
rless-implies-rless,
rinv_wf2,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
req-iff-rsub-is-0,
rsub_wf,
req_wf,
req_weakening,
rless_functionality,
uiff_transitivity,
req_functionality,
req_inversion,
rmul-assoc,
rmul_functionality,
rmul_comm,
req_transitivity,
rmul-ac,
rmul-rdiv-cancel,
rmul-one-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
lemma_by_obid,
independent_isectElimination,
inlFormation,
because_Cache,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
promote_hyp
Latex:
\mforall{}x,y,z:\mBbbR{}. ((y < r0) {}\mRightarrow{} (x < z \mLeftarrow{}{}\mRightarrow{} (z * y) < (x * y)))
Date html generated:
2017_10_03-AM-08_35_08
Last ObjectModification:
2017_07_28-AM-07_28_50
Theory : reals
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