Nuprl Lemma : rmul_reverses_rleq_iff
∀[x,y,z:ℝ]. uiff(x ≤ z;(z * y) ≤ (x * y)) supposing y < r0
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
label: ...$L... t,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
less_than'_wf,
rsub_wf,
rmul_wf,
real_wf,
nat_plus_wf,
rleq_wf,
rless_wf,
int-to-real_wf,
rmul_reverses_rleq,
rdiv_wf,
rinv-negative,
rleq_functionality_wrt_implies,
rinv_wf2,
rleq_weakening_rless,
rless-implies-rless,
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,
rleq_weakening_equal,
rleq_weakening,
req_wf,
req_weakening,
uiff_transitivity,
rleq_functionality,
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,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
isect_memberEquality,
independent_isectElimination,
lemma_by_obid,
inlFormation,
independent_functionElimination,
computeAll,
int_eqEquality,
intEquality,
voidEquality
Latex:
\mforall{}[x,y,z:\mBbbR{}]. uiff(x \mleq{} z;(z * y) \mleq{} (x * y)) supposing y < r0
Date html generated:
2017_10_03-AM-08_35_00
Last ObjectModification:
2017_07_28-AM-07_28_45
Theory : reals
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