Nuprl Lemma : rmul_preserves_rleq
∀[x,y,z:ℝ]. uiff(x ≤ z;(x * y) ≤ (z * y)) supposing r0 < y
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
Lemmas referenced :
less_than'_wf,
rsub_wf,
rmul_wf,
real_wf,
nat_plus_wf,
rleq_wf,
rless_wf,
int-to-real_wf,
rmul_functionality_wrt_rleq,
rleq_weakening_rless,
rinv_wf2,
rinv-positive,
rleq_functionality,
req_transitivity,
req_inversion,
rmul-assoc,
rmul_functionality,
req_weakening,
rmul-rinv,
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,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
isect_memberEquality,
independent_isectElimination,
independent_functionElimination,
inrFormation
Latex:
\mforall{}[x,y,z:\mBbbR{}]. uiff(x \mleq{} z;(x * y) \mleq{} (z * y)) supposing r0 < y
Date html generated:
2016_05_18-AM-07_12_28
Last ObjectModification:
2015_12_28-AM-00_40_50
Theory : reals
Home
Index